Quantum Structures and the Nature of Reality

QUANTUM STRUCTURES AND THE NATURE OF REALITY EINSTEIN MEETS MAGRITTE: An Interdisciplinary Reflection on Science, Nature, Art, Human Action and Society Series Editor Diederik Aerts, Center Leo Apostel, Vrije Universiteit Brussel, Belgium Volume 1 Einstein Meets Magritte: An Interdisciplinary Reflection The White Book of 'Einstein Meets Magritte' Edited by DiederikAerts, Jan Broekaert and Ernest Mathijs Volume 2 Science and Art The Red Book of 'Einstein Meets Magritte' Edited by Diederik Aerts, Ernest Mathijs and Bert Mosselmans Volume 3 Science. Technology. and Social Change The Orange Book of 'Einstein Meets Magritte' Edited by DiederikAerts, Serge Gutwirth, Sonja Smets and Luk Van Langenhove Volume 4 World Views and the Problem of Synthesis The Yellow Book of 'Einstein Meets Magritte' Edited by Diederik Aerts, Hubert Van Belle and Jan Van der Veken Volume 5 A World in Transition; Humankind and Nature The Green Book of 'Einstein Meets Magritte' Edited by DiederikAerts, Jan Broekaert and Willy Weyns Volume 6 Metadebates on Science The Blue Book of 'Einstein Meets Magritte' Edited by Gustaaf C. Cornelis, Sonja Smets, Jean Paul Van Bendegem Volume 7 Quantum Structures and the Nature of Reality The Indigo Book of 'Einstein Meets Magritte' Edited by Diederik Aerts and Jarosl'aw Pykacz Volume 8 The Evolution of Complexity The Violet Book of 'Einstein Meets Magritte' Edited by Francis Heylighen, J ohan Bollen and Alexander Riegler VOLUME 7 Quantum Structures and the Nature of Reality The Indigo Book of "Einstein Meets Magritte" Edited by DiederikAerts and Jaroslaw Pykacz Brussels Free University, University of Gdansk VUB UNIVERSITY PRE S S VRIJE UNIVERSITEIT BRUSSEL BELGIUM SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of congress Cataloging-in-Publication Data ISBN 978-90-481-5243-8 DOI 10.1007/978-94-017-2834-8 ISBN 978-94-017-2834-8 (eBook) Sold and distributed in Belgium by VUB University Press, Waversteenweg 1077, B-l160 Brussels, Belgium Printed an acid-free paper AlI Rights Reserved © 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permis sion from the copyright owner. Table of contents General Introduction Diederik Aerts vii Editorial Introduction: Quantum Structures and the Nature of Reality Diederik Aerts and J aroslaw Pykacz xv 1. A Half-Century of Quantum Logic. What Have We Learned? D.J. Foulis 1 2. Quantum Mechanical Measurements S. Gudder 37 3. From Logic to Physics: The Logico-Algebraic Foundations of Quantum Theory G. Cattaneo and F. Laudisa 53 4. Non-Classical Logics, Non-Classical Sets, and Non-Classical Physics J. Pykacz 67 5. Against "Paradoxes": A New Quantum Philosophy for Quantum Mechanics C. Garola 103 6. Quantum Mechanics: Structures, Axioms and Paradoxes D. Aerts 141 7. Orthogonality Relations: From Classical to Quantum T. Durt 207 8. Quantum Logical Semantics, Historical Truths and Interpretations in Art M.L. Dalla Chiara and R. Giuntini 225 Index 235 v DIEDERIK AERTS THE GENERAL INTRODUCTION OF EINSTEIN MEETS MAGRITTE The series of books 'Einstein meets Magritte' presented here originates from an international interdisciplinary conference with the same title, which took place in Brussels in spring 1995. On the eve of the third millennium, we assembled scientists and artists to reflect together on the deep nature of reality and the knowledge and skill humankind has gathered in this field. We had decided to call this meeting 'Einstein meets Magritte' because we believed that meaningful keys could be found at the place where the two meet. It is the way of the world that has made Einstein and Magritte into icons of our culture. The purpose of the conference was to reflect and debate without fear on the most profound and timeless questions. On one of those evenings, when the talks and discussions were long and exhausting and the press were doing all they could to get Albert Einstein and Rene Magritte in front of the microphones and cameras, a few of my most loyal aides and myself succeeded in getting them safely and quietly to a taxi, which then carried us off into the Brussels night. We got out at Manneken Pis, since that was on Einstein's list, and we concealed ourselves among the many tourists who were coming and going, expressing their wonder in every language under the sun at the famous little statue. And one of us was taking pictures: Einstein and Magritte leaning against the railings, with us beside them, and one more, arm in arm, and then another in case the first was no good, when suddenly I felt a heavy slap on my shoulder: "How you doing, mate?" It was Jacky and his inseparable girlfriends Nicole and Sylvie, and everyone embraced everyone else. I introduced Albert and Rene, and interest was immediately shown, and I had my heart in my mouth, because Jacky was a painter, poet and urban philosopher. We walked together through the alleys of Brussels in dismal Belgian rain, over cobblestones that glistened in the street lamps. When we had provided for the inner man with 'Rabbit in Beer' and 'Mussels with Fries', and finally a 'Dame Blanche' topped with warm chocolate sauce as apotheosis, Jacky enticed us to his house in the Rue Haute where we threw ourselves into deep, soft armchairs. Albert and Rene were offered the best places and as always Jacky told the story VB © 1999 Kluwer Academic Publishers. viii DIEDERIK AERTS of his life and discussed his rightness, as he did repeatedly, with a confidence and suppleness that distinguished him so sharply from modern science. Albert listened enthralled and Rene was fascinated, and once more my heart was in my mouth, but Nicole winked reassuringly, and Sylvie brought us snacks on cushions of Brussels lace and sweet white wine in tall, old-fashioned crystal glasses. The topic of discussion for the evening turned out to be 'the doubts of modern science'. In science there is not a single hypothesis for which one cannot find two groups of hard-working scientists, one of which can 'prove' the hypothesis while the other can 'prove' its negation. And the more fundamental and important the question is, the more clearly the situation turns out like this. "It's crazy," maintained Jacky, "In fact science states that one doesn't know anything anymore." "That's right," said Albert, "Truth is not a simple concept, and I believe that the history of science makes it clear how often erroneous hypotheses have been believed over the centuries." "A good thing too," replied Rene, "Things can only happen as a result of the movement brought about by that constant doubt." Meanwhile Sylvie came to join us and handed round pictures of the exhibitions of Jacky's paintings and poems. Jacky suddenly got very excited, as if something had inspired him, and he leapt up and vanished into his studio. A few minutes later he returned with his palette and brush poised. Before I could stop him he had started painting violently right at the spot where Albert and Rene were sitting. A large, gossamerthin piece of Brussels lace gradually took shape and Albert and Rene vanished. Fortunately, my young assistants, Jacky's girlfriends and myself got away with just a few vicious daubs of paint in the face. The series of eight volumes introduced here are not just the results of the conference, as would be the case with a record of the proceedings. The authors were invited to write with the events at the conference in the back of their mind, so that the books would form a second phase in the process of thought set in motion at the conference. A second ph,ase/ more clearly crystallised than the self-organising forum that arose during the conference, but one which focuses on the same timeless questions and problems. The whole ensemble was already streamlined at the conference into a number of main topics named after the colours of the rainbow - red, orange, yellow, green, blue, indigo and violet, as well as white, the synthesis of all colours. This order was maintained and led to eight separate books in the series. EINSTEIN MEETS MAGRITTE ix Volume 1: Einstein meets Magritte: an Interdisciplinary Reflection The White Book of Einstein meets Magritte The white book contains more fully developed versions of the contributions made by the keynote speakers at the conference. So this white book covers various scientific topics. In his article, 'Basically, it's purely academic', John Ziman asks himself what 'basic research' really is in today's world. In his contribution, 'The manifest image and the scientific image', Bas Van Fraassen analyses the considerable differences between the theoretical scientific description of the world and the way it appears to us. He argues that most formulations of this problem may themselves be tendentious metaphysics, full of false contrasts, and that insistence on a radical separation between science and what we have apart from science, and on the impossibility of accommodating science without surrender, may be a way of either idolising or demonising science rather than understanding it. In the 'Microdynamics of incommensurability: philosophy of science meets science studies', Barbara Herrnstein-Smith examines the bemusing but instructive logical, rhetorical and cognitive dynamics of contemporary theoretical controversy about science. In his contribution 'Subjects, objects, data and values', Robert Pirsig proposes a radical integration of science and value that does no harm to either. It is argued that values can exist as a part of scientific data, but outside any subject or object. This argument opens a door to a 'metaphysics of value' that provides a fundamentally different but not unscientific way of understanding the world. Ilya Prigogine discusses in 'Einstein and Magritte: a study of creativity', the global transformation of a classical science which was based on certainties into a new science that takes possibilities as its basic concepts. Constantin Piron demonstrates in his contribution 'Quanta and relativity: two failed revolutions' that none of the two great revolutions in physics, quantum mechanics and relativity theory, have actually been digested by the physics community. He claims that the vast majority of physicists still cling to the idea of a non-existent void full of little particles, in the spirit of Leibniz or Descartes. Rom Harre reflects on the significance of the theory of relativity. In his article 'The redundancy of spacetime: relativity from Cusa to Einstein', he defends the hypothesis that relativity theory is best interpreted as a grammar for coordinating narratives told by different observers. In his contribution 'The stuff the world is made of: physics and reality', Diederik Aerts analyses the consequences of the recent advances in quantum mechanics, theoretically as well as experimentally, for the nature of reality. He analyses the deep conceptual paradoxes in the light of these recent data and tries to picture a coherent model of the world. In his contribution 'Da- x DIED ERIK AERTS sein's brain: phenomenology meets cognitive science', Francisco Varela puts forward the hypothesis that the relation between brain processes and living human experience is the really hard problem of consciousness. He argues that science needs to be complemented by a deep scientific investigation of experience itself to move this major question beyond the sterile oppositions of dualism and reductionism. In his contribution 'What creativity in art and science tell us about how the brain must work' William Calvin defends the prospects for a mental Darwinism that operates on the milliseconds to minutes time scale, forming novel ideas and sentences never previously expressed. Adolf Griinbaum in his article 'The hermeneutic versus the scientific conception of psychoanalysis: an unsuccessful effort to chart a via media for the human sciences' argues that the so called 'hermeneutic' reconstruction of psychoanalytic theory and therapy proposed by Karl Jaspers, Paul Ricoeur and Jiirgen Habermas fails both as a channel and as alleged prototype for the study of human nature. In his article 'Immortality, biology and computers', Zygmunt Bauman analyses the shift that post modern society has provoked regarding the concept of immortality. He points out that strategies of collective and individual immortality have shifted from the modern deconstruction of death to a postmodern deconstruction of immortality, and points out that the possible consequences of this process need to be taken into consideration. Brian Arthur, in his article 'The end of certainty in economics', points out that our economy is very non-classical, meaning that it is based on essentially self-referential systems of beliefs about future economic conditions. He argues that our economy is inherently complex, subjective, ever-changing, and to an unavoidable degree ill-defined. Volume 2: Science and Art The Red Book of Einstein meets M agritte And then Magritte comes in. Many obvious differences exist between science and art. But the Science and Art volume of this series addresses not only these differences but also the possibilities of crossing several of the gaps between science and art. Several contributions deal with sociological and philosophical elaborations of the similarities and differences between science and art, while others approach science from an artistic point of view and art from a scientific point of view. The volume also considers several approaches that attempt to go beyond the classical dichotomy between the two activities. In a special section, attention is paid to the particular role played by perception in both science and art as a regulator of human understanding. Together, these contributions strive for an intensive interaction between science and art, and to a con- EINSTEIN MEETS MAGRITTE Xl sideration of them as converging rather than diverging. It is to be hoped that both science and art will benefit from this attempt. Volume 3: Science, Technology and Social Change The Orange Book of Einstein meets Magritte The major subject of the orange book is that society as a whole is changing, due to changes in technology, economy and the changing strategies and discourses of social scientists. The collected articles in the orange stream discuss a range of specific societal problems related to the subject of social change, the topics of the articles range from the scale of for instance sociology of health and psychohistory to more specific social problems like for instance anorexia nervosa, art academies and the information superhighway. Although the authors approach different subject matters from dissimilar perspectives and work with various methods, all the papers are related to the theme of science, technology and social change. In the orange book the reader will find a lot of arguments and hints pertaining to questions like: To what exactly will this social change lead in the 21st century? What kind of society lies ahead? She/he will be confronted to a plethora of enriching conceptions of the relationships between social sciences and social changes. Volume 4: World Views and the Problem of Synthesis The Yellow Book of Einstein meets M agritte A rapidly evolving world is seen to entail ideological, social, political, cultural and scientific fragmentation. Many cultures, subcultures and cultural fragments state their views assertively, while science progresses in increasingly narrowly defined areas of inquiry, widening not only the chasm between specialists and the layman, but also preventing specialists from having an overall view of their discipline. What are the motive forces behind this process of fragmentation, what are its effects? Are they truly inhospitable to the idea of synthesis, or do they call out, more urgently than ever before, for new forms of synthesis? What conditions would have to be met by contemporary synthesis? These and related questions will be addressed in the yellow book of Einstein meets Magritte. Volume 5: A World in Transition; Humankind and Nature The Green Book of Einstein meets Magritte IA World in Transition. Humankind and Nature' is appropriately entitled after its aim for an intrinsic property of reality: change. Of major concern, in this era of transformation, is the extensive and profound interaction of humankind with nature. The global scaled, social and technological project of humankind definitely involves a myriad of changes of xii DIEDERIK AERTS the ecosphere. This book develops, from the call for an interdisciplinary synthesis and respect for plurality, acknowledging the evolving scientific truth, the need for an integrated but inevitably provisional world view. Contributors from different parts of the world focus on four modes of change: i) Social change and the individual condition, ii) Complex evolution and fundamental emergent transformations, iii) Ecological transformation and responsibility inquiries, iv) The economic-ecological and socio-technical equilibria. Primarily reflecting on the deep transformations of humankind and on the relationship between humans and nature it addresses major points of contemporary concern. Volume 6: Metadebates The Blue Book of Einstein meets Magritte This book provides a meta-disciplinary reflection on science, nature, human action and society. It pertains to a dialogue between scientists, sociologists of science, historians and philosophers of science. It covers several topics: (1) the relation between science and philosophy, (2) new approaches to cognitive science, (3) reflections on classical thinking and contemporary science, (4) empirical epistemology, (5) epistemology of quantum mechanics. Indeed, quantum mechanics is a discipline which deserves and receives special attention here, for it still is a fascinating and intriguing discipline from a historiographical and philosophical point of view. This book does not only contain articles on a general level, it also provides new insights and bold, even provocative theories on the meta-level. That way, the reader gets acquainted with 'science in the making', sitting in the front row. Volume 7: Quantum Structures and the Nature of Reality The Indigo Book of Einstein meets M agritte This book refers to the satellite symposium that was organised by the International Quantum Structure Association (IQSA) at Einstein meets Magritte. The IQSA is a society for the advancement and dissemination of theories about structures based on quantum mechanics in their physical, mathematical, philosophical, applied and interdisciplinary aspects. The book contains several contributions presenting different fields of research in quantum structures. A great effort has been made to present some of the more technical aspects of quantum structures for a wide audience. Some parts of the articles are explanatory, sketching the historical development of research into quantum structures, while other parts make an effort to analyse the way the study of quantum structures has contributed to an understanding of the nature of our reality. EINSTEIN MEETS MAGRITTE XIll Volume 8: The Evolution of Complexity The Indigo Book of Einstein meets Magritte The violet book collects the contributions that consider theories of evolution and self-organisation, on the one hand, and systems theory and cybernetics, on the other hand. Both can add to the development of an integrated world view. The basic idea is that evolution leads to the spontaneous emergence of systems of higher and higher complexity or "intelligence": from elementary particles, via atoms, molecules, living cells, multicellular organisms, plants, and animals to human beings, culture and society. This perspective makes it possible to unify knowledge from presently separate disciplines: physics, chemistry, biology, psychology, sociology, etc. The volume thus wishes to revive the transdisciplinary tradition of general systems theory by integrating the recently developed insights of the "complex adaptive systems" approach, pioneered among others by the Santa Fe Institute. Even these books only signify a single phase in the ever-recurring process of thought and creation regarding the basic questions on the reality that surrounds us and our place in it. Brussels, July 17, 1998. D. AERTS & J. PYKACZ QUANTUM STRUCTURES AND THE NATURE OF REALITY This book grew out of the Satellite Symposium "Quantum Structures and the Nature of Reality" organised by the International Quantum Structures Association (IQSA) during the conference "Einstein Meets Magritte: an Interdisciplinary Reflection on Science, Nature, Art, Human Action, and Society" which was organised in Brussels at the Vrije Universiteit Brussel from May 29 to June 3, 1995. The purpose of the Symposium (and of the present book) was to acquaint the possibly widest audience consisting of people interested in foundations of quantum physics but not necessarily physically or even mathematically experienced with the variety of subjects considered within the IQSA, aims, problems, and methods used to solve them. The International Quantum Structures Association was established in 1991 and it gathers researchers interested in studying various aspects of logico-algebraic structures encountered in the very foundations of quantum physics. Quantum structures (formerly called quantum logics) are situated on the Map of Science in that place where quantum physics (the foundations of quantum mechanics), mathematics (mathematical logic, abstract algebra, theory of ordered structures, measure theory, probability theory, and fuzzy set theory), logic (many-valued, modal, intuitionistic, and paraconsistent), and philosophy (esp. philosophy of science) meet. Therefore, activities of the members of the IQSA are really interdisciplinary, which was one more reason for including the IQSA Satellite Symposium within the scope of the Interdisciplinary Conference "Einstein meets Magritte" . Although the present book consists of papers written independently by various authors which unavoidably leads to some repetitions, we tried to organise it in such a way that it could serve as a possibly self-contained introduction to the theory of quantum structures oriented towards an inexperienced reader. Therefore, the book begins with the review paper by D. Faulis (University of Massachusetts, Amherst, USA) A HalfCentury of Quantum Logics- What Have We Learned, which in its first part contains a brief exposition of the historical development of quantum structures from their prehistory traced back to Leibniz and Boole till the most recent papers concerning effect algebras, with the exclusion, however, of the "parallel stream" within the theory of quantum strucxv © 1999 Kluwer Academic Publishers. D. AERTS & J. PYKACZ XVI tures connected with many-valued logics and fuzzy setsl. In the main part of his paper Foulis illustrates practically all fundamental notions encountered within the theory of quantum structures, and relations between them on the very elementary example of a firefly in the box with windows using only the simplest mathematical tools, i.e., sets and functions. Almost an equally elementary example is used in the next paper Quantum Mechanical Measurements by S. Gudder (University of Denver, USA) to illustrate the notion of an effect algebra. Studying effect algebras became nowadays popular within the IQSA since these structures are more general, therefore, also more "flexible" than traditional "quantum logics", i.e., orthomodular partially ordered sets or lattices. Their introduction allowed a lot of "fresh air" to come into the theory of quantum structures2 , which, after the period of a rapid development in the sixties and the seventies, in the eighties was seen by many as a "decaying" theory without much future. The paper Fmm Logic to Physics: The Logico-Algebmic Foundations of Quantum Theory by G. Cattaneo and F. Laudisa (Universita di Milano and Universita di Firenze, Italy) also contains a brief description of the historical development of quantum logic ideas. In particular, the reader will find here a more detailed outline of the seminal Birkhoff and von Neumann's paper The Logic of Quantum Mechanics generally regarded as a cornerstone of this branch of Science. The Cattaneo and Laudisa paper is finished with the notion of an effect introduced (already encountered in the paper by S. Gudder), however, with the aid of Zadeh's idea of a fuzzy set, i.e., a set that admits gradual rather than abrupt transition from membership to non-membership. Fuzzy set theory is as closely related to Lukasiewicz infinite-valued logic as traditional set theory is related to the classical, two-valued logic. These relations are exposed in the paper Non-Classical Logics, NonClassical Sets, and Non-Classical Physics by J. Pykacz (Uniwersytet Gdanski, Gdansk, Poland) in which it is also shown how these two "nonclassical" branches of mathematics can be utilised in the "non-classical" branch of physics, i.e., in quantum mechanics. Also this paper contains a brief survey of the historical development of all "non-classical" theories that it deals with. 1 2 The brief history of this "stream" is contained in the paper by Pykacz. and also was one of the reasons of changing the name of this branch of Science from "theory of quantum logics" to the more general "theory of quantum structures" . QU ANTUM STRUCTURES AND REALITY xvii The paper by C. Garola (Universita di Lecce, Italy) Against "Paradoxes": A New Quantum Philosophy for Quantum Mechanics, shows that changing the basic philosophical premises (in this case abandoning the verificationist theory of truth in favour of the classical (Tarskian) theory of truth and introducing a new conception of physical laws) implies changes in the interpretation of quantum theory and provides new solutions to some old "paradoxes" encountered in the very foundations of quantum physics. The paper by D. Aerts (Brussels Free University, Belgium) Quantum Mechanics: Structures, Axioms and Paradoxes, gives an overview of some of the results in quantum structures research that have been obtained in the Brussels group. Working within a realist approach to quantum mechanics-following the inspiration that was outlined in the Geneva approach-the Brussels group tries to elaborate a generalised quantum mechanics, where some of the basic axioms of standard quantum mechanics are abandoned (e.g., the superposition principle). The reason for this generalisation is standard quantum mechanics' impossibility to describe the situation of two separated quantum entities, a shortcoming directly linked to the Einstein-Podolsky-Rosen paradox. Aerts shows the different steps of this generalised axiomatic quantum mechanics and the related problems to be solved tentatively by means of a simple macroscopic model entailing a quantum mechanical structure. Together with this alternative quantum mechanics an explanation for the probabilities of quantum mechanics-their origin residing in a lack of knowledge about the interaction between the measuring apparatus and the system-is exposed by means of the simple example. The paper Orthogonality Relations and the f.-Model by T. Durt (Vrije Universiteit Brussel, Belgium) is also written in the spirit of the Brussels approach and devoted to various orthogonality relations that are quantum-logical generalisations of the classical negation. These relations are studied within the f.-model of quantum spin-measurements developed recently in Brussels, where a continuous transition from quantum to classical mechanics can be modelled. This description also needs the more general framework, developed in Brussels and presented in the paper of Aerts, where some of the standard quantum mechanical axioms are not fulfilled. The final paper of the book Quantum Logical Semantics, Historical Truths, and Interpretations in Arts by M.L. Dalla Chiara and R. Giuntini (Universita di Firenze, Italy) shows that notions and constructions typical to the theory of quantum structures, and non-classical logics implied by them can find application even in such remote fields XVlll D. AERTS & J. PYKACZ as History and Art. This shows that the methods elaborated within the theory of quantum structures do not apply exclusively to microworld and quantum physics and justifies once more the idea of including the IQSA Symposium within the very broad scope of the Interdisciplinary Conference "Einstein Meets Magritte" . Diederik Aerts J aroslaw Pykacz DAVID J. FOULIS A HALF-CENTURY OF QUANTUM LOGIC WHAT HAVE WE LEARNED? 1. INTRODUCTION This expository paper comprises my personal response to the question in the title. Before giving my answers to this question, I discuss the utility of quantum logic in Section 2, offer a succinct review of the history of quantum logic in Section 3, and present in Section 4 two simple thought experiments involving a firefly in a box. The two thought experiments are pursued in Sections 5 through 9, where they give rise to natural and (I hope) compelling illustrations of the basic ideas of quantum logic. In Section 10, I replace the firefly by a "quantum firefly," and in Section 11, I summarize the lessons that we have (or should have) learned. This paper is written for so-called laypersons (although some of the ideas presented here have yet to be fully appreciated even by some expert quantum logicians). Thus, in using the firefly box to motivate and exemplify the fundamental notions of quantum logic, I need only the simplest mathematical tools, i.e., sets and functions. This does not necessarily imply that a casual reading of the narrative will guarantee an adequate understanding of the basic principles-a certain amount of attentiveness to detail is still required. 2. QUA N TUM LOG I C: W HAT GOO DIS IT? Before proceeding, I should address a question germane to any meaningful discussion of what we have learned, namely the related question what good is quantum logic? Until now, quantum logic has had little or no impact on mainstream physics; indeed some physicists go out of their way to express a contempt for the subject l , Whether or not the insights achieved by quantum logicians contribute directly to an achievement of 1 Witness the following ill-natured remark, arrogantly inserted in the review (Mathematical Reviews of the American Mathematical Society, Nov.-Dec., 1991, 91k 81010) of a paper written by a well-known Polish physicist who (in the opinion of the reviewer, a "mainline physicist" of some repute) had the temerity to concern himself with matters pertaining to the foundations of physics: " ... a small but persistent core of diehards who find fault with quantum mechanics is still active today. The journal Foundations of Physics serves to give them somewhere to publish." 1 © 1999 Kluwer Academic Publishers. 2 DAVID J. FOULIS whatever the Holy Grail of contemporary or future physicists happens to be 2 , quantum logic has already made significant contributions to the philosophy of science and to both mathematical and philosophical logic. Prior to Galileo's celebrated declaration that the Great Book of Nature is written in mathematical symbols, what we now call physical science was commonly referred to as natural philosophy. Quantum logic offers the possibility of reestablishing some of the close bonds between physics and philosophy that existed before the exploitation of powerful techniques of mathematical analysis changed not only the methods of physical scientists, but their collective mindset as well. The hope is that quantum logic will enable the mathematics of Descartes, Newton, Leibniz, Euler, Laplace, Lagrange, Gauss, Riemann, Hamilton, Levi-Civita, Hilbert, Banach, Borel, and Cartan, augmented by the mathematical logic of Boole, Tarski, Church, Post, Heyting, and Lukasiewicz to achieve a new and fertile physics/philosophy connection. The mathematics of quantum mechanics involves operators on infinite dimensional vector spaces. Quantum logic enables the construction of finite, small, easily comprehended mathematical systems that reflect many of the features of the infinite dimensional structures, thus considerably enhancing our understanding of the latter. For instance, a finite system of propositions relating to spin-one particles constructed by Kochen and Specker [50] settled once and for all an aspect of a long standing problem relating to the existence of so-called hidden variables [5J. Another example is afforded by the work of M. KHiy in which a finite model casts considerable light on the celebrated paradox of Einstein, Podolski, and Rosen [47J. One facet of quantum logic, yet to be exploited, is its potential as an instrument of pedagogy. In introductory quantum physics classes (especially in the United states), students are informed ex cathedra that the state of a physical system is represented by a complex-valued wavefunction 7/J, that observables correspond to self-adjoint operators, that the temporal evolution of the system is governed by a Schrodinger equation, and so on. Students are expected to accept all this uncritically, as their professors probably did before them. Any question of why is dismissed with an appeal to authority and an injunction to wait and see how well it all works. Those students whose curiosity precludes blind compliance with the gospel according to Dirac and von Neumann are told that they have no feeling for physics and that they would be better off studying 2 Just now, the desired consummation of theoretical physics seems to be a so called "theory of everything," i.e., a master theory encompassing both quantum mechanics and the general theory of relativity A HALF-CENTURY OF QUANTUM LOGIC 3 mathematics or philosophy. A happy alternative to teaching by dogma is provided by basic quantum logic, which furnishes a sound and intellectually satisfying background for the introduction of the standard notions of elementary quantum mechanics. Quantum logic is a recognized, autonomous, and rapidly developing field of mathematics3 and it has engendered related research in a number of fields such as measure theory [11, 17, 18, 20, 23, 26, 34, 36, 45, 70,71,74, 76] and functional analysis [4, 11, 19, 22, 23, 25, 35, 72]. The recently discovered connection between quantum logic and the theory of partially ordered abelian groups [29, 33] promises a rich cross fertilization between the two fields. Also, quantum logic is an indispensable constituent of current research on quantum computation and quantum information theory [24]. 3. A B R I E F HIS TOR Y 0 F QUA N TUM LOG I C In 1666, G.W. Leibniz envisaged a universal scientific language, the characteristica universalis, together with a symbolic calculus, the calculus ratiocinator, for formal logical deduction within this language. Leibniz soon turned his attention to other matters, including the creation of the calculus of infinitesimals, and only partially developed his logical calculus. Nearly two centuries later, in Mathematical Analysis of Logic (1847) and Laws of Thought (1854), G. Boole took the first decisive steps toward the realization of Leibniz's projected calculus of scientific reasoning4. From 1847 to the 1930's, Boolean algebra, which may be considered as a classical precursor of quantum logic, underwent further development in the hands of De Morgan, Jevons, Peirce, Schroder, et al, and received its modern axiomatic form thanks to the work of Huntington, Birkhoff, Stone, et al. Nowadays, Boolean algebras are studied either as special kinds of lattices [9], or equivalently as special kinds of rings 5 . In 1933, Kolmogorov, building upon an original idea of Frechet, established the modern theory of probability using Boolean sigma-algebras of sets as a foundation [51]. 3 Papers on quantum logic are reviewed in Sections 03G and 81P of the American Mathematical Society's Mathematical Reviews. 4 One of Boole's primary motivations was to construct a mathematical foundation for a theory of probability. Indeed, the full title of his 1854 masterpiece is An Investigation Into the Laws of Thought, On Which are Founded the Mathematical Theories of Logic and Probabilities. 5 A Boolean algebra can be defined either as a complemented distributive lattice or as a ring with unit in which every element is idempotent. 4 DAVID J. FOULIS The genesis of quantum logic is Section 5, Chapter 3 of J. von Neumann's 1932 book on the mathematical foundations of quantum mechanics [59]. Here von Neumann argued that certain linear operators, the projections defined on a Hilbert space6 , could be regarded as representing experimental propositions affiliated with the properties of a quantum mechanical system. He wrote, " ... the relation between the properties of a physical system on the one hand, and the projections on the other, makes possible a sort of logical calculus with these." In 1936, von Neumann, now in collaboration with G. Birkhoff, published a definitive article on the logic of quantum mechanics [10]. Birkhoff and von Neumann proposed that the specific quantum logic of projection operators on a Hilbert space should be replaced by a general class of quantum logics governed by a set of axioms, much in the same way that Boolean algebras had already been characterized axiomatically. They observed that, for propositions P, Q, R pertaining to a classical mechanical system, the distributive law P & (Q or R) = (P & Q) or (P & R) holds, they gave an example to show that this law can fail for propositions affiliated with a quantum mechanical system, and they concluded that, " ... whereas logicians have usually assumed that properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic." Birkhoff and von Neumann went on to argue that a quantum logic ought to satisfy only a weakened version of the distributive law called the modular law7 ; however, they pointed out that projection operators on a Hilbert space can fail to satisfy even this attenuated version of distributivity. Much of von Neumann's subsequent work on continuous geometries [60J and rings of operators [61 J was motivated by his desire to construct logical calculi satisfying the modular law. In 1937, K. Husimi [41J discovered that projection operators on a Hilbert space satisfy a 6 A Hilbert space is a vector space (over the reals, the complexes, or the quaternions) equipped with an inner product, and complete with respect to the metric arising from the inner product. 7 A lattice L satisfies the modular law iff, for p, q, r E L, p S r implies that p V (q 1\ r)=(pVq)l\r. A HALF-CENTURY OF QUANTUM LOGIC 5 weakened version of the modular law, now called the orlhomodular identit" . From 1937 until 1955, all research on quantum logic ceased as scientists turned their attention to military applications of physics. In 1955, L. Loomis [53] and S. Maeda [57] independently rediscovered Husimi's orthomodular identity in connection with their efforts to extend von Neumann's dimension theory for rings of operators to more general structures. The structures studied by Husimi, Loomis, and Maeda are now called orthomodular lattices9 . In 1957, G. Mackey wrote an expository article on quantum mechanics [55] based on lectures he was giving at Harvard. In 1963, he published an expanded version of these lectures in the form of an influential monograph [56] in which he referred to propositions affiliated with a physical system as questions. Under fairly reasonable hypotheses, it is easy to show that Mackey's questions form an orthomodular lattice. The simplicity and elegance of Mackey's formulation and the natural and compelling way in which it gave rise to a system of experimental propositions inspired a renewed interest in the study of quantum logic, now identified with the study of orthomodular lattices. Could it be that these lattices provide a basis for Leibniz's long awaited calculus ratiocinator? Thus motivated, a small but devoted group of researchersCatlin, Finch, Foulis, Greechie, Gudder, Holland, Janowitz, Jauch, Kotas, MacLaren, Maeda, Piron, Pool, Ramsay, Randall, Schreiner, Suppes, Varadarajan, et al-began in the early 1960's the task of working out a general mathematical theory of orthomodular lattices. A comprehensive account of the resulting theory and an extensive bibliography up to about 1983 can be found in [44]. In 1964, C. Piron introduced an alternative to Mackey's approach in which questions again band together to form an orthomodular lattice, but this time possessing more of the special features of the lattice of projection operators on a Hilbert space [62]. In fact, Piron was able to show that his questions could be represented as actual projection operators on a so-called generalized Hilbert space. Piron's work raised the issue of how to characterize the standard Hilbert spaces among the class of generalized Hilbert spaces. A list of more or less "natural conditions" on generalized Hilbert spaces was soon proposed in the hopes of singling out the "true" Hilbert spaces. 8 A lattice L with an orthocomplementation p >--+ p' satisfies the orthomodular identity iff, for p, q E L, p ~ q implies that q = p V (q 1\ p'). 9 The terminology "orthomodular lattice" was suggested by 1. Kaplansky because, in such a lattice, orthogonal pairs are modular pairs. 6 DAVID J. FOULIS In 1980, H. Keller dashed these hopes by constructing an example of a generalized Hilbert space satisfying all of the proposed natural conditions, but that is not a standard Hilbert space [46]. In 1995 M. Soler showed that Keller's counterexample could be bypassed by adding just one more natural condition to the previous list [72]. Thanks to Soler's work, we are now in possession of a satisfactory axiomatic approach to Hilbert-space based quantum mechanics [40]. As early as 1962 [27], it was noticed by some of the aforementioned researchers that, even without the imposition of additional hypotheses, Mackey's questions form an intriguing structure called an orthomodular poset. For this reason, orthomodular posets were also considered as possible candidates for quantum logics and were studied in parallel with orthomodular lattices, especially by S. Gudder [34, 35] and his students. A comprehensive account of orthomodular lattices and posets as models for quantum logics can be found in [65]. In orthodox quantum mechanics, when systems are combined or coupled to form composite systems [3, 28, 43], the combined system is represented mathematically by a so-called tensor product of Hilbert spaces. Even in the early 1960's, researchers realized that the entire quantum logic program would falter unless a suitable version of tensor product could be found for the more general logical structures then under consideration. After many unsuccessful attempts to formulate a suitable tensor product for orthomodular lattices and posets, it was discovered in 1979 that all such attempts were doomed to failure owing to the fact that the category of orthomodular posets is too small to admit a tensor product [67]. C. Randall and D. Foulis showed that, to accommodate the construction of tensor products, a larger category of mathematical structures called orthoalgebras has to be employed [8,31,49,66,68]. For a while, it seemed that orthoalgebras were the true quantum logics [8, 20, 26, 30, 36, 71]. Composite physical systems were studied from the perspective of quantum logic in an important and influential sequence of papers by D. Aerts [1, 2, 3]. In these studies, Aerts introduced the crucial notion of an entity which, roughly speaking, consists of a quantum logic of questions or propositions affiliated with a physical system together with a related system of properties, or attributes, of the system. Among other things, Aerts showed convincingly that a proper representation of a composite system requires consideration of the way in which properties of a total system depend on the properties of its constituents. In parallel with the development of quantum logic, and starting as early as 1970 [19, 38, 39], Davies, Lewis, Holevo, Ludwig, Prugovecki, Ali, Busch, Lahti, Mittelstaedt, Schroeck, Bujagski, Beltrametti, et al worked out a theory of A HALF-CENTURY OF QUANTUM LOGIC 7 quantum statistics and quantum measurement based on so-called effect operatorslO on a Hilbert space [54]. Every projection operator is an effect operator, but not conversely, and the effect operatorsdo not even form a lattice, let alone an orthomodular lattice, or even an orthoalgebra. In 1989, R. Guintini and H. Greuling introduced axioms for a generalized orthoalgebra and argued that effect operators on a Hilbert space form such a structure [32]. The generalized orthoalgebras of Guintini and Greuling, which turned out to be mathematically equivalent to the Dposets of Kopka and Chovanec [52], have come to be called effect algebras [29]. It can be argued that fuzzy or unsharp propositions are properly represented as elements of an effect algebra [13, 14, 15, 16, 52]. In 1994, Bennett and Foulis [29] discovered a connection between effect algebras and partially ordered abelian groups. In subsequent papers, they went on to show that virtually every structure previously proposed for a quantum logic, and indeed every Boolean algebra, can be represented as an interval in a such a group. An interval in a partially ordered abelian group, organized in a natural way into an effect algebra, is called an interval effect algebra. As a mathematical theory, quantum logic is thus subsumed by the theory of partially ordered abelian groups. Because there was no serious work on quantum logic per se during the years 1938-1957, I consider that quantum logic has been under development for roughly half a century. The history of quantum logic has been a story of more and more general mathematical structures-Boolean algebras, orthomodular lattices, orthomodular posets, orthoalgebras, and effect algebras-being proposed as basic models for the logics affiliated with physical systems. Whether effect algebras are the end of the line remains to be seen. Those wishing to read more about quantum logic and its connections with quantum physics are encouraged to consult the following standard references [6, 12, 21, 35, 42, 43, 54, 55, 56, 59, 62, 65]. 4. THE FIR E FLY BOX AND ITS EVE N T LOG I C Now I invite you to contemplate with me some "thought experiments" involving a firefly in a box (Figure 1). The box is to have two translucent (but not transparent) windows, one on the front and one on the side. The remaining four sides of the box are opaque. At any given moment, the firefly might or might not have its light on. If the light is on, it can be seen as a blip by looking at either of the two windows. 10 An effect operator is a self-adjoint operator A such that 0 :::; A :::; 1 8 DAVID J. FOULIS b 1 r f Figure 1. The Firefly Box Looking directly at the front window of the box when the light is on, one can tell by the position of the blip whether the firefly is in the left (1) or right (r) half of the box. Likewise, looking directly at the side window when the light is on, one can tell whether the firefly is in the front (f) or back (b) half of the box. Because the windows are not transparent, one cannot rely on depth perception to determine from the front window whether the firefly is in the front or back half of the box, nor from the side window, whether the firefly is in the left or right half of the box. Now consider two experimental procedures F and S. Procedure F is conducted by looking directly at the front window and recording I, r, or nF according to whether the blip is on the left, on the right, or there is no blip, respectively. Procedure S is conducted by looking directly at the side window and recording f, b, or ns according to whether the blip is in the front, the back, or there is no blip, respectively. One cannot conduct both procedures F and S at the same time because of the necessity of looking directly at one window or the other. Indeed, if one stands in a position to see both windows, parallax could spoil the accuracy of the observationl l . 11 The situation is quite analogous to the fact that attempts to make simultaneous measurements of noncommuting quantum-mechanical observables lead to interference effects that spoil the accuracy of the measurements. This does not mean that such simultaneous measurements cannot or should not be made. It simply means that, when they are made, one has to deal with a certain amount of fuzzyness or unsharpness. A HALF-CENTURY OF QUANTUM LOGIC 9 Imagine that we plan to conduct an experimental study of the firefly's habits using the only means available to us, namely the two experimental procedures F and S. Our work will be guided by an emerging "firefly box theory" (FBT) that may have to be amended as we collect more and more experimental data or change our mind about what is going on inside the box. To begin with, let us provisionally incorporate into our FBT the simplifying assumption that there are no baffles within the box behind which the firefly might hide from either window. If this is so, then a blip would be seen on the front window if and only if it would be seen on the side window. (Note the use of the subjunctive here-we can not meaningfully perform both F and S simultaneously!) This assumption is implemented simply by identifying outcome nF of F with outcome ns of S. Thus, we set n := nF = ns. (The notation := means equals by definition.) Let EF := {l, r, n} and Es := {f, b, n} be the mutually exclusive and exhaustive outcome sets for the experimental procedures F and S, respectively. Execution of F will yield one and only one outcome l, r, or n; execution of S will yield one and only one outcome f, b, or n. By an event for F we will mean a subset A of EF. Including the empty set and EF itself, there are 8 such events, namely 0, {l}, {r}, {n}, {l,r}, {l, n}, {r, n}, {l, r, n}. If F is executed and the resulting outcome is e E EF, say that an event A ~ EF occurs if e E A and that it nonoccurs if e tt. A. Likewise, an event for S is understood to be one of the 8 subsets 0, {J}, {b}, {n}, {J,b}, {J,n}, {b,n}, {J,b,n} of Es. If S is executed, then an event A for S occurs or nonoccurs according to whether the outcome belongs or does not belong to A, respectively. An event A can only occur or nonoccur when tested by the execution of either F (if A ~ EF) or S (if A ~ Es). The null event 0, the event {n}, and only these two events, are tested by both F and S. Let E:= {0,{l},{r},{l,r},{l,n},{r,n},{l,r,n}, {n}, {J}, {b}, {J, b}, {J, n}, {b, n}, {J, b, n}} (1) be the collection of all events and let E:= EF uEs = {l,r,n,f,b} (2) be the set of all outcomes of the available experimental procedures. There are 32 subsets of E, but only 14 events in E. For example, {r, f} is a subset of E, but it is not an event since there is no conclusive way to test it. For instance, if S is executed and b is the outcome, we would hardly say that {r, J} failed to occur since F was not executed and it is meaningless to ask whether or not the outcome r was secured. 10 DAVID J. FOVLIS The set £ of all events can be organized into a rudimentary logical structure as follows: Let A, B, C E £. Say that A and B are compatible iff they are simultaneously testable in the sense that A, B ~ EF or A, B ~ Es. (We abbreviate if and only if as iff.) Call A and B orthogonal iff they are compatible and disjoint. (Two sets are disjoint iff they have no elements in common.) Say that A and C are local complements iff they are orthogonal and their union is either EF or Es. If A and B share a common local complement C, say that A and B are perspective with axis C. For instance, the events {l, r} and {r, n} are compatible, since they are both tested by F, but they are not orthogonal since they have a common outcome r. If two events are orthogonal, they can be tested simultaneously and, when so tested, at most one of them can occur. The events {l, r} and {n} are local complements, since they are disjoint and their union is E F. When tested by F, one and only one of them will occur. The events {I, b} and {n} are also local complements, and both are tested by S. Therefore, {l, r} and {j, b} are perspective with {n} as an axis. If A E £, then A has at least one local complement C in £. Indeed, if A ~ EF, then the complement C := EF\A of A in EF is a local complement of A. Likewise, if A ~ Es, then D = Es\A is a local complement of A. Therefore, every event A is perspective to itself (with any local complement of A as an axis). Note that the two events EF and Es are perspective, with 0 as an axis. There is an obvious sense in which perspective events are "logically equivalent." For instance {l, r} occurs iff the firefly's light is on, and likewise for {j, b}. Also, EF = {l, r, n} always occurs (when tested, of course), and so does Es = {j,b,n}. The collection £ is partially ordered by set containment ~. If A and B are events and A ~ B, there is an obvious sense in which A "implies" B. For instance, {n} ~ {r, n} and, if {n} occurs, the firefly's light is out, and presumably {r, n} would have occurred too, had it been tested. (Again, note the use of the subjunctive. Indeed, if S is executed and {n} occurs, then {r, n} was not tested, so it neither occurred nor nonoccurred.) Figure 2 shows a diagram (called a Hasse diagram) of the event logic for the firefly box. The 14 events in £ are shown as nodes in this diagram, and event A is a subset of an event B iff either A = B or it is possible to go upward from A to B along a sequence of connecting line segments. The perspective events {l, r} and {j, b}, as well as {l, r, n} and {j, b, n} are enclosed in shaded ellipses on the diagram. A HALF-CENTURY OF QUANTUM LOGIC ( ~ ~- ------------ -~ {f, b, n} ") {l, r, n} ------------ {r, n} {I } 11 ~ {f, n} {r} {n} {f} {b} Figure 2. Logic of events 5. THE LOG l e o F EX PER I MEN TAL PRO P 0 SIT ION S As we have seen, in the event logic £ of the firefly box, two perspective events are, in some sense, logically equivalent. The temptation to identify logically equivalent events is irresistible, and we do so now by collapsing the diagram in Figure 2 as indicated by the shaded ellipses. Call the elements of the collapsed diagram experimental propositions, denote the experimental proposition corresponding to an event A E £ by 7r(A) , and let II be the set of all7r(A) as A runs through £. For simplicity, if A = {e} is an event with only one outcome, we write 7r( e) rather than 7r( {e} ). Denote the proposition 7r(0) by 0 := 7r(0) E II. Define the proposition 1 E II by 1 := 7r({l,r,n}) = 7r({J,b,n}). The proposition 7r({l,r}) = 7r( {J, b}), which can be regarded as asserting that the firefly's light is on, is denoted by 7r(n'). Likewise, the proposition 7r( {l, n}) can be regarded as asserting, "it is false that the firefly is in the right half of the box with its light on," so we denote it by 7r(r'), etc ... The resulting Hasse diagram for the logic II of experimental propositions is shown in Figure 3. 12 DAVID J. FOVLIS 1 1t (I ') 1t (l) 1t (r') n (r) 1t (n') n (n) 1t (f) n(f) 1t (b') n (b) Figure 3. Logic of experimental propositions The partial order on II, depicted in Figure 3, is denoted by ::; and called implication, or entailment. Note that, if A,B E £, then 7r(A) ::; 7r(B) iff there is an event BI such that A ~ BI and BI is perspective to B. Evidently, 0::; 7r(A) ::; 1 for all 7r(A) ElI. To test an experimental proposition 7r(A) E II, we select any event Al (including A itself) such that 7r(A) = 7r(A I ), we choose a test for AI, and we carry out the test. If Al occurs, we say that the proposition 7r(A) is confirmed, otherwise, we say that it is refuted. Thus, confirmation and refutation of experimental propositions is linked to occurrence and nonoccurrence of events. For instance, to test whether 7r( n') is confirmed (i.e., whether the light is on), we can execute either F or S and conclude that the light is on iff the outcome n is not secured. In reporting that a proposition 7r(A) is confirmed or refuted it is not necessary to specify which test was executed. There is a natural notion of "logical negation" for the experimental propositions in the logic II. Indeed, if A E £, define 7r(A)' := 7r(C), where C E £ is any local complement of A. The proposition 7r(A)', which is A HALF-CENTURY OF QUANTUM LOGIC 13 easily seen to be well defined, is regarded as a logical negation, or denial, of 7f(A). Evidently, 0' = 1 and l' = O. In Figure 3, the logical negations of each proposition in the first row above 0 are located directly above that proposition in the first row below 1. For instance, 7f( l)' = 7f( l'). The partially ordered set II depicted in Figure 3 is actually a lattice; that is, any pair of propositions 7f(A) and 7f(B) have a least upper bound, or join, 7r(A)V7r(B) and a greatest lower bound, or meet, 7r(A)I\7f(B) with respect to the implication relation ::;. For instance, 7r(l) V7r(r) = 7r(n') and 7f(r) 1\ 7f(J) = O. The mapping 7f(A) f--+ 7f(A)' is an orthocomplementation on the lattice II in that it has the following properties for all experimental propositions p, q E II: (i) p I\p' = (ii) p V p' (iii) pI! ° =1 =P (iv) p::; q => q' ::; p'. As a consequence, II satisfies the De Morgan Laws: (v) (p /\ q)' = p' V q' and (vi) (p V q)' = p' /\ q'. Furthermore, the following orthomodular identity holds in II: (vii) p::; q => q = p V (q /\p'). Therefore, II forms a so-called orthomodular lattice [27, 44]. Say that propositions 7f(A) and 7f(B) in II are compatible iff there are compatible events Al and BI with 7f(A) = 7f(A I ) and 7f(B) = 7f(BI)' Note that a common test for the events Al and BI is then a common test for the propositions 7f(A) and 7f(B). For instance, 7f(l) and 7f(n) are compatible, 7f(n) and 7f(b) are compatible, but 7f(l) and 7f(b) are incompatible. Say that propositions 7f(A) and 7f(0) in II are orthogonal iff there are orthogonal events Al and BI with 7f(A) = 7f(A I ) and 7f(B) = 7f(BI)' Note that orthogonal propositions are necessarily compatible and that 7f(A) is orthogonal to 7f(C) iff 7f(A) ::; 7f(C)'. If 7f(A) ::; 7f(B) and if 7f(A) is confirmed, it is understood that 7f(B) is also confirmed and that every proposition 7f( 0) that is orthogonal to 7f(A) is refuted. Note that 7f(A) is confirmed iff 7f(A)' is refuted. Evidently, 1 is always confirmed, and is always refuted. In classical (Boolean) logic, the meet p/\ q of two propositions p and q is effective as their logical conjunction p&q. This is certainly not the case in the logic II; for instance, 7f( l) /\ 7f(J) = 0, whereas 7f( l) & 7f(J) would be the proposition asserting that the firefly is in the left front quadrant ° 14 DAVID J. FOULIS of the box with its light on. What is happening here is perfectly clearthe conjunction 7f( l) & 7f(f) is not in the logic II because there is no way to test it! How does one account for the nonclassical nature of the firefly box logic II, given that its source, a box with windows and a firefly, is utterly classical? The answer, as we shall see in Section 10 below, is that there are pairs of nonclassical quantum-mechanical experiments that yield the same event logic [;, and therefore the same experimental logic II, as the firefly box. The event logic [; does not "know" the difference between the firefly box and the quantum-mechanical system, so it produces an experimental logic II compatible with both. 6. THE LOGIC OF ATTRIBUTES Usually there are certain properties, or attributes, associated with a physical system S, such as "S is green," or "S carries an electric charge of 1.60217733 x 10- 19 coulomb," or "S has a spin component +~ in the z direction." An attribute can be either actual or potential. Those attributes that are always actual, such as the color of a raven or the charge of an electron, are said to be intrinsic. Attributes that can be either actual or potential, such as the color of a chameleon or the spin component of an electron, are called accidental. An actual attribute ex of a physical system S can manifest itself experimentally only in terms of outcomes of experimental procedures. In fact, ex induces a division of the set E of all outcomes of experimental procedures into two disjoint parts: P = those outcomes that are possible when ex is actual, and E\P = those outcomes that are impossible when ex is actual. What are the attributes associated with our firefly box? Attributes such as "the temperature in the box is 18° C" or "the box weighs 15 kg" do not concern us here since they are unrelated to the only experimental procedures at our disposal, namely F and S. However, consider the attribute ex = "either the firefly's light is off, or else it is on and the firefly is in the left front quadrant of the box." If ex is actual, outcomes rand b are impossible and the set of possible outcomes is P = {l, j, n}. Conversely, given that the possible outcomes when ex is actual are l, j, and n, one can easily identify the original attribute ex. More generally, those attributes ex of the firefly box that can manifest themselves by way of the experimental procedures F and S can always be recaptured as soon as we know the subset P of E consisting of outcomes that are possible when the attribute is actual. Therefore, the set P S;;; E A HALF-CENTURY OF QU ANTUM LOGIC 15 provides a perspicuous mathematical representation of the attribute a, and in what follows, we shall simply identify a with P. Suppose that P ~ E is an attribute of our firefly box and let A E [; be an event. If An P = 0, then A consists entirely of outcomes that are impossible when P is actual. Thus, if An P = 0 and P is actual, then A is impossible in the sense that it must nonoccur when tested. Recall that the two events {l, r} and {f, b} are logically equivalent; hence if {l, r} is impossible, so is {J, b} and vice versa. Consequently, P n {l, r} = 0 {=:} P n {f, b} = 0. (3) Of the 32 subsets of E = {l, r, n, f, b}, only 20 satisfy condition (1), and each of these can be interpreted as a meaningful attribute of the firefly box. Even the empty set 0 can be interpreted as an attribute, albeit one that is never actual. On the other hand, the set E of all outcomes satisfies (1) and represents an attribute that is always actual12 . Let us denote by A the collection of all attributes P ~ E, so that PEA iff P ~ E and P satisfies (1). The set A is partially ordered by the relation ~ of set containment. Furthermore, if P, Q E A, then P ~ Q iff Q is actual whenever P is actual. Therefore, ~ can be regarded as a kind of implication relation on A and, in this sense, A becomes a logical system called the attribute logic of the firefly box. It is easy to see that, if P, Q ~ E and both P and Q satisfy condition (1), then so does P U Q. Therefore, A is closed under the formation of unions. Thus, if P, Q E A, then P and Q have a join P V Q = P U Q in A. Also, if P, Q E A, then P and Q have a meet P A Q in A; in fact P A Q is the union of all attributes in A that are contained in both P and Q. Although A is closed under unions, it is not closed under intersections. For instance, P := {l, f, n} E A and Q := {l, b, n} E A, but P n Q = {l,n} r:J. A. In fact, P A Q = {n} -=f. P n Q. In general, if P, Q E A, then the attribute P A Q is a subset of the set P n Q, but P A Q -=f. P n Q unless it happens that P n Q E A. Suppose PEA and P is actual. If the experimental procedure F is executed, one of the outcomes l, r, or n in EF must be secured and, since the outcomes in EF \P are impossible, it follows that one of the outcomes in P n EF must be secured. In other words, the event P n EF necessarily occurs when P is actual. Likewise, if P is actual, the event pnEs necessarily occurs when tested. Furthermore, if P is actual, A E t:, and one of the conditions P n EF ~ A or P n Es ~ A holds, then A necessarily occurs when tested. Let us say that P guarantees A iff one 12 All of the intrinsic attributes are thus identified with E. 16 DAVID J. FOULIS of the conditions P n Ep ~ A or P n Es ~ A holds. The fact that the empty attribute 0 guarantees every event is harmless since the empty attribute is never actual. If A E E, let [A] denote the union of all attributes PEA that guarantee A. In other words, [A] is the largest attribute that, when actual, necessitates the occurrence of A when tested. It is easy to check that, if A,B E E, then 1I"(A) ::; 1I"(B) ~ [A] ~ [B], (4) so we can and do define [1I"(A)] := [A] for all events A E E. For simplicity, if e E E, we write [e] rather than [{ e}] and we write [e'] rather than [11" (e )']. An attribute of the form [A] is called a principal attribute. In view of (2), the mapping 1I"(A) I---> [A] embeds the experimental logic II in the attribute logic A, whence 12 of the 20 attributes in A are principal, and 8 are nonprincipal. Although the embedding 1I"(A) I---> [A] preserves joins, it fails to preserve meets. For instance, 11"( l) /\ 11"(1) = 0 in II, but [l] = {l,J, b}, [f] = {l, r, J} and [l] /\ [f] = {l, J} i- 0 in A. Each of the 8 non principal attributes in A can be written as a meet of principal attributes; for instance {l, j, n} is a nonprincipal attribute in A and {l,J, n} = [r'] /\ [b']. For simplicity, we write [r'b'] rather than [r'] /\ [b']. Similarly, we write [l f] rather than [l] /\ [f]' and so on. With this notation, the Hasse diagram for the attribute logic is shown in Figure 4. 7. ST A TES AND IRREDU eIB LE A TTRIB UTES Nearly every scientific theory utilizes, explicitly or implicitly, the notion of the state of a physical system. The usual understanding is that, at any given moment, the system is in a particular state 'l/;. All information about outcomes of experimental procedures executed on the system in state'l/; are supposed to be encoded into 'l/;. The state of the system can change in time under a deterministic or stochastic dynamical law , it can change because an experimental procedure is executed, or it can change spontaneously. Until now, our firefly box theory (FBT) has recognized only one explicit principle, namely n = np = ns. (However, one could argue that much of the discussion in Section 6 regarding the attributes of the firefly box constitutes a further evolution of the FBT). Now we have to face the issue of incorporating into our FBT a suitable mathematical representation for the set \)i of all possible states of the firefly box. 17 A HALF-CENTURY OF QUANTUM LOGIC 1 [r'] [l ] [b'] [r' b'] [l f] [n'] [f] [r' f] [l b] [f] [I'] [l ' b'] [b] [l' f] [n] [r f] [r] [r b] Figure 4. Logic of attributes We cannot see inside the box, but we are formulating our FBT under the supposition that the blips of light on the windows are caused by a firefly. The firefly could be located in anyone of the four quadrants of the box, and its light could be on or off, so there seem to be eight different possible states of the firefly box. However, when the light is off our available experimental procedures F and S provide no information about the location of the firefly. In view of this experimental limitation, it seems more reasonable to restrict our state space W to five possible states, namely (in Dirac's "ket" notation) \II = {Il!), Ilb), Ir!), Irb), In)}. (5) 18 DAVID J. FOULIS The first four states correspond to the location of the firefly in the leftfront, left-back, right-front, and right-back quadrant with its light on. In the fifth state In), the light is off. Notice in Figure 4 that there are exactly five minimal nonempty attributes in A, namely [If]' [lb], [r f], [rb], and [n]. These are the attributes that are irreducible in the sense that they cannot be decomposed into more elementary attributes, and they are in obvious one-to-one correspondence with the five states in \]I. Two new principles suggest themselves, and we now incorporate them into our FBT: Principle of Irreducible Attributes To each state 'lj; E \]I there corresponds a uniquely determined irreducible attribute P,p E A. The attribute P,p is actual iff the system is in state 'lj;. Principle of Actuality for Attributes An attribute PEA is actual iff the system is in a state 'lj; for which P,p ~ P. Thanks to the principle of irreducible attributes, one and only one of the irreducible attributes is actual at any given moment. As a consequence of both principles, this unique irreducible attribute is the meet of all the attributes that are actual at that moment. Suppose P, Q E A. In spite of the fact that P /\ Q is not necessarily P n Q, it turns out (and is not difficult to verify) that P /\ Q is actual iff both P and Q are actual. Thus, P /\ Q is effective as a true logical conjunction of P and Q in the attribute logic A. That the join P V Q = P U Q of two attributes is not necessarily a logical disjunction of P and Q is a profound observation first made by D. Aerts [1]. For instance, a glance at Figure 4 shows that [If] V [rb] = [n'l; yet the attribute [n'] can be actual (i.e., the light is on) in a state (e.g., Irf)) for which neither [If] nor [rb] is actual. The fact that a join of attributes need not be a logical disjunction of the attributes can and should be regarded as the true basis for the notion of "superposition of states." Say that a state 'lj; E \]I is a proper superposition of states ct, {3 E \]I iff P,p ~ Pa V P{3 but 'lj; =I- ct, (3. For instance, Ir f) is a proper superposition of Ilf) and Irb). Since the publication in 1930 of Dirac's seminal monograph on the mathematical foundations of quantum mechanics [21], it has been an article of faith among physicists that a fundamental distinction-if not the fundamental distinction-between quantum and classical mechanics is that there are proper superpositions of states in the former, but not in the latter. If this is so (and I am not entirely convinced that it is [7]), then our firefly box is already exhibiting quantal behavior! A HALF-CENTURY OF QU AN TUM LOGIC 19 8. PROBABILITY MODELS The system of real numbers is denoted by the symbollR, and the closed interval of real numbers between 0 and 1 is written as [0, 1]. By a probability model for the firefly box, we mean a function w : E ----t [0, 1] ~ JR mapping each outcome e E E into a real number w(e) between 0 and 1 in such a way that w(l) + w(r) + w(n) = 1 and w(j) + w(b) + w(n) = 1. (6) If e E E, then w(e) is to be interpreted as the probability, according to the model w, that the outcome e will be secured when an experimental procedure (F or S) is conducted for which e is a possible outcome. Denote by n the set of all probability models w for the firefly box. If A E £ is an element of the event logic and wEn is a probability model, we define w(A) := L w(e) (7) eEA and interpret w(A) as the probability, according to the model w, that the event A will occur if tested. In this way, probability models wEn can be "lifted" to the logic £ of events. If A, B E £ with A ~ B, it is clear that w(A) :S w(B). If A, C E £ and A is orthogonal to C, then w(A U C) = w(A) + w(C). Therefore, (6) implies that w(A) + w(C) = 1 for local complements A, C E £. If wEn and A and B are perspective events with axis C, then w(A) + w(C) = 1 = w(B) + w(C), and it follows that w(A) = w(B). Hence, for an experimental proposition 7r(A) we can and do define w(7r(A)) := w(A). In this way, probability models wEn can be lifted to the logic II of experimental propositions. Naturally, w(7r(A)) is interpreted as the probability, according to the model w, that 7r(A) will be confirmed if tested. If wEn, then, regarded as a function w : II ----t [0, 1] ~ JR, w is a probability measure in the sense that w(l) = 1 and, for orthogonal propositions 7r(A) and 7r(C), w(7r(A) V 7r(C)) = w(7r(A)) + w(7r(C)). For the firefly box, n provides a so-called full, or order determining, set of probability measures in the sense that, if w(7r(A)) :S w(7r(B)) for all wEn, then 7r(A) :S 7r(B). If WI, W2, ... , wn E nand tl, t2, ... tn are positive real numbers such that tl + t2 + ... + tn = 1, the function w : E ----t JR defined for all e E E by (8) 20 DAVID J. FOULIS is called a convex combination, or mixture, of WI,W2, ... ,Wn with mixing coefficients tl, t2, ... , tn. It is not difficult to see that such a mixture takes on values between 0 and 1 and satisfies 0, so it is again a probability model wEn. In other words, n is a convex set, i.e., it is closed under the formation of convex combinations. Each wEn is completely determined by the three real numbers (9) x := w(l), y := w(f), and z := w(n). Indeed, as a consequence of (6), w (r) = 1- x - z and w (b) = 1- y - z. (10) Of course, the numbers x, y, z are subject to the conditions o ::; x, y, z ::; 1, x + z ::; 1, and y +z ::; 1. (11) The set of all points (x, y, z) in coordinate 3-space IR3 that satisfy (11) is a pyramid with a square base (Figure 5). A point (x, y, z) in the pyramid may be identified with the corresponding state w by (9) and (10), so the pyramid provides a geometric representation of the space n of probability models for the firefly box. The five vertices WII!) := (1,1,0), Wllb) := (1, 0, 0), Wlrf) := (0, 1, 0), Wlrb) := (0, 0, 0), and win) := (0, 0, 1) of the pyramid correspond in an obvious way to the five states in W. For instance, x = Wllf)(l) = 1, y = WII!) (f) = 1, and z = WII!)(n) = 0 for the probability model Wllf). In the geometric representation of n as a pyramid (Figure 5), the five vertices correspond to extreme points of the convex set n, that is, points W that cannot be written in the form (3) unless WI = W2 = ... = w. For a polytope, such as n, there are only finitely many extreme points, and every point is a convex combination of extreme points. If wEn, we define the support of w, in symbols supp(w) by supp(W) := {e E EIO < w(e)}, (12) noting that supp(w) is the set of all outcomes e E E that are possible according to the model w. In view of our discussion in Section 7, it should come as no surprise (and it is easy to check) that supp(w) E A. Furthermore, the supports of extreme points produce irreducible attributes corresponding to states, just as one would expect. For instance, supp( WI If) ) = {l, J} = [l fJ which corresponds to the state Ilf) . Note that the support of a convex combination (8) is the union of the supports of WI, W2, ... , Wn E n from which it is formed. The five vertices, eight edges, four triangular faces, and the square base of the pyramid in Figure 5 are called faces of n. In addition it is A HALF-CENTURY OF QU ANTUM LOGIC 21 z 1 y x Figure 5. Geometric representation of n convenient to include the empty set 0 and fl itself as (improper) faces, making a total of twenty faces in all. The four triangular faces and the square base-that is, the maximal proper faces-are called facets. The five vertices are the minimal proper faces. The intersection of two faces is again a face, and, given any two faces there is a unique smallest face containing both. Therefore, partially ordered by inclusion ~, the faces form a lattice, called the face lattice of fl, and denoted by F. The fact that both the attribute logic A and the face lattice F have twenty elements is no accident. In fact there is a natural one-to-one correspondence P f---t <I> between attributes PEA and faces <I> E F given by <I> := {w E fl\supp(w) ~ P} and P := UWEi[>SUpp(w). Furthermore, 22 DAVID J. FOULIS the correspondence P f-----t <P is a lattice isomorphism in that it preserves meets and joins. Thus, the face lattice :F of [2 provides an alternative representation for the attributes of the firefly box. 9. TESTING AND INFERENCE For the firefly box we now have three related logical structures, namely £, II, and A, as well as the state space W, the convex set [2 of probability models, and the face lattice :F of fl, which is isomorphic to A. For A E £ we have three truth values: occur, nonoccur, and not tested. For 7r(A) E II we again have three truth values: confirmed, refuted, and not tested. By their very definitions, these truth values can be determined by executing an appropriate experimental procedure, either F or S. For an attribute PEA we have two truth values: actual and potential. Unlike the truth values for events and experimental propositions, it might not be possible to determine the truth value of an attribute P by conducting a single experiment. If P = [7r(A)] is a principal attribute, we can test P by testing 7r(A). If 7r(A) is refuted, then P cannot have been actual since its actuality guarantees 7r(A). If 7r(A) is confirmed, we have evidence that P might have been actual, but it may not be conclusive. If PEA is not principal, it can be written as a conjunction of principal attributes, one of which can be selected and tested, again supplying (usually inconclusive) evidence that P was either actual or only potential [63]. Apparently, inferences about which properties of a physical system are actual and which are only potential will have to depend on evidence gathered from repeated testing, either under circumstances in which one has reason to believe that the state of the system remains unchanged, or on a sequence of replicas of the underlying system all of which are presumed to be in the same state. As of now, however, there seems to have been no serious attempt to develop a mathematical theory of formal scientific inference regarding the attributes of a physical system. For each state 't/J E W, we have two truth values, in and not in. To test the state 't/J, we can test the corresponding irreducible attribute P'Ij; as indicated above. But then state testing will be as inconclusive as attribute testing and it will be hindered by the same lack of a theory of inference. For certain physical systems (if not for our firefly) it is possible to prepare a preassigned state 't/J, that is, to bring the system into the state 't/J by carrying out suitable procedures. When state preparation is possible, it may render moot the question of how to test states (and perhaps attributes as well). A HAL F - C E N T U R Y 0 F QUA N TUM LOG I C 23 Testing probability models is quite another matter-indeed this is what statistical inference is all about! The usual idea is that there exists a "true probability model" w* E n representing the habits of the firefly. Although we might not know which probability model is w*, we might be able to make some (perhaps tentative) conclusions about w* by repeatedly executing our procedures F and S and processing the experimental data thus obtained. It is often assumed that the repeated trials of F and S are "independent" in the sense that the firefly's habits are unaffected by our experiments. It is easy to challenge this assumption, but not so easy to design reliable strategies of statistical inference to take into account observation-induced changes in the firefly's behavior patterns. Two useful mathematical tools conventionally employed in statistical investigations are statistical hypotheses and parameters. By a statistical hypothesis is meant a subset A of n, usually subject to a condition that it be measurable in some appropriate sense (e.g., Borel or Lebesgue measurable). Let I denote the set of all statistical hypotheses. By a statistical parameter is meant a real valued function), : n ---+ JR satisfying the condition that, for every interval I ~ JR, the set ),-1(1) := {w E n I ),(w) E I} (13) is a statistical hypothesis in I. A statistical hypothesis of the form)' -1 (I) is called a parametric hypothesis. A statistical hypothesis A E I is understood to represent the proposition w* E A asserting that the true probability model belongs to A. Partially ordered by ~, I forms a logical system called the inductive logic, and in I the meet A 1\ r = A n r and join A V r = A u rare effective as the conjunction and disjunction, respectively, of statistical propositions A, rEI. Under these operations, I forms a Boolean algebra. The branch of statistics known as hypothesis testing is concerned with the problem of deciding whether to (tentatively) accept or reject a statistical hypothesis in the face of experimental data, or to hold it in abeyance. Thus, statistical hypotheses acquire three truth values: accepted, rejected, and held in abeyance. The "true value" of a statistical parameter)' is of course ),* := ),(w*) and parameter estimation is the branch of statistics devoted to the problem of estimating ), * on the basis of experimental data. A point estimation of ),* produces a real number Xthat one has reason to believe is a good approximation to ), *. An interval estimation of ), * yields a confidence interval I ~ JR with the understanding that the statistical hypothesis), -1 (1) is to be accepted. For our firefly box, the components x, y, z of the geometric point (x, y, z) representing the probability model 24 DAVID J. FOULIS wEn as in Figure 5 form a complete set of statistical parameters in the sense that knowledge of x*, y*, and z* would determine w*. Suppose the experimental procedure F is executed TF times and that the outcomes l, r, and n are secured N(l), N(r), and NF(n) times, respectively, during these trials. Likewise, suppose S is executed Ts times and that the outcomes f, b, and n are secured N(f), N(b), and Ns(n) times, respectively, during these trials. Thus, TF = N(l) + N(r) + NF(n) (14) and + N(b) + Ns(n) := TF + Ts trials. If we Ts = N(f) (15) for a total of T assume that the habits of the firefly are unaffected by our observations, then the sequential order in which F and S are executed is presumably irrelevant. We could carry out the TF trials of F first, then perform the Ts trials of S-or vice versa. We could alternate trials of F and S. We could even flip a coin after each trial to see whether to perform F or S on the next trial. In any case, all pertinent information derived from the T = TF + Ts trials is encoded in the observed frequency vector rJ := (N(l), N(r), NF(n), Ns(n), N(f), N(b)). (16) If ,\ is a statistical parameter, an estimator for ,\ is a function X (rJ) that provides a numerical estimate ,\* ~ ,\1rJ) of ,\* based on the experimentally observed frequencies. Statisticians have developed several techniques and conditions to assess and compare various proposed estimators. For instance, an estimator X is said to be unbiased iff, whenever the observed frequency vector rJ conforms exactly to a probability model w in the sense that N(l) = w(l)TF, N(r) = w(r)TF, NF(n) = w(n)TF, Ns(n) = w(n)Ts, N(f) = w(f)Ts, and N(b) = w(b)Ts, then ,\1rJ) = '\(w). A weaker, and perhaps more realistic condition is that the estimator be asymptotically unbiased in the sense that '\1w) approaches '\(w) as a limit when TF and Ts become larger and larger. 9.1 Example Let N(l, r) := N(l) + N(r), N(f, b) := N(f) + N(b), N n := NF(n)+Ns(n), and T:= TF+TN. Then the maximum likelihood estimators [48] for the statistical parameters x, y, z are given by x ~= y ~= N(l) N(l,r) [1 _N(n)] T' (17) N(f) N(f,b) [1 _N(n)] T' (18) A HAL F - C E N T U R Y 0 F QUA N TUM LOG I C ZA= ~. 25 (19) As is easily checked, the estimators in Example 9.1 are unbiased. 10. THE QUA N TUM FIR E FLY BOX Associated with a quantum-mechanical system S is a vector-like quantity called spin. However, it turns out that measurements of the spin component in a fixed direction can produce only finitely many different numerical outcomes rather than the continuum of possible components that would be expected for an ordinary vector quantity. In other words, the spin components of S in a given direction are "quantized." The behavior of S in regard to its spin is characterized by a number j which can be 0, a positive integer, or half of a positive integer. The spin component of a spin-j system S, measured in a fixed spatial direction d, can take on only 2j + 1 different values: -j, -j + 1, ... ,j -1, or j. For instance, the spin component of a spin-! system, measured in a given direction d, can only be -! or !. For a spin-! system, the outcomes -! and ! are called spin down and spin up in the direction d. An electron, for example, is a spin-! particle. Suppose S is a spin-! system and we have a spin detecting apparatus that will measure the spin component of S in the direction of a unit vector d = (d 1 , d2, d3), di + d§ + d§ = 1. It turns out that the possible states of the system S are represented by vectors 'l/J = ('l/Jl, 'l/J2, 'l/J3) with 'l/Jr + 'l/J~ + 'l/J~ ~ 1. Therefore, the state space \[1 of S can be visualized as a solid sphere of radius 1. According to the rules of quantum mechanics, the probability of spin-up in direction d when S is in state 'l/J E \[1 is given by Probv,,(spin-up in dir. d) = For the special case in which 'l/Jr + 'l/J~ the unit vectors d and \[1, ~(1 + 'l/Jldl + 'l/J2d2 + 'l/J3d3)' (20) + 'l/J§ = 1 and 'Y is the angle between Probv,,(spin-up in dir. d) = cos2(~). (21) experimental apparatus is rarely 100 percent efficient, and the probabilities given by (20) and (21) have to be regarded as conditional probabilities, given that the detector actually produces a response. Imagine now that our firefly is really a spin-! system, that a spin detector is inside the box, and that it signals spin-up or spin-down by 26 DAVID J. POVLIS producing a blip of light on the front or side window. For the front window, with the same symbols used in Section 4, suppose the detector is set so that spin-up in direction dp produces outcome l, spin-down in direction dp produces outcome r, and outcome np (no blip on the front window) simply means that the detector failed to respond. Likewise, for the side window, spin-up in direction ds produces outcome j, spindown in direction ds produces outcome b, and outcome ns means that the detector failed to respond. As before, we must look directly at one window or the other 13 . Suppose the spin detector has the same detection efficiency, say 100E%, o :::; E :::; 1, in anyone direction dp as in any other direction ds. Then we are (almost literally) "in the dark" about the spin of S when we look at the front window about as often as when we look at the side window. In this case, there seems to be no harm in setting n := np = ns as we did in Section 4. Then, for either window, the probability of outcome n is 1 - E. Let us choose our coordinate system so that dp = (1, 0, 0) and ds = (cos a, sin a, 0) with 0 < a :::; 7r /2. The state vector 'ljJ can then be written in terms of spherical coordinates 0 :::; p :::; 1, 0 :::; 0 :::; 27r, and 0:::; <1>:::;7r as o 'ljJ = (p cos 0 sin <1>, psin sin <1>, pcos <1». (22) Then, with the same notation as in Section 8, x = Prob?fJ(l) = ~E(l y = Prob?fJ(r) = ~E(l + pcosOsin<1», (23) + pcos(a - (24) 0) sin <1», and z = 1- E. (25) For the quantum firefly box, not every probability model w in the pyramid n of Section 8 corresponds to a possible state 'ljJ E \Ii. For instance, the four vertices on the bottom square are unattainable from (23), (24) and (25), even if E = 1. In fact, for E = 1 and 0 < a :::; 7r /2, the set r a of all points in n that correspond to possible states 'ljJ of the quantum firefly box is a convex region in the square base of n bounded by an ellipse with 13 Although there is no Heisenberg principle of uncertainty for spin measurements in different directions, the Hilbert-space operators for such measurements fail to commute. A HAL F - C EN T U R Y 0 F QUA N TUM LOG I C center (~, ~), 27 major axis along the line y = x, and having semi-major 1 1 and semi-minor axes of lengths 2-2 cos(a/2) and 2-2 sin(a/2), respectively. If a = 7r /2, then r a is a circular disk tangent to the boundary of the square base of n at the four midpoints (~, 0), (1, ~), (~, 1), and (0, ~ ). For our original firefly box, r a is a statistical hypothesis asserting that the firefly is behaving like a quantum firefly. 11. CONCLUSION---WHAT HAVE WE LEARNED? Our thought experiments with the firefly box have provided illustrations of some of the more important ideas and tools employed in the scientific study of physical systems: outcomes, events, experimentally testable propositions, states, attributes, probability models, tests, statistical hypotheses, and statistical parameters. Of course, this list is far from complete-what about observables, dynamics, symmetries, invariants, conservation laws, amplitudes, coupled systems, relativistic physics, causality, and so on? Although most of these ideas can also be illustrated and studied in the context of the firefly box (or firefly boxes), it is already possible to discuss what I consider to be the main lessons of quantum logic, and I propose to do that now. Here then are my personal candidates for the answers to the question posed in the title of this article: Logical Connectives. In dealing with propositions associated with a physical system, one must question the meaning and even the existence of the basic connectives of classical logic-and, or, denial, and implication. In our firefly example: (1) The event logic £ is not even a lattice, a fact which warns us not to try forming the logical disjunction-let alone the join-of event propositions such as {l} and {j} that cannot be tested simultaneously. There is no meaningful denial connective on the event logic £. For instance, what would be the denial of the event {n}? Is it {l, r}? Or is it {j, b}? (2) The logic II of experimental propositions is a lattice, but the meet (respectively, join) of experimental propositions is not their logical conjunction (respectively, disjunction) unless the propositions are simultaneously testable For instance 7r( l) 1\ 7r(J) = 0, which in no way corresponds to a logical conjunction 7r(l) & 7r(J) of 7r(l) and 7r(J). However, the logic II does carry a rather perspicuous logical negation p 1--7 pi (which in fact is an orthocomplementation). (3) The logic A of attributes is again a lattice, and the meet P 1\ Q of attributes P and Q is effective as their logical conjunction. But, if there are irreducible attributes contained in P and Q that admit proper 28 DAVID J. FOULIS superpositions, then the join P V Q cannot be construed as a logical disjunction of P and Q. Also, the logic A, does not admit any reasonable denial connective. In particular, there is no orthocomplementation on A. (4) The implication connective (p, q) f---7 P :J q is even more conspicuously absent in quantum logic than the conjunction and disjunction connectives. The material implication connective p :J q := p' V q of classical (Boolean) logic has been the subject of considerable philosophical criticism and debate; in quantum logics modelled by orthomodular lattices, one has to forfeit even this suspect connective and make do, if at all, with severely attenuated versions thereof [37]. Note, however, that all of the logical systems £, II, A, and I admit perspicuous implication relations, namely their respective partial order relations <;::;, :s;, <;::;, and <;::;. (5) It is only when one reaches the level of the inductive logic I of statistical hypotheses (a Boolean algebra) that one encounters a logical system with a secure and well-understood meaning of conjunction, disjunction, and denial as well as a (material) implication connective. (6) Even at the level of the inductive logic, a conditional hypothesis qA (i.e., r given A) cannot be construed as a material implication A :J r, and the logic I has to be enlarged to a Heyting algebra I I I to accommodate this important alternative notion of implication [73]. Conditional events, conditional propositions, and conditional hypotheses are currently under intense study by electronic engineers and computer scientists because of the necessity of codifying conditional information in expert systems. A Hierarchy of Logical Systems. There is a hierarchy of related but distinct logical structures affiliated with a physical system. These include, but are not limited to, the event logic £, the logic II of experimental propositions, the attribute logic A, the face lattice F, the inductive logicI, and the logic I I I of conditional hypotheses. Propositions in the various logics are different in kind, are tested in different ways, and have their own distinct modalities. I have a coin. (i) I can toss the coin, observe the outcome and determine whether or not the event "heads" occurred. (ii) I can make a prediction that the coin will fall "heads" on the very next toss. (iii) I can assert that the coin is fair and that I am willing to bet at odds of 1:1 on either "heads" or "tails" on the next toss. (iv) I can claim that, in a sufficiently long sequence of independent tosses, the proportion of "heads" will be very close to 0.5. It is patently obvious that the observation (i), the prediction (ii), the assignment (iii) of betting odds, and the claim (iv) regarding long run frequency are four propositions of completely different characters. The folly of attempting to formulate a single "unified logic" comprising all propositions affiliated with a physical system is manifest. A HAL F - C EN T U R Y 0 F QUA N TUM LOG I C 29 Perhaps the source of all such unfortunate attempts is the use of the definite article the in the title of the seminal paper of Birkhoff and von Neumann [10]. Events vs. Experimental Propositions. One cannot necessarily formulate quantum logic purely on the basis of the logic II of experimental propositions and higher-level logics built upon II. Indeed, some of the information implicit in the observation that a certain outcome was obtained (or that a certain event occurred) may be lost in the passage from events to experimental propositions. For the firefly box, the loss of information in the passage from E to IT is of little concern. Neither is it particularly serious in Hilbert-space based quantum mechanics, provided that one is dealing with a single isolated observation, e.g., a measurement of a spin-component with a Stern-Gerlach apparatus. In such a case, one can safely use elements of the standard quantum logic of Hilbert-space projection operators to carry the pertinent experimental information. However, if one has to deal with sequential or compound observations, e.g., iterated Stern-Gerlach measurements [75], the phase and amplitude information encoded in the complex wavefunction becomes critical. In the passage from an orthonormal set ('lj;i) of Hilbert-space state vectors to the corresponding projection operator P onto the closed linear subspace spanned by these vectors, all phase and amplitude information is wiped out! Experimental Propositions vs. Attributes. experimentally testable propositions about a physical system are one thing; attributes or properties of that system are quite another. By universal agreement, the genesis of what is now called quantum logic is von Neumann's Grundlagen der Quantenmechanik [59]. Nowhere in the Grundlagen does von Neumann refer to propositions about a physical system; he refers only to properties (i.e., what we have been calling attributes) of that system. However, four years after the publication of the Grundlagen, von Neumann (in collaboration with Birkhoff), writes only of experimental propositions and propositional calculi-there is no further mention of properties. I do not know how to account for von Neumann's abrupt transition from a logic of properties to a logic of experimental propositions. I do know that, since then, many (but not all [1, 2, 3, 58, 62, 63, 69]!) quantum logicians have routinely identified experimental propositions about a physical system and attributes of that system. This is a mistake, and a serious one! For our firefly box, we have seen that the logic II of experimental propositions and the logic A of attributes are separate, distinct, and 30 DAVID J. FOULIS nonisomorphic logical systems, albeit linked by the mapping 7r(A) 1-+ [AJ. I know of no more compelling illustration of the error of confusing experimental propositions and attributes. States and Probability Models. Whereas it may be useful to assume, as is often done, that there is a probability model w'IjJ E n corresponding to each physical state 't/J E \]i, there is no a priori reason that the mapping 't/J 1-+ w'IjJ has to be either injective or surjective. In the literature of pure mathematics, certain linear functionals, measures, or homomorphisms are referred to as states because they or their analogues do in fact represent physical states in some conventional theory of mathematical physics. Quantum logicians need to be more careful! Even in conventional Hilbert-spaced based quantum theory, where (pure) states are represented by vectors 't/J in the unit sphere, there is a distinction between a state 'ljJ and the corresponding probability measure w'IjJ on the logic II of projection operators. Indeed, the probability measure w'IjJ, defined for P E II by w'IjJ(P) = (P't/J, 't/J), determines 't/J only up to a phase factor, and the identification of 't/J with w'IjJ would wipe out all phase information incorporated in the state vector. Thus, in conventional quantum mechanics, the mapping 't/J 1-+ w'IjJ is surjective (by a celebrated theorem of Gleason [23]), but not injective. For our quantum firefly in Section 10, the mapping 't/J 1-+ w'IjJ from state vectors 't/J to probability models wEn is injective, but not surjective. For the quantum firefly, the lack of surjectivity of't/J 1-+ w'IjJ can be ascribed to the fact that we can only measure the spin component in two different directions. If we append spin component measurements in additional directions to our list of available experimental procedures, we find that n gets smaller and smaller until, finally, the mapping 't/J 1-+ w'IjJ becomes surjective. An assumption that the mapping 't/J 1-+ w'IjJ from the state space 't/J of a physical system to the convex set n of all probability models for the system is injective, surjective, or both constitutes a significant physical assumption about the system and the available experimental procedures for its study. Hidden Variables. Quantum logic enables the construction of simple finite models that can help us understand the so-called problem of hidden variables. The question of whether apparent quantal behavior can be explained by classical experimental procedures that are currently unknown or unavailable is called the problem of hidden variables. For instance, our (non quantal) firefly box certainly admits hidden variables in the sense that a third window on top of the box would remove all apparent quantal behavior! A HALF-CENTURY OF QU ANTUM LOGIC 31 A mathematical proof showing that the quantal behavior of a particular physical system cannot be accounted for by hidden variables is called a no go proof. The first convincing no go proof was given by von Neumann himself in the Grundlagen [59]. An excellent survey of no go proofs up until about 1973 can be found in [5]. Complementarity. Quantum logic also enables the construction of simple models that can help us appreciate the so-called principle of complementarity. In the writings of a number of philosophers and scientists, not the least of whom was Bohr himself, Bohr's principle of complementarity has often been burdened with confusing metaphysical embellishments. Stripped of these encumbrances, the principle seems to affirm that there may be different experimental procedures, each of which can reveal aspects of a physical system necessary for a complete determination of its state, but whose conditions of execution are mutually exclusive. Our firefly box with the two experimental procedures F and S provides a perfect example of this situation, and also exposes the strong connection between complementarity and the problem of hidden variables. At roughly the same time that scientists were confronted by a breakdown in the Newtonian mechanical view of the physical world, artists were discovering a "principle of complementarity" in their own world. I doubt that many artists had any direct understanding of the physical principle of complementarity in quantum mechanics. Nevertheless, there was a sympathetic resonance between the two worlds, which I leave it to the reader to contemplate after perusing the following words of the art critic Marco Valsecchi [64]. "The idea was to arrange the forms in a plane so that an object or figure could be recognized not through perspective illusion, but through an analysis of its form, and also so that it could be seen from several points of view. These multiple analyses of total vision were put into a single image, thus giving an immediate unity to what has been seen, deduced and imagined. . .. to bring together all the multiple aspects of an object and to reduce them to the plane of the painting, like a summation all at the same time of all the different instances of poetic and rational perception." 32 DAVID J. FOULIS AFFILIATION David J. Foulis Department of Mathematics and Statistics University of Massachusetts Amherst, MA 01003, USA [email protected]. edu REFERENCES [1] Aerts, D., The One and the Many, Doctoral Dissertation, Brussels Free University, 1981. [2] Aerts, D., "Description of many physical entities without the paradoxes encountered in quantum mechanics", Found. Phys., 12, 1982, pp. 1131-1170. [3] Aerts, D. and Daubechies, I., "Physical justification for using tensor product to describe two quantum systems as one joint system", Helv. Phys. Acta, 51, 1978, pp. 661-675. [4] Alfsen, E.M. and Shultz, F.W., Non-commutative Spectral Theory for Affine Function Spaces on Convex Sets, Memoirs of the Amer. Math. Soc., 172, 1976. [5] Belinfante, F.J., A Survey of hidden variables Theories, Pergamon Press, Oxford, 1973. [6] Beltrametti, E. and Cassinelli, G., The Logic of quantum mechanics, Addison-Wesley, Reading, MA, 1981. [7] Bennett, M.K. and Foulis, D.J., "Superposition in quantum and classical mechanics", Found. 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[37] Herman, L., Marsden, E., and Piziak, R., "Implication connectives in orthomodular lattices", Notre Dame J. Formal Logic, XVI, 1975, pp. 305-328. [38] Holevo, A.S., "Statistical decision theory for quantum systems", J. of Multivariate Analysis, 3, 1973, pp. 337-394. [39] Holevo, A.S., "Probabilistic and Statistical Aspects of Quantum Theory", in: Krishnaiah, P.R. and Rao, C.R. (eds.), North-Holland Series in Statistics and Probability, Vol. 1, North Holland, Amsterdam/New York/Oxford, 1982. [40] Holland, S.S., "Orthomodularity in infinite dimensions; a theorem of M. Soler", Bull. Amer. Math. Soc., 32, 1995, pp. 205-232. [41] Husimi, K., "Studies on the foundations of quantum mechanics I", Proc. Physico-Mathematical Soc. Japan, 19, 1937, pp. 766-78. [42] Jammer, Max, The Philosophy of quantum mechanics, Wiley, New York, 1974. [43] Jauch, J.M., Foundations of quantum mechanics, Addison-Wesley, Reading, Mass., 1968. [44] Kalmbach, G., Orthomodular lattices, Academic Press, N.Y., 1983. A HAL F - C E N T U R Y 0 F QUA N TUM LOG I C 35 [45] Kalmbach, G., Measures on Hilbert lattices, World Scientific, Singapore, 1986. [46] Keller, H., "Ein Nicht-klassischer Hilbertscher Raum", Math. Zeit., 172, 1980, pp. 41-49. [47] Klay, M.P., "Einstein-Podolsky-Rosen experiments: The structure of the probability space. I" , Found. Phys. Letters, 1, No.3, 1988, pp. 205244. [48] KUiy, M.P. and Foulis, D.J., "Maximum likelihood estimation on generalized sample spaces: An alternative resolution of Simpson's paradox", Found. Physics, 20, No.7, 1990, pp. 777-799. [49] Klay, M.P., Randall, C.H., and Foulis, D.J., "Tensor products and probability weights", International Jour. of Theor. Phys., 26, No. 3, 1987, pp. 199-219. [50] Kochen, S. and Specker, E.P., "The problem of hidden variables in quantum mechanics", J. Math. Mech., 17, 1967, pp. 59-87. [51] Kolmogorov, A.N., Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933, English Translation, Foundations of the Theory of Probability, Chelsea, New York, 1950. [52] Kopka, F., "D-posets of fuzzy sets", Tatra Mountains Mathematical Publications, 1, 1992, pp. 83-87. [53] Loomis, L., The lattice Theoretic Background of the Dimension Theory of operator Algebras, Memoirs of the Amer. Math. Soc., 18, 1955. [54] Ludwig, G., Foundations of quantum mechanics Vols. I and II, Springer, New York, 1983/85. [55] Mackey, G., "Quantum mechanics and Hilbert space", Amer. Math. Monthly, 64, 1957, pp. 45-57. [56] Mackey, G., The Mathematical Foundations of quantum mechanics, Benjamin, N.Y., 1963. [57] Maeda, S., "Dimension functions on certain general lattices", J. Sci. Hiroshima Univ., A19, 1955, pp. 211-237. [58] Mielnik, B., "Quantum logic: Is it necessarily orthocomplemented?", in: Flato, M. et al (eds.), Quantum mechanics, Determinism, Causality, and Particles, Reidel, Dordrecht, 1976. [59] von Neumann, J., Grundlagen der Quantenmechanik, Springer Verlag, Berlin, Heidelberg, New York, 1932; English translation: Mathematical Foundations of quantum mechanics, Princeton University Press, Princeton, NJ, 1955. [60] von Neumann, J., Continuous Geometry, Princeton Univ. Press, Princeton, NJ, 1960. 36 DAVID J. FOULIS [61] von Neumann, J. and Murray, F.J., "On rings of operators III", Ann. of Math., 41, 1940, pp. 94-161, in: J. von Neumann Collected Works, III, Pergamon Press, Oxford, 1961. [62] Piron, C., Foundations of Quantum Physics, Mathematical Physics Monograph Series, W.A. Benjamin, Reading, MA, 1976. [63] Piron, C., "Ideal measurement and probability in quantum mechanics", Erkenntnis, 16, 1981, pp. 397-401. [64] Porzio, D. and Valseechi, M., Pablo Picasso, Man and His Work, Chartwell Books, Inc., Secaucus, NJ, 1974. [65] Ptak, P. and Pulmannova, S., Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht/Boston/London, 1991. [66] Randall, C. and Foulis, D., New Definitions and Theorems, University of Massachusetts Mimeographed Notes, Amherst, MA, 1979. [67] Randall, C. and Foulis, D., "Tensor products of quantum logics do not exist", Notices Amer. Math. Soc., 26, No.6, 1979, A-p. 557. [68] Randall, C.H., and Foulis, D.J., "operational statistics and tensor products", in: Neumann, H. (ed.), Interpretations and Foundations of Quantum Theory, Band 5, Wissenschaftsverlag, Bibliographisches Institut, Mannheim/Wien/Ziirich, 1981, pp. 21-28. [69] Randall, C.H. and Foulis, D.J., "Properties and operational propositions in quantum mechanics", Found. Physics, 13, No.8, 1983, pp. 843-863. [70] Riittimann, G.T., Non-commutative Measure Theory, Habilitationsschrift, Universitat Bern, 1980. [71] Riittimann, G.T., "The approximate Jordan-Hahn decomposition", Canadian J. of Math., 41, No.6, 1989, pp. 1124-1146. [72] Soler, M.P., "Characterization of Hilbert space by orthomodular spaces", Comm. Algebra, 23, 1995, pp. 219-243. [73] Walker, E.A., "Stone algebras, conditional events, and three valued logic", IEEE Transactions on Systems, Man, and Cybernetics, 24, No. 12, 1994, pp. 1699-1707. [74] Wilee, A., "Tensor products in generalized measure theory", International J. Theor. Phys., 31, No. 11, 1992, pp. 1915-1928. [75] Wright, R., "Spin manuals", in: Marlow, A.R. (ed.), Mathematical Foundations of Quantum Theory, Academic Press, N.Y., 1978. [76] Younce, M.B., Random Variables on Non-Boolean Structures, Ph.D. dissertation, University of Massachusetts, 1987. STANLEY GUDDER QUANTUM MECHANICAL MEASUREMENTS 1. INTRODUCTION One of the main purposes of physics is to gain an understanding of the natural universe. Such an understanding cannot be achieved all at once so the physicist concentrates on a limited physical system S (sayan electron). The experimentalist performs experiments or measurements on S in an attempt to discover various properties. The theoretician uses these properties to try to find regularities and patterns in the behavior of S under prescribed conditions. In this way, a theory may be developed that predicts the behavior of S and this predicted behavior may be checked by the experimentalist in the laboratory. If the predictions are experimentally verified, the theory stands for the time being and if they are refuted the theory must be altered or abandoned. Since the main enterprise of physics (and practically all the basic natural sciences) is performing and predicting the results of measurements, it is important to understand the structure of a set of measurements. For example, can measurements be decomposed into simpler ones? Conversely, how can measurements be combined to form new measurements and what are the properties of these combinations? Investigators who have studied measurement theory have developed mathematical models for the description of measurements. In this article, we shall discuss one of these models in very simple terms. Although we shall primarily be concerned with quantum mechanical measurements, what we say can be applied to practically any natural science. Moreover, there is the intriguing possibility that similar models may be employed in the arts, humanities and social sciences. Although it has evolved from more restrictive models over a period of many years, the model that we shall present has been developed only recently. Quantum measurement theory is not a closed subject but is still intensively studied and discussed. One can usually reduce the study of measurements or experiments to a consideration of simple two-valued measurements. Such simple experiments are frequently called yes-no measurements. For example, suppose M is a real-valued measurement such as the determination of the energy of a particle and A is a subset of the real numbers R If M is executed under certain prescribed experimental conditions, we can ask whether M results in a value in A. If this experiment gives a value for M in A, 37 © 1999 Kluwer Academic Publishers. 38 STANLEY GUDDER then the answer to our question is yes, if not, the answer is no. In this way, M can be replaced by a set of simpler yes-no measurements, each of which corresponds to a subset of JR.. Following G. Ludwig, we shall call a yes-no measurement an effect [8]. An effect can be thought of as corresponding to a physical phenomenon that either occurs or does not occur. There are two main types of effects, those that are sharp and those that are unsharp (fuzzy). A sharp effect corresponds to a perfectly accurate measurement. These are impossible to attain in practice, although experimentalists attempt to improve their techniques and equipment to approach such ideal measurements as closely as possible. Sharp effects are theoretical idealizations that have a simpler mathematical structure than unsharp effects and orthodox quantum mechanics usually only considers such effects. However, since unsharp effects give more realistic models for experimental practice, they have recently drawn considerable attention. Moreover, their more general nature has important mathematical consequences [1, 2, 6, 7, 8]. 2. PARTICLE DETECTORS An important example of an effect is a measurement utilizing a particle detector such as a Geiger or photon counter. Let us denote such a detector by d. Now d will have a certain domain of sensitivity given by a subset D of three-dimension space JR.3. If a particle (sayan electron) enters the domain D, then the particle triggers a response in d. We shall then say that d clicks. For each run of an experiment involving such a particle, either d clicks or it does not, so d is an effect. What we have just described is a sharp detector, but in practice, no detector is perfectly accurate. For example, the particle may enter D and on rare occasions d may not click, or d may click on rare occasions when the particle does not enter D. Moreover, D will never have a perfectly sharp boundary and d may not be as sensitive to a particle near the edge of D. In order to describe the behavior of d, we need a method for measuring its sensitivity at points in three-dimensional space. One technique for measuring the accuracy of d is given by a confidence function k We use the standard notation fd: JR.3 --+ [0, 1] which means that fd is a function that takes points in JR.3 to a number between and 1. Assuming that d is at a fixed location, D will encompass a fixed set within JR.3. Then for any point x in ]R3, fd(X) gives the confidence, on a scale between and 1, of an experimenter that d will click when the particle is located at x. The function fd is an intrinsic property of the ° ° qu ANTUM MECHANICAL MEASUREMENTS 1"":'" • • " . " ·-,....---0"""""·,, " " 0: u 39 "0------41",,, v r Figure 1. particle detector d and can be determined by independent, preliminary calibration experiments. In general, different detectors will have different confidence functions. Presumably, fd(X) is close to 1 when x is in D and x is not very close to the boundary aD of D. This indicates that in this case the confidence is high that d will click. However, for x in D and close to aD, fd(x) may not be close to 1 and the confidence of a click is lower. Similarly, for x outside of D but close to aD, fd(x) may be positive so there is a nonzero confidence of a click. The characteristic function of a set D is the function ID: IR3 - 7 [0,1] that has the value 1 on D and the value 0 on the complement DC of D. Of course, DC is the set of all points in IR3 that are not in D. If d is sharp, there is complete confidence that d will click when the particle is in D and that d will not click otherwise. Hence, for a sharp detector d, fd will coincide with ID and the closer fd is to ID, the more accurate d will be. To illustrate this discussion in a simple way, let us pretend that the detector experiment is carried out on an interval [0, r] of the real line R Suppose the experiment consists of a particle producing apparatus that is located at the origin O. The apparatus injects a particle to the right at time 0 and we attempt to detect the particle's location at time 1. We imagine that a is a one-dimensional detector which is placed so that its sensitivity domain is the interval [u, v] ~ [0, r]. If a is sharp, then fa = I[u,vl as shown in Figure 1. However, in practice, a will be unsharp and fa takes a form as illustrated in Figure 2. In this case, there is high confidence that a will click if the particle is near the center of [u, v] but low confidence if it is near the boundary u or v. We think of [u, v] as the physical shape of detector a and this is the domain in which a would faithfully respond to the presence of a particle if it were perfectly accurate. N ow we can assume the existence of an imaginary detector denoted by 1 whose confidence function is f1 = I[O,rl. We do not have to actually construct this detector, we just record a click no matter what happens. Similarly, we can assume there is a detector 0 whose confidence function fo has the constant value o. Again, we need not construct this detector, 40 STANLEY GUDDER u v r Figure 2. 1...;----0 0: •u 0>__----- .' v +, r Figure 3. we just record a nonclick no matter what happens. Moreover, for any detector d, we can imagine another "detector" d' whose confidence function is fd = fl - fd, That is, for any x in [0, r], we have fd(x) = 1 - fd(X). Notice that we can also write this as fd + fd = kNow d' may not be an actual detector but its equivalent is provided by d itself. Once d is in place, we just register a click if d does not click and a nonclick if d clicks. We call d' the orthosupplement of d. The confidence functions for the orthosupplements of Figure 1 and Figure 2 are shown in Figures 3 and 4. As far as this discussion is concerned, an actual detector d is determined by its confidence function fd. This function describes the accuracy of d and its domain of sensitivity and thus where d is placed. Moreover, fd determines the orthosupplement d' whose confidence function is fd =!1-fd, We now discuss various ways that detectors can be combined to form new detectors or new effects. Since for our purposes, a detector is com- r Figure 4. 41 QUANTUM MECHANICAL MEASUREMENTS u v w x r Figure 5. pletely described by its confidence function, we can accomplish this by discussing ways in which confidence functions can be combined to form new confidence functions. In our one-dimensional situation a confidence function will be any function f: [0, r] --+ [0,1]. Let a and b be detectors with corresponding confidence functions fa, fb respectively. It is clear that fa + fb is still a confidence function if and only if fa + fb ::; iI (that is, fa(x) + fb(X) ::; 1 for all x E [0, rD. If fa + fb ::; iI, we say that a EEl b exists and consider c = a EEl b to be an effect with confidence function fe = fa + fb. In this case, we also write fe = fa EEl!b- Roughly speaking, we may think of c as the effect (two-valued measurement) that clicks if a or b (or both) click and does not click otherwise. Now c may not correspond to an actual detector that can be constructed in practice. For example, if the sensitivity domains for a and b overlap, then such a detector c might be difficult or impossible to construct. However, we can still think of c as an idealized combination of a and b that has the confidence function fe = fa EEl !bIt is possible that a EEl b may not exist. For example, there may be a point Xo E [0, r] at which fa(xo) + fb(XO) > 1. In this case, fa + fb would not be a confidence function. On the other hand, if fa and fb are not simultaneously positive, then a EEl b does exist and c = a EEl b corresponds to a constructible detector. We merely place a and b in their designated positions and consider c = a EEl b to be the single detector obtained by combining the two detectors a and b. For example, let a be the detector illustrated in Figure 2 and let b be a similar detector that is placed so that its sensitivity domain is the interval [w, x] (Figure 5). Then c = aEElb can be considered to be a detector with confidence function fe as illustrated in Figure 5. However, if a and b are right next to each other (that is, v = w), then we have the situation shown in Figure 6. This combination of a and b may not give a EEl b because a particle located at v may very well not result in a click in either a or b which contradicts the fact that fa(v) + fb(V) = 1. But it is quite possible that a detector c = aEElb can be constructed. It would have a configuration similar to that of a and b but with a larger sensitivity domain. In this case fe appears as in Figure 7. 42 STANLEY GUDDER 0 u v w x r x r Figure 6. ILL 0 u ~ .. I v w Figure 7. Notice that for any detector d, the "detector" d' always exists (if only in our imagination) and dEB d' = 1. Also, note that if 1 EB d exists, then d = O. Indeed, we would then have !I + fd ::; !I so that fd = fo· In certain situations, we can even combine overlapping detectors. As long as fa + fb ::; !I, the effect c = a EB b exists and corresponds to the confidence function fe = fa EB fh. Carrying this to the extreme, we might be able to combine detector a with itself. If fa(x) ::; 1/2 for all x E [0, rJ, then the effect 2a = a EB a exists with confidence function ha = fa EB fa = 2fa. If 2a exists, then a is called an isotropic detector. This method of combining detectors can be continued to three or more detectors. For example, if a, band c are detectors (or more generally, effects) whose confidence functions satisfy fa +fb+ fe ::; !I, then d = aEBbEB c exists and is an effect whose confidence function satisfies fd = fa + fb +fc. To be more precise, if a EB band (a EB b) EB c exist, then b EB c and a EB (b EB c) exist and a EB (b EB c) = (a EB b) EB c. This last equation is called the associative law and is interpreted as meaning that the two sides of the equation are effects with the same confidence function. We conclude that parentheses are not needed when we consider EB combinations of three or more effects. Another important property of EB is that if a EB b exists, then bEBa exists and bEBa = aEBb. We thus say that EB is commutative. We call EB a partial binary operation on the set of effects (or confidence function). This is because EB is only defined for certain pairs of effects, namely those that satisfy fa + fb ::; !I. This is in contrast to a total binary operation such as the ordinary addition + for numbers or functions which is defined for all pairs of elements. In the type of mathematical model QU ANTUM MECHANICAL MEAS UREMENTS 43 that we are considering here, E9 is taken as the basic partial operation and all properties of the system are derived from the properties of EEl. To make a systematic study of the properties of EEl in our one-dimensional detector example, let C be a collection of functions from [0, r] into [0,1]. We say that C is a confidence algebra if C satisfies the following conditions. (1) Io,iI E C. I' = h - IE C. (3) If I, 9 E C and I + 9 ~ iI, then I + 9 E C. (2) If IE C, If f then f, 9 E C and f + 9 ::; EEl 9 is defined or that iI, we define f EEl 9 = f + 9 and say that f EEl 9 exists. Thus, a confidence algebra is contains fo, iI, is closed under I and is closed a set of functions that under EEl whenever f EEl 9 is defined. Naturally, we call the elements of C confidence functions. Confidence algebras exist because the set of all functions from [0, rJ into [0,1 J is a confidence algebra. A simpler example is the confidence algebra consisting of only fo and k It is easy to check that the intersection of any collection of confidence algebras is again a confidence algebra. Now let D be a set of detectors on [0, rJ which we identify with their corresponding confidence functions. As we have noted previously, if fa,fb E D with fa + fb ~ iI, then fa EEl fb need not be in D so D may not be a confidence algebra. However, the intersection C(D) of all the confidence algebras containing D is a confidence algebra and C(D) is the smallest confidence algebra containing D. We call C(D) the confidence algebra generated by D. It can be shown that any element of C(D) can be obtained from elements of D in a finite number of steps by performing EEl combinations and orthosupplements '. Thus, the elements of D correspond to detectors and the other elements of C(D) correspond to effects that can be derived from basic combinations of detectors. When we speak of effects in the remainder of this section and in Section 3, we are referring to elements of C(D). We can employ the partial operation EEl to define a natural order for effects. For effects f and g, we write f ~ 9 if there exists an effect h such that fEEl h = g. In particular, if a and b are detectors, we write a ~ b if fa ~ fb. It is interesting that this order f ::; 9 is the same as the usual order for functions given by f(x) ::; g(x) for all x E [0, rJ. However, we shall not use this fact here because we are adhering to the philosophy that the properties of effects should be given in terms of the basic partial operation EEl. Moreover, there are more general examples of effects that cannot be represented as functions so the function order is not applicable there. 44 STANLEY GUDDER What are the properties of the order S:.? Since f EB fo = f, we have f S:. f for any effect f. This is called reflexivity of S:.. Suppose that f S:. 9 and 9 S:. f· Then there exist effects hand i such that f EEl h = 9 and gEEli = f. Hence, fEElhEEli = f and it follows that hEEli = fo. Since h and i have nonnegative values, we conclude that h = i = fo. Therefore, f = g. In particular, a S:. band b S:. a imply that a = b. This is called the antisymmetry property of S:.. Finally, suppose that f S:. hand h S:. g. Then there exist effects i, j such that fEEl i = hand hEEl j = g. We then have fEEl (i EEl j) = (J EEl i) EEl j = hEEl j = 9 so that f S:. g. In particular, if a S:. band b S:. c, then a S:. c. This property is called the transitivity of S:.. These three properties show that S:. is a partial order relation for the set of effects. The relation S:. has many of the properties of the usual order for numbers. However, the dichotomy property of numbers does not hold for effects. Dichotomy states that for any a and b, either a S:. b or b S:. a. A partial order relation that satisfies dichotomy is called a total order relation. The orthosupplementation operation ' is order inverting which means that f S:. 9 implies that g' S:. f'. To show this, suppose that f S:. g. Then there exists an effect h such that fEElh = g. Hence, 11- fEElh = l1-g and we have g'+h=fl-g+h=l1-f=j' Thus g' EEl h = f' and g' S:. f'· Also, since 11 - (11 - J) = f, we conclude that f" = f for any effect f. We then say that ' is an involution. Moreover, we can show that f EEl 9 exists if and only if f S:. g'. Indeed, if f S:. g', then there exists an effect h such that f EEl h = g'. But then f + 9 = 11 - h = h' Hence, f EEl 9 exists and f EEl 9 = h'. Conversely, suppose that h exists. Then h' = fl - f - 9 so that f + h' = It follows that fEEl h' = g' so that 11 - = f EEl 9 9 = g' f S:. g'. 3. OTHER WAYS OF COMBINING EFFECTS In the previous section we began with a set of detectors V and formed the confidence algebra C(V) generated by V. The elements of C(V) were interpreted as effects that can be derived from detectors in V using QU ANTUM MECHANICAL MEASUREMENTS 0: u w v x 45 r Figure 8. l~ 0: u 0 v w "------11-------11··· . x r Figure 9. the "plus" partial operation EB and the orthosupplement operation '. By employing this EB partial operation, we can also define a "minus" partial operation. For f, 9 E C(V), if f :S g, then there exists an effect h such that f EB h = 9 and we write h = 9 e f. In this way, we have fEB(geJ) = g. Of course, h(x) = g(x)- f(x) for all x E [0, r] so gef is the usual difference of functions. However, e is a partial operation because it is only defined when f :S g. Just as EB has some of the properties of the usual plus + for numbers, e has some of the properties of the usual minus - for numbers. We shall pursue these properties in the next section, but now we consider other ways of combining effects. For effects f, 9 E C(V), we define the function h = max(J, g) by h(x) = max(J(x),g(x)) for every x E [O,r]. In other words, h(x) is the larger of f(x) and g(x) for every x E [0, r]. (If f(x) = g(x), then either one will do.) Although h: [0, r] ~ [0,1]' h need not be an effect even when fEB 9 exists. If f :S g, then max(J, g) = 9 which, of course, is an effect. For the detectors in Figure 5, max(fa,fb) = fa EB fb is an effect. However, these are very special cases. For the detectors in Figure 6, max(fa,fb) has the form illustrated in Figure 8. In general, this would not correspond to a detector or even an effect. In a similar way, we define i = min(J, g) by i(x) = min(J(x) , g(x)) for every x E [0, r]. Again, min(J, g) need not be an effect even when fEB 9 exists. For the detectors in Figure 6, min(fa,fb) has the form illustrated in Figure 9. Since C(V) need not be closed under max and min, these give partial binary operation on C(V). For effects f, 9 E C(V), the least upper bound f V 9 of f and 9 is the smallest effect that is greater than or equal to f and 9 in terms 46 STANLEY GUDDER of the order :S. In general, f V 9 need not exist so again V is a partial binary operation on C(V). In symbols, if h = f V 9 exists, then f :S h, 9 :S h and if i E C(V) satisfies f :S i, 9 :S i, then h :S i. There is a relationship between f V 9 and max(f, g). In fact, it is easy to check that if max(f, g) E C(V), then max(f, g) = fV g. However, if fV 9 exists, then f V 9 need not equal max(f, g). In a similar way, we define the greatest lower bound f !\ 9 of f and 9 as the largest effect that is less than or equal to f and 9 relative to the order :S. Again, f !\ 9 need not exist and !\ is a partial binary operation on C(V). As before, if min(f, g) E C(V), then min(f, g) = f !\ 9 but if f !\ 9 exists, then f !\ 9 need not equal min(f, g). A simple argument shows that if f V 9 exists, then f' !\ g' exists and f' !\ g' = (f V g)'. Moreover, if f !\ 9 exists, then f' V g' exists and f' V g' = (f !\ g)'. These equations are called De Morgan's laws. De Morgan's laws also hold for max(f, g) and min(f,g). Recall that a detector d is sharp if fd is the characteristic function of a set or equivalently if fd has only the values 0 or 1. In a similar way, we say that an effect f is sharp if f has only the values 0 or 1. It is clear that f is sharp if and only if f = f2. (This means that f(x) = (f(x))2 for every x E [0, rJ.) However, for our later work, a more useful characterization of sharpness is given in the following theorem. Theorem. For an effect f, the following statements are equivalent. (i) f is sharp. (ii) min(f, f') = fa. (iii) max(f, f') = fI. Proof. Suppose that f is sharp. Then f(x) = 0 or 1 for all x E [0, rJ. Moreover, f(x) = 1 if and only if f'(x) = 1 - f(x) = o. It follows that min(f, f') = fa. Conversely, suppose that min(f, f') = fa. If f(x) =f. 0, then 1 - f(x) = f'(x) = 0 so that f(x) = 1. Hence, f(x) = 0 or 1 for all x E [0, rJ so f is sharp. This shows that (i) and (ii) are equivalent. The equivalence of (iii) follows from De Morgan's laws. 0 4. EFFECT ALGEBRAS In Section 2, we discussed a method for combining two detectors in terms of a partial binary operation EB. In order to obtain a mathematical model with well behaved properties, we were forced to extend the definition of EB to effects that were derived from combinations of detectors. We also noticed that EB possesses various properties that the usual + operation of numbers possess. Now a detector corresponds to a position measurement for a particle, but there are many other measurements that an experimentalist may want to perform. For example, there could be measurements of energy, momentum, angular momentum, spin, charge, etc. QUANTUM MECHANICAL MEASUREMENTS 47 Moreover, the experimentalist may want to perform more complicated measurements involving the scattering or decay of many particles. For this reason, investigators have studied general sets of measurements and ways of combining them. We now present a mathematical framework or model for such studies [3, 4, 5]. An effect algebra is a system (P, 0,1, EB), where P is a set of elements called effects, 0 and 1 are distinct elements of P, and EB is a partial binary operation on P that has the following properties. (1) If a EB b is defined, then b EB a is defined and b EB a = a EB b. (Commutativity) (2) If a EB b and (a EB b) EB c are defined, then b EB c and a EB (b EB c) are defined and a EB (b EB c) = (a EB b) EB c. (Associativity) (3) For every a in P there exists a unique a' in P such that a EB a' is defined and a EB a' = 1. (Orthosupplementation) (4) If 1 EB a is defined, then a = O. (0 - 1 Law) For brevity, we frequently denote an effect algebra by the single letter P. We have seen in Section 2 that a confidence algebra is an effect algebra. For a and b in P, we write a ~ b if there exists a c in P such that b = a EB c. If a ~ b', we write a 1- b and say that a and b are orthogonal. We denote the least upper bound and greatest lower bound of a and b, if they exist, by a V b and a 1\ b, respectively. The next theorem summarizes some of the basic properties of effect algebras and its proof may be found in [4]. Theorem. Let P be an effect algebra. (i) ~ is a partial order relation on P such that 0 ~ a ~ 1 for all a in P. (ii) 0' = 1, l' = 0 and aEBO = a for all a in P. (iii) a EB b is defined if and only if a 1- b. (iv) a" = a for all a in P and a ~ b implies that b' ~ a'. (v) If a V b exists, then a' 1\ b' exists and a' 1\ b' = (a Vb)'. If a 1\ b exists, then a' V b' exists and a' V b' = (a 1\ b)'. When we discussed detectors, it was clear how sharp detectors should be defined. However, in the context of a general effect algebra P, such a definition is not at all obvious. We cannot say that a E P is sharp if a is a characteristic function because the elements of P may not be real-valued functions. For the same reason, the partial operations max and min do not make sense for a general effect algebra so the characterization given in the theorem in Section 3 cannot be used to define sharpness. In fact, there is a considerable controversy among researchers concerning the best definition of sharpness in an effect algebra. We shall consider one of the several definitions that have been proposed. One way of motivating this definition is to consider the theorem in Section 3 and 48 STANLEY GUDDER replace max by V and min by /\. This is legitimate because V and /\ make sense in an arbitrary effect algebra. To differentiate this definition from our previous definition of sharpness, we shall use the term crisp instead of sharp. Unfortunately, crispness and sharpness are not equivalent for confidence algebras. Indeed, our discussion in Section 3 shows that if an element f of a confidence algebra is sharp, then it is crisp, but f may be crisp without being sharp. In the most important mathematical model for quantum mechanics, namely the Hilbert space model mentioned in Section 5, crispness is strong enough to describe perfectly accurate measurements. An element a in P is crisp if a V a' exists and equals 1. This is equivalent to a /\ a' exists and equals o. Does an effect algebra have a stronger structure if all of its elements are crisp? To answer this question we introduce a definition and prove a theorem. An orthoalgebra is a system (P, 0,1, EEl) that satisfies conditions (1), (2), (3) and (4') if aEEla is defined, then a = O. Condition (4') states that 0 is the only isotropic element of P. Theorem (i) An orthoalgebra is an effect algebra. (ii) An effect algebra is an orthoalgebra if and only if all of its elements are crisp. Proof (i) Suppose P is an orthoalgebra. To show that property (4) holds, assume that 1 EEl a is defined. Since a' EEl a = 1, (a' EEl a) EEl a is defined. By property (2) a EEl a is defined so by (4'), a = O. (ii) Suppose P is an orthoalgebra and a is in P. Assume that b 2: a, a'. Then b' ::; a' ::; b = b" Hence, b' -1 b' so b' EEl b' is defined. Therefore, b' = 0 and b = 1. Thus, a V a' = 1 and a is crisp. Conversely, suppose P is an effect algebra in which every element is crisp. To show that P is an orthoalgebra, suppose that a EEl a is defined. Then a ::; a' so 1 = a V a' = a'. Hence, a = 0 so P is an orthoalgebra. 0 We can also define a "minus" operation on an effect algebra. If a ::; b, then there exists a c such that a EEl c = b. We then write c = be a. It is not hard to show that e satisfies the following three conditions for all a,b,c in P. (5) bea::; b (6)be(bea)=a (7) a ::; b ::; c implies c e b ::; c e a and (c e a) e (c e b) = be a A partially ordered set P with a greatest element 1 and a partial binary operation e such that a e b is defined if and only if a ::; b and that satisfies (5), (6) and (7) is called a difference poset [3]. It can be shown QU ANTUM MECHANICAL MEASUREMENTS 49 that an effect algebra and a difference poset are equivalent mathematical structures. In this way, we can use either EB or e to define a mathematical model for measurements. 5. EXAMPLES An effect algebra P is said to be a lattice if a V b and a 1\ b exist for all a, bin P. We call P distributive if P is a lattice and the distributive law al\(bVc) = (al\b)V(al\c) holds for all a, band c in P. We first present some examples of distributive effect algebras. Example 1. Let P be the unit interval [0,1] in R. For a, b in [0,1]' aEBb is defined if a + b :::; 1 and in this case a EB b = a + b. Then ([0,1]' 0,1, EB) is a distributive effect algebra with a' = 1 - a. In this example, :::; is the usual order of numbers so :::; is a total order relation. Notice that a is crisp if and only if a = or 1. Hence, the crisp elements of [0,1] form a suborthoalgebra of [0,1]. That is, if a and b are crisp and a EB b exists, then a EB b is crisp. 0 ° Example 2. Let X be a nonempty set and let P be the collection of all subsets of X. Let be the empty set, 1 = X, and for a, b in P, a EB b is defined if their intersection a n b = and in this case a EB b is their union aU b. Then (P, 0,1, EB) is a distributive effect algebra with a' equal to the complement a C • In this example, :::; is the same as set-theoretic inclusion <;;:;. Every element a in P is crisp because a V a' = a U a' = 1. Hence, P is an orthoalgebra (in fact, P is a Boolean algebra). 0 ° ° Example 3. Let X be a nonempty set and let P be the set of all functions f: X ---t [0,1] <;;:; R. Let o(x) = 0, l(x) = 1 be the constant and 1 functions, respectively, and for f, 9 in P, f EB 9 is defined if f(x) + g(x) :::; 1 for all x in X and in this case fEB 9 = f + g. Then (P, 0,1, EB) is a distributive effect algebra with l' = I-f. In this example, :::; is the usual order f :::; 9 if and only if f(x) :::; g(x) for all x in X. It follows that f V 9 = max(f, g) and f 1\ 9 = min(f, g). Hence, f is crisp if and only if f is the characteristic function of a subset of X. Thus, the crisp elements form a suborthoalgebra (even a sub-Boolean algebra) of P. The crisp elements may be identified with the collection of all subsets of X which reduces to Example 2. The structure of P corresponds to a fuzzy set theory or fuzzy logic and is closely related to our discussion of detectors in Sections 2 and 3. Moreover, P may be thought of as a ° STANLEY G UDDER 50 confidence algebra on the set X and in this case crispness and sharpness are equivalent. 0 In these last three examples, the crisp elements of an effect algebra formed a suborthoalgebra. This also applies to the Hilbert space effect algebras that we shall mention later. Is this always true? That is, in an effect algebra, if a and b are crisp and a EEl b is defined, is a EEl b necessarily crisp? The answer is no and the following counterexample is due to D. Foulis and R. Greechie. This is also an example of a nondistributive effect algebra. Example 4. Let P be the effect algebra with elements 0,1, a, b, c, a', b', c' and the following EEl table. In this table we do not include the trivial elements 0, 1 and dash indicates that the corresponding EEl is not defined. EEl a a' b b' a 1 b b' c' - 1 - - - c' - - - b' 1 a' - - a - C b' c' - - - c b c' - - - at - - - - - - 1 1 1 - Then a and c are crisp and a EEl c = b'. However, b' is not crisp because b' V b = b' =1= 1. Note that P is not a lattice and hence is not distributive. This is because a V band c V b do not exist. For example, c' and b' are minimal upper bounds for a and b. 0 The most important example of a nondistributive effect algebra is called a Hilbert space effect algebra. Hilbert space effect algebras are frequently employed as mathematical models for quantum mechanics. However, their definition involves some rather sophisticated mathematics and we shall not pursue them here. The interested reader can consult any of the following references. ACKNOWLEDGMENTS The author would like to thank J. Pykacz for a very careful reading of the preliminary version of this paper and for some useful suggestions for improving it. These suggestions were incorporated in the final version. QUANTUM MECHANICAL MEASUREMENTS 51 AFFILIATION Stanley Gudder Department of Mathematics and Computer Science University of Denver U.S.A. [email protected] REFERENCES [1] Busch, P., Lahti, P. and Mittelstaedt, P., The Quantum Theory of measurements, Springer Verlag, Berlin, 1991. [2] Davies, E.B., Quantum Theory of Open Systems, Academic Press, London, 1976. [3J Dvurecenskij, A. and Pulmannova, S., "Difference posets, effects and quantum measurements", Intern. J. Theor. Phys., 33, 1994, pp. 819-850. [4] Foulis, D. and Bennett, M.K., "Effect algebras and unsharp quantum logics", Found. Phys., 24, 1994, pp. 1331-1352. [5] Greechie, R. and D. Foulis, D. "The transition to effect algebras" , Intern. 1. Theor. Phys., 34, 1995, pp. 1-14. [6] Holevo, A.S., Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, Amsterdam, 1982. [7] Kraus, K., States, Effects, and Operations, Springer-Verlag, Berlin, 1983. [8] Ludwig, G., Foundations of quantum mechanics Vol. I and II, Springer-Verlag, Berlin, 1983 and 1985. G. CATTANEO AND F. LAUDISA FROM LOGIC TO PHYSICS: THE LOGICO-ALGEBRAIC FOUNDATIONS OF QUANTUM THEORY 1. INTRODUCTION The relevance of the axiomatic method to the science and philosophy of our century need not be emphasized. The introduction and the development ofaxiomatization did not prove immensely useful only in the field of foundations of mathematics at the turn of the century, but also in the general analysis of the foundations of sciences. With particular reference to physics, axiomatic formulations of physical theories provide a framework in which all assumptions--especially the physical ones-are explicitly stated, a framework which enables one to clearly assess different approaches to particular problems of the theories and-by a specification of a mathematical model-to avoid mathematical inconsistencies: most important by a philosophical viewpoint, the common "universe of discourse" provided by the axiomatic framework enables one to analyse the conceptual foundations of a physical theory. This task appears to be especially important in the case of quantum mechanics, a physical theory full of interpretational difficulties in spite of its spectacular success by the experimental viewpoint. In this paper we will survey the quantum-logical approach to quantum mechanics, both in its historical and conceptual motivations and in some of its more recent formal developments and generalizations. In Section 2, the historical origins of the subject are reviewed, whereas in Section 3 the idea of sharp quantum logic and its formal structure are introduced. Finally, in Section 4, we will outline the main elements of the generalization of sharp quantum logic into the unsharp or fuzzy quantum logic, based on the possibility of defining degrees of membership of a value to a set (typically, a number interval). 2. THE ORIGIN S 0 F QUANTUM LOGIC: A SHORT HISTORY The investigations on the logical nature of quantum-mechanical statements historically arise from the attempt to apply the three-valued logic, 53 © 1999 Kluwer Academic Publishers. 54 G. CATTANEO AND F. LAUDISA introduced by Lukasiewicz [23] in 1920, to the domain of quantum phenomena. The suggestion of employing this new logic in the analysis of the formal structure of quantum theory derived essentially from the assumption that a logic satisfying the general principle of bivalence could not ground a physical theory such as quantum mechanics, that embodies Heisenberg uncertainty relations and therefore indeterminism. Although the first step was taken in 1931 by the Polish logician Zawirski, (cf. [16, p. 344]) the most systematic attempt to construct a suitable three-valued logic of quantum mechanics was to be developed in Reichenbach's book Philosophic Foundations of quantum mechanics [33]1. Relevant it might have been by a historical viewpoint, however, the rejection of a principle of bivalence was not the conceptual source of a possible logic of quantum theory. The successful development of a quantum logic lies in the rejection of the distributivity law, a condition that qualifies the algebraic structure of classical logic. This step, that gives rise officially to the logical (or logico-algebraic) approach to quantum mechanics can be said to develop from the same source as the standard mathematical formulation of quantum mechanics, namely from von Neumann's Die Mathematische Grundlagen der Quantenmechanik [25]. In this book, von Neumann proposes to take into account, in addition to the class of physical quantities relevant to a given physical system S, mathematically represented by self-adjoint operators on the Hilbert space H associated with S, the class of the properties of the states of the system: To each property t: we can assign a quantity which we define as follows: each measurement which distinguished between the presence or absence of t: is considered as a measurement of this quantity, such that its value is 1 if t: is verified, and zero in the opposite case [25, p. 249]. Since each property of this kind is shown to be represented by a projection operator on H, the relation between the properties of a physical system on the one hand, and the projections on the other, makes possible a sort of logical calculus with these. However, in contrast to the concepts of ordinary logic, this system is extended by the concept of "simultaneous decidability" which is characteristic for quantum mechanics. 2 Again the Heisenberg uncertainty relations force upon the theory a limitation, but this time the lack of overall "simultaneous decidability" will 1 As Max Jammer describes [16, pp. 364-370]' this book was subject to heavy criticisms, notably by Carl Hempel and Ernest Nagel. A defense of Reichenbach in [29] was likewise criticized by Feyerabend [8]. 2 von Neumann [25, p. 253], our emphasis. FROM LOGIC TO PHYSICS 55 yield not the rejection of bivalence but that of distributivity. Opening their seminal paper "The logic of quantum mechanics" [3]' Birkhoff and von Neumann explicitly state: One of the aspects of quantum theory which has attracted the most general attention is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S (Heisenberg's Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S simultaneously (Principle of Non-commutativity of Observations). The object of the present paper is to discover what logical structure one may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic. Our main conclusion, based on admittedly heuristic arguments, is that one can reasonably expect to find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces with respect to set products, linear sums, and orthogonal complements--and resembles the usual calculus of propositions with respect to and, or and not". Let us briefly outline then the essential lines of the Birkhoff-von Neumann approach. First of all we must stress that in the above quoted paper the authors take into account both the classical situation and the quantum one. They realize that these two cases are characterized by a propositional language based on the set 0 of all possible observations one can measure on a physical system. The concept of a physically observable "physical system" is present in all branches of physics, and we shall assume it. It is clear that an observation of a physical system 6 can be described generally as a writing down of the readings from various compatible measurements. Thus if the measurements are denoted by the symbols a1, ... , an, then an observation of 6 amounts to specifying numbers Xl, ... ,Xn corresponding to the different ak. It follows that the most general form of prediction concerning 6 is that the point (Xl' ... ' xn) determined by actually measuring (a1, ... , an) will lie in a subset t:. of (Xl' ... ' xn)-space. Hence, if we call the (Xl' ... ' xn)-spaces associated with 6, its "observationspaces, " we may call the subsets of the observation-spaces associated with any physical system 6, the "experimental propositions" concerning 6. In a footnote, Birkhoff and von Neumann point out that: One may regard a set of compatible measurements [(a1, ... , an)) as a single composite measurement [(which we denote by A))-and also admit non-numerical readings-without interfering with subsequent arguments. 3 [3, p. 1] of the reprinted version in [15] 56 G. CATT ANEO AND F. LA UDISA Among conspicuous observables in quantum theory are position, momentum, energy, and (non-numerical) symmetry. Therefore, in Birkhoff-von Neumann point of view the most general statement, or experimental proposition, which one can formulate about a physical system is that the value ofa physical magnitude (shortlyobservable) A which can be observed on it lies in a subset ~ of the set of possible values (shortly observation space) associated with A. A statement of this kind is represented by a pair (A,~) and corresponds to the elementary statement: val(A) E ~ = "the value of the observable (physical magnitude) A lies in the subset ~ of the observation space of A." This agrees with the following quotation of Varadarajan [34J: "The center of the stage of the present discussion is occupied by a physical system and the experimental propositions that are associated with it. ( ... ) if A is an observable, to each Borel set ~ of the real line lR is associated the proposition the value of A lies in ~." In the literature the "experimental proposition" of Birkhoff and von Neumann [3] and Varadarajan [34] is also called the "physical statement", "question", "theoretical sentence" , "elementary statement". experimental propositions (A, ~l)' (B, ~2)" .. which can be stated for a physical system are for instance "the particle has passed through the slit 1 of the screen", "the spin of the particle along the z direction is up", "the position of the particle is between 100Km and 250Km from a reference point of its trajectory", "the energy of the photon is between lOMe V and lOOMe V" and so on. Making use of the connectives "and" (&), "or" (Q), "not" ("""), it is possible to construct on the basis of elementary sentences r = (A, ~l)' s = (B, ~2), ... the complex sentences "r & s" , "rQs" , ".....,r" of the sentential language of the involved physical theory. For instance the sentence "the particle has passed through the slit 1 or through the slit 2", is a complex sentence of the form "( Q, 1) Q (Q, 2)"; or the sentence "the particle has position in the interval [Xl, X2] and velocity in the interval [VI, V2J" , is a complex sentence of the form "(Q, [XI,X2]) & (P, [VI.V2])". As another example involving a system consisting of two separated particle (1 + 2), the statement "the spin of particle 1 along the z direction is "up" in region ~1 and the spin of particle 2 along the x direction is "down" in region ~2", is described by a complex sentence of the form "(8}1), ~r) & (812 ), ~)". In the case of a classical physical system (J"c with n degrees of freedom it is possible to associate a set 8 (called the phase space of (J"c): this set is representable as the Cartesian product M x lRn (~ lR 2n ), where M (~ lRn) is the set of all the possible configurations of the system and lRn is the set of momentum vectors. The points in 8 are in one-to-one cor- FROM LOGIC TO PHYSICS 57 respondence with the mechanical states of (jc, since this is characterized in each state by a particular configuration and a particular momentum vector. Let us now denote by £(S) a (j-algebra of subsets of S, whose elements are interpreted as classical events. The set-theoretical operations of intersection, union, and complementation on the Boolean (j-algebra £(S) resembles the usual calculus of proposition with respect to and, Of, and not. The physical quantities A are mathematically represented, first of all, by real valued functions fA on S associating with any classical state XES the real number fA (x) E JR.; this real number represents the value possessed by A in the classical state x. For instance, the energy of a classical system is a function associating with any state of the system a real number: the energy of the particle in this state. But, in order to be a representation of an observable A, for any Borel set f::. of the real line the collection f"Al(f::.) of all states xES in which the observable A takes values fA (x) in ~ must be a collection of classical events: formally for any ~, the collection of classical states fA I (~) = {x E S : fA (x) E ~} must be an event from £ (S); for example, the set of all classical states in which a particle possesses an energy between 10 and 25 Joule must be an event from £(S); similarly, both the set of all classical states in which it possesses a position in the interval [Xl, X2] and in which it possesses a velocity in the interval [VI, V2] must be events. In this way the map fAI associating with any subset of possible values ~ E 8(JR) the collection E £(S) is an event-valued measure, realizing the obof states fAI(~) servable A. Since the physical quantity A is represented by a real-valued function fA on Sand fAI is an event-valued measure, it is possible to set up a correspondence between certain physically relevant statements about (jc, e.g., (A, ~), and certain subsets of the phase space S, e.g., (classical events) fAI(~). In this way the propositional language ofpossible statements about (j c (closed with respect the connectives "and", "or", "not") is realized by the logic of (j c, namely, the set of possible properties (events) of the system (closed with respect to set theoretic operations of "intersection", "union", "complement"). When a quantum system (jq is taken into account, the "quantum phase space" is a complex Hilbert space H, whose non-zero vectors represent states of (jq. The relevant physical quantities are represented by selfadjoint operators on H, whereas the quantum events are represented by orthogonal projectors on H. The role of the algebra of subsets of the phase space-as structure of the possible properties of a quantum system-is played by a more general structure than a Boolean algebra, i.e., an orthomodular complemented lattice. The key point is that this structure is more general exactly in that it fails to satisfy the distribu- 58 G. CATTANEO AND F. LAUDISA tivity law. 4 The Birkhoff and von Neumann paper was reviewed in 1937 by Alonzo Church, who described the authors' position as founded on the assumption that "the advance of an experimental science may some day require revision of the system of logic on which the mathematical theory of the science is based". However, such paper remained virtually ignored for twenty-five years, until the physicists of the "Geneva group" (J.M. Jauch, G. Emch, C. Piron and others) and, on the other side of the ocean, G.W. Mackey re-discovered it, giving rise to the area of research properly known today as quantum logic. 5 The possible philosophical import of the quantum-logical ideas was emphasized in some Finkelstein's papers of the sixties and early seventies, according to which the inquiry on the laws governing the physical world can be pursued at three levels of increasing abstraction and generality: the mechanical, the geometrical and finally the logical level. Physics has many layers, and one of the deeper layers, that of world geometry, has undergone a profound upheaval in our times. As a result of this revolution, certain geometric ideas of Riemann have been generally accepted, and it is entirely in order for a physicist to contribute his present conception of the world geometry. [... ] We are presently in the midst of an analogous development of the logic of physical systems, a stratum that conceptually underlies even that of physical geometry. I think that besides mathematical logic there is now also a physical or world logic, different in principle and describing at a very deep and general level the way inanimate physical systems interact. To the extent that there are physical systems, let them be men or machines, that behave the way symbolic systems are supposed to behave, much of mathematical logic recurs as a special or limiting case of physical logic, but more general physical systems may in principle and do in fact obey more general laws. [... ] It is therefore appropriate to indicate how it is at all possible. let alone necessary, to discover laws of physical logic from experience [10, pp. 48-49] (See also [9]). The analogy with non-Euclidean geometry actually was the starting point of the widely discussed Putnam's paper "Is Logic Empirical?" [30] where, with reference to the quantum-logical approach to quantum mechanics, he argued that some "necessary truths" of logic might turn out to be false for empirical reasons: empirical data or results might force us to deny some logical laws and reinforce the idea that logic too is an empirical science. 6 4 5 [3], pp. 9-10 of the reprinted version in [15]. Twenty-five years of silence explain also why Popper [28] could argue as late as 1968 for no less than the logical inconsistency of the Birkhoff and von Neumann paper. For a recent, thorough analysis of Popper's misinterpretation of Birkhoff and von Neumann paper, cf. [6]. 6 For a review of the debate engendered by this Putnam's claim, see [32]. FROM LOGIC TO PHYSICS 59 3. THE FORMAL STRUCTURE OF SHARP QUANTUM LOGIC After more than two decades of silence about the quantum-logical approach to the foundations of quantum mechanics, G.W. Mackey gave a twist to the field in the early sixties. In his Mathematical Foundations of quantum mechanics [24], Mackey assumed the notions of state and observable for a given physical system (J' as primitive, and defined a probability measure expressing the probability distribution of a given observable in a given state: if 0 is the set of the observables of (J', S the set of the states of (J' and B(lR) is the set of all Borel subsets of the real line, a (probability) function P : S x 0 x B(lR) f---t [0,1] is defined such that the quantity between 0 and 1, for 0: E S and A E 0 fixed, P(o:, A, ~), is the probability that a measurement of the observable A in the state 0: yield a value lying in ~, for any ~ E B(lR). The two sets of states and observables and the function P had to satisfy six preliminary axioms according to which: - for 0: E S and A E 0 fixed, P is a probability measure on the Borel subsets of the real line; - a pair of observables, in order to be distinguishable, have to have a different probability distribution in at least one state (principle of indistinguishability for observables); - a pair of states, in order to be distinguishable, have to ascribe different probability distributions to at least one observable (principle of indistinguishability for states); - any (Borel) function of an observable is still an observable (corresponding to the same measuring apparatus with a change of its reading scale according to the Borel function); - the set of states is convex, namely any convex combination of different states is still a state (corresponding to the possibility of a statistical mixing of states); - for any sequence of orthogonal questions there always exists their "sum". Let us remark that if A is an observable with set of possible real values KA ad f : lR f---t IR. is any Borel real function, then f(A) is the observable whose set of possible real values is {J(k) : k E KA}, as a result of the change of the reading scale of the instrument which measure both A and f(A). For example, we can observe the "deflection angle" (in radians) of a beam of electrons under the influence of a uniform magnetic field; with a change of the reading scale of the instrument we can read the same experimental observation as "charge over mass" (in coulombs/gram). 60 G. CATTANEO AND F. LAUDISA The last axiom of the list refers to the notion of a question, precisely a particular kind of observable introduced by Mackey, whose set of values includes only 0 and 1. The set [; of these particular observables was singled out from the whole set of the observables of (]" via the application of characteristic functions Xl> : JR f-+ [0, 1] of Borel subsets .6. of JR, whose only values can be 1 (when the point x from JR belongs to.6.) and 0 (when the point x does not belong to .6.). To be precise, if A is any observable with set of possible values KA and Xl> is the characteristic function of a Borel set .6., then the question Xl>(A) is an observable whose set of possible values {Xdk) : k E KA} consists only of 0 or 1. From the above axioms it turns out that for any state a E S, P(a, A,.6.) = P(a, Xl>(A), {I}), namely, for any ~ E B(lR) the probability that a measurement of the observable A in the state a yields a value lying in ~ is just the probability that a measurement of the question Xl>(A) in the state a yield the value 1. The set of the questions obtained in this way can be ordered with respect to a suitable order relation based on the probability measures on the questions (intuitively, a question el is "less or equal" to a question e2 if and only if the probability that a measurement of el in any state a yield 1 is less or equal to the probability that a measurement of e2 in the same state a yield 1; formally, for any state a, P(a, el, {I}) :::; P(a, e2, {I} )). The resulting structure is that of a partially ordered set, which can be endowed with several important mathematical properties provided one makes some further assumptions, though very natural by a mathematical viewpoint. 4. DOES MEMBERSHIP TO A SET ADMIT DEGREES? THE UNSHARP GENERALIZATION OF QUANTUM LOGIC The physical idea underlying the notion of a question is simple. An experimental apparatus can be associated with the pair composed by the physical quantity A and the numerical interval .6. such that the reading scale of A is covered leaving open only the particular portion of the scale corresponding to .6.. Given that the physical system is prepared in a state a, in the event of a measurement of A in the state a, the value of Xl> (A) will be 1 if the pointer is found in the portion of the scale left open by the cover, and 0 otherwise [2, Ch. 13]. With a terminology that will be useful later, a macroscopic device of sharp localization can be said to correspond to the characteristic function introduced above. This 61 FROM LOGIC TO PHYSICS device can be represented as a test of the membership of a value to the fixed numerical interval ~. It is now worth pointing out a simple mathematical fact. When we have defined the characteristic function, we have stated that the only possible values were 0 and 1, whereas the proper range was the interval [0,1]. The generalization toward an unsharp (or fuzzy) generalized formulation of quantum logic can be summarized in the idea of "taking seriously" all the numbers in the interval as possible values: the notion of membership would then admit degrees, associated with all the numbers between 0 and 1. The unsharp quantum logic originates then with the application of some suggestions coming from fuzzy logic and fuzzy set theory to the logico-algebraic foundations of quantum mechanics. Fuzzy logic and fuzzy set theory was originally developed in the sixties by Lotfi Zadeh, in order to deal with approximate reasoning by a logical and cognitive viewpoint: according to him "[ ... ] the notion of a fuzzy [device] is a convenient point of departure for the construction of a conceptual framework [... ] which provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership" [36]. As emphasized in a later paper, Traditionally logical systems have aimed at the construction of exact models of exact reasoning-models in which there is no place for imprecision, vagueness or ambiguity. In a sharp break with this deeply entrenched tradition, the model of reasoning embodied in fuzzy logic aims, instead, at an accommodation with the pervasive imprecision of human thinking and cognition. [... ] To provide an appropriate conceptual framework for approximate reasoning, fuzzy logic is based on the premise that human perceptions involve, for the most part, fuzzy sets, that is, classes of objects in which the transition from membership to non-membership is gradual rather than abrupt." [1, p. 105] Let us return then to the set of questions and let us briefly investigate the consequences of its "fuzzification". To this aim, we generalize the notion of macroscopic device of sharp localization into the notion of concrete macroscopic device of localization, i.e., when a ~ E 8(JR) is fixed, any mapping W,6. : JR 1---+ [0,1] whose range consists not only of the two numbers and 1, but of the set of all real numbers in the real unit interval [0,1]. For any point x the real number w,6.(x), between and 1, represents the degree of membership of x in ~ The application of the sharp localization mappings X,6. gives rise to the (partially ordered) set of questions, so that the question naturally arises of what is the set of objects generated by the application of the unsharp localization mappings W,6.. Such objects turn out to be the so-called effects [19] [20] [21] [22], that may be intuitively seen as the mathematical ° ° 62 G. CATTANEO AND F. LAUDISA representatives of approximate properties instantiated by the physical system in the framework of a concretely realizable experiment. In analogy with the Zadeh's quotations concerning logical systems, the spectrum of the properties of physical systems that we can investigate can be much wider if we allow approximation into our commonly idealized descriptions of physical systems. As far as quantum mechanics itself is concerned, the unsharp or fuzzy approach (sometimes called also operationa0 was originally developed in different areas of fundamental quantum physics such as, respectively, the quantum field theory [14J and the quantum theory of open systems [7J. In a more abstract and axiomatic vein, however, an unsharp generalization of the main elements of the standard Hilbert space formulation of orthodox quantum mechanics has been elaborated in more recent times, usually in the context of a realistic and individual interpretation of quantum theory (see, e.g., [4]). If we agree to consider the notions of state and observable as the fundamental ones, the unsharp approach to Hilbert space quantum mechanics has generalized the notion of observable as represented by a PV (projection-valued) measure into the notion of POV (positive operator-valued) measure, since the already available generalization of notion of the state vector into the notion of density matrix has been shown to be the most general mathematical representation of a quantum state. As recalled above, a sharp property of a quantum system is represented in a Hilbert space framework by a projection operator, i.e., an idempotent self-adjoint bounded operator whose spectrum includes only the values 0 and 1. The set of these operators has a lattice structure and this lattice can be considered as the totality of the sharp properties of the system. However, if approximation has to be taken into account, the notion of sharp property of a quantum system has to be relaxed, in order to envisage a class of generalized events that instantiate properties testable on the system in an unsharp and approximate way. Each element of this class of approximate properties is mathematically represented by an effect operator, namely a (non-idempotent) self-adjoint bounded operator whose spectrum includes all values between 0 and 1. The effects represent the events of particular measurement outcomes, and the expectation values of the effects represent the probabilities of these events. Unlike the case with sharp properties, one is no more entitled to state that, after a certain measurement on a quantum system, a property either holds or it does not hold, but rather that such property holds approximately, with the degree of approximation given by the probability of occurrence of the effect that is tested in the experiment. FROM LOGIC TO PHYSICS 63 AFFILIATION Gianpiero Cattaneo Dipartimento di Scienze dell' InJormazione Universita di Milano Italy Frederico Laudisa Dipartimento di Filosofia Universita di Firenze Italy REFERENCES [1J Bellman, R.E., Zadeh, L.A., "Local and fuzzy logics", in: Dunn, M., Epstein, G. (eds.), Modern Uses oj Multiple- Valued Logic, Dordrecht, Reidel, 1977. [2] Beltrametti, E., Cassinelli, G., The Logic oj quantum mechanics, Cambridge, Mass., Addison-Wesley, 1981. [3] Birkhoff, G., von Neumann, J., "The logic of quantum mechanics", Annals oj Mathematics, 37, 1936, pp. 823-843 (reprinted in [15])). [4] Busch, P., Grabowski, M., Lahti, P., Operational quantum mechanics, Berlin, Springer, 1996. [5] Cattaneo, G., Laudisa, F., "Axiomatic unsharp quantum theory", Foundations oj Physics, 24, 1994, pp. 631-681. [6] Dalla Chiara, M.L., Giuntini, R., "Popper and the logic of quantum mechanics", 1995, preprint. [7] Davies, E.B., Quantum Theory oj Open Systems, London, Academic Press, 1976. [8] Feyerabend, P.K., "Reichenbach's interpretation of quantum mechanics", Philosophical Studies, 9, 1958, pp. 49-59. [9] Finkelstein, D., "Matter, space and logic", Boston Studies Philosophy oj Science, vol. 5, 1969. [10] Finkelstein, D., "The physics of logic", in: Colodny, R.G. (ed.), Paradigms and paradoxes, University of Pittsburgh Press, 1972. [11] van Fraassen, B.C., "Semantical analysis of quantum logic", in: Hooker, C.A. (ed.), Contemporary Research in the Foundations oj Quantum Theory, Reidel, Dordrecht, 1973. [12] van Fraassen, B.C., "The labyrinth of quantum logics", in: Cohen, R. S. and Wartofsky, M.W. (eds.), Boston Studies in the Philosophy oj Science, Vol. XIII, Reidel, Dordrecht, 1974. 64 G. CATTANEO AND F. LAUD IS A [13] Gudder, S.P., Quantum Probability, Academic Press, NY, 1988. [14] Haag, R., Kastler, D., "An algebraic approach to quantum field theory", Journal of Mathematical Physics, 5, 1964, pp. 846-86l. [15] Hooker, C.A. (ed.), The Logico Algebraic Approach to quantum mechanics, vol. I, Dordrecht, Reidel, 1975. [16] Jammer, M., The Philosophy of quantum mechanics, New York, Wiley and Sons, 1974. [17] Jauch, J .M., Foundations of quantum mechanics, Addison-Wesley, Reading, Mass., 1968. [18] Jauch, J.M., and Piron, C., "On the structure of quantal proposition systems", Helvetica Physica Acta, 42, 1969, p. 842. [19] Ludwig, G., "The measuring process and an axiomatic foundation of quantum mechanics", in: d'Espagnat, B. (ed.), Foundations of quantum mechanics, New York, Academic Press, 1971. [20] Ludwig, G., "Measuring and preparing processes", in: Hartkamper, A., Neumann, H. (eds.), Foundations of quantum mechanics and Ordered Linear Spaces, Berlin, Springer, 1974. [21] Ludwig, G., "A theoretical description of single microsystem", in: Price, W.C., Chissick, S.S. (eds.), The Uncertainty Principle and Foundations of quantum mechanics, New York, Wiley and Sons, 1977. [22] Ludwig, G., Foundations of quantum mechanics, New York, Springer, 1983. [23] Lukasiewicz, J., "0 logice trojwartosciowej" (On three-valued logic), Ruch FilozoJiczny, 5, 1920, pp. 169-171 (reprinted in: Selected Works, Amsterdam, North-Holland, 1970, pp. 87-88). [24] Mackey, G.W., The Mathematical Foundations of quantum mechanics, New York, Benjamin, 1963. [25] Neumann, von J., Die Mathematische Grundlagen der Quantenmechanik, Berlin, Springer, 1932. (English translation The Mathematical Foundations of quantum mechanics, Princeton, Princeton University Press, 1955). [26] Piron, C., "Axiomatique quantique", Helvetica Physica Acta, 37, 1964, p. 439. [27] Piron, C., Foundations of Quantum Physics, Benjamin, Reading, Mass., 1976. [28] Popper, K.R., "Birkhoff and von Neumann's interpretation of quantum mechanics", Nature, 219, 1968, pp. 682-685. [29] Putnam, H., "Three-valued logic", Philosophical Studies, 8, 1957, pp. 73-80. FROM LOGIC TO PHYSICS 65 [30] Putnam, H., "Is logic empirical?", Boston Studies Philosophy of Science, vol. 5, 1969, pp. 216-24l. [31] Redhead, M.L.G., Incompleteness, Nonlocality and Realism, Oxford, Clarendon Press, 1987. [32] Redhead, M.L.G., "Logic, quanta and the two-slit experiment", in: Hale, R., Clark, P. (eds.), Reading Putnam, Oxford, Blackwell, 1994. [33] Reichenbach, H., Philosophic Foundations of quantum mechanics, Berkeley-Los Angeles, University of California Press, 1944. [34] Varadarajan, V.S., "Probability in physics and a theorem on simultaneous observability", Communications in Pure and Applied Mathematics, 15, 1962, p. 189. [35] Varadarajan, V.S., Geometry of Quantum Theory, Vol. I, Van Nostrand, Princeton, 1968. [36] Zadeh, L.A., "Fuzzy Sets", Information and Control, 8, 1965, pp. 338-353. J AROSLA W PYKACZ NON-CLASSICAL LOGICS, NON-CLASSICAL SETS, AND NON-CLASSICAL PHYSICS 1. INTRODUCTION Quantum physics and many-valued logics were born nearly simultaneously in the third decade of the XX Century. However, the early attempts at identifying logic able to describe quantum systems with some versions of a three-valued logic failed and the opinion that "quantum logic", although non-classical, is a two-valued logic prevailed. The recently observed revival of interest in applying many-valued logics to the description of quantum physical systems is closely connected with a new and rapidly developing branch of mathematics: the fuzzy set theory. Fuzzy sets, which remain in the same relation to the infinite-valued logic as traditional sets to the classical two-valued logic, form a bridge by which one can pass from the "orthodox" quantum logic in the Birkhoffvon Neumann sense to the infinitely-valued Lukasiewicz logic. The paper is organized as follows: A brief introduction to many-valued logics is given in Section 2 while Section 3 gives rudiments of the fuzzy set theory and Section 4 shows its links with infinite-valued Lukasiewicz logic. The aim of Section 5 is to amuse and surprise the reader by some mysteries of the microworld based on the most standard example of Young double-slit experiment and finishes with some indications that many-valued logic could be better-suited for the description of this experiment. Sections 6 and 7 contain short historical survey of already made attempts at applying non-classical logics to the description of quantum phenomena out of which only Birkhoff-von Neumann proposal of using two-valued but non-distributive logic gained wide popularity and is still in use nowadays. The paper is concluded by Section 8 in which further indications for using many-valued logics in quantum physics are given and it is shown that it is possible to represent Birkhoff-von Neumann experimental propositions as propositional functions that belong to the domain of Lukasiewicz infinite-valued logic. After such "translation" Birkhoff-von Neuman quantum logic becomes a (partial) infinitevalued Lukasiewicz logic which solves the long-lasting problem of finding the proper models for disjunction and conjunction, and unifies two competing approaches: many-valued, and two-valued but non-distributive, existing in the quantum logic theory since its very beginning. 67 © 1999 Kluwer Academic Publishers. 68 JAROSLAW PYKACZ 2. NON - C LAS SIC ALL 0 G I C S The classical, two-valued logic deals exclusively with statements which can be unambiguously classified as being either true or false. Other statements simply do not belong to the domain of classical logic. In particular this refers to the statements concerning future events and this problem was noticed already by Aristotle who considered since then numerously quoted statement: "There will be a sea battle tomorrow". It should be mentioned that despite the tradition of naming the classical two-valued logic "Aristotelian logic" there are indications [1] that Aristotle himself classified "future contingents", i.e., statements about future events which are not yet decided, as neither true nor false!. In the Middle Ages the problem of future contingents was also discussed and it seems that such thinkers as Duns Scotus and William of Ockham in the thirteenth and the fourteenth centuries and Peter de Rivo in the fifteenth century considered such statements as indeterminate. Modern attempts at establishing non-classical logical systems, mostly three-valued ones, begun in the end of the nineteenth century. In 1897 Hugh MacColl investigated so called "three-dimensional logic" and in 1909 Charles Peirce considered "triadic logic" as a possible basis for "trichotomic mathematics". In 1919 Nicolai Vasil'ev in Kazan, Russia, built a system of three-valued "imaginary (non-Aristotelian) logic" whose name obviously referred to "imaginary (non-Euclidean) geometry" presented for the first time at the same university 84 years earlier by Nicolai Lobachevskij. Jan Lukasiewicz is generally recognized as a founding father of the modern theory of many-valued logics and his numerous papers on this subject published since 1920 until his death in 1956 are well-known. Contrary to these papers, his booklet "Die logischen Grundlagen der Wahrscheinlichkeitsrechnung" [3] published in 1913 although evaluated as "one of Lukasiewicz's most valuable works" in the foreword to Lukasiewicz's "Selected Works" [2], is relatively less-known. In this booklet he considered statements containing a variable, e.g., "x is an Englishman" and he attributed to them truth-value equal to the ratio of the number of values of a variable for which this statement is true to the total number 1 Lukasiewicz in many of his papers [2] claimed that the law of bivalence is actually due to the Stoics, especially Chrisippus who " ... appears to have been the first logician to consciously set up and stubbornly defend the theorem that every proposition is either true or false" (quotation from [1]). Therefore, Lukasiewicz proposed to call his many-valued logic "non-Chrisippean" rather than "non-Aristotelian". NON-CLASSICAL LOGICS 69 of values of this variable. Since he assumed the total number of values of a variable to be finite, the logic thus obtained is n-valued with n being a natural number depending on the particular situation described by a sentence. Lukasiewicz's principal aim in his 1913 paper [3] was to give the logical background to the notion of probability which at that time was much more alien to the rest of mathematics than it is now. An n-valued non-classical logic, which nowadays can be classified as a probability logic was only a kind of a by-product of these efforts and never gained such popularity as his later versions of many-valued logics investigated since 1920. The year 1920 is generally recognized as the year of the birth of the modern theory of many-valued logics. Actually, in this year two seminal papers on this theory were published independently by Jan Lukasiewicz in Poland [4] and by Emil Post in the USA [5]. Lukasiewicz arrived at his construction of a three-valued logic after the long period of philosophical investigations concerning the problem of determinism (cf. his numerous papers collected in [2], especially [6]), and of modal propositions, i.e., propositions of the form it is possible (impossible, contingent, necessary) that... [1]. He openly declared himself as devoted adherent of indeterdeclared a spiritual war minism who, according to his own words [7], upon all coercion that restricts man's free creative activity". Chrisippean law of bivalence which states that every proposition is necessarily either true or false occurred in this war to be a fortress which had to be blown up since it blocked the way towards indeterminism: Lukasiewicz argued that determinism follows necessarily from the law of bivalence, not from the law of excluded middle which only states that the disjunction of any proposition and its negation, e.g., "there will be a sea battle tomorrow OR there will not be a sea battle tomorrow" is always a true proposition. According to Lukasiewicz, who also claimed that this had been the original position taken by Aristotle, such disjunction remains true even though both its constituents can be neither true nor false. Lukasiewicz usually interpreted the third truth value as "indeterminacy,,2 and in the majority of his papers on three-valued logic he denoted this additional truth-value by the number 1/2. It is possible that in the beginning he did it just because 1/2 lays between 0 and 1- the generally accepted symbols of falsehood and truth, but later on this choice occurred to be very fortunate since it made the generalization of his three-valued logic to an n-valued or infinite-valued logic almost straightforward. (I••• 2 In the beginning [4] he interpreted it as "possibility" but later on he abandoned this idea probably in order not to confuse this name with the name of the possibility functor of modal logic. 70 JAROS LAW PYKACZ Lukasiewicz basic idea was to supplement two-valued logic with a third truth value in such a way that the obtained three-valued logic should deviate least from the ordinary logic. He did it by adopting the following truth table for implication: -+ 0 "2 1 1 o 1 1 1 1 1 1 "2 "2 1 1 1 0 "2 1 Table 1. Truth values of implication p -+ q (p implies q or if p then q) in Lukasiewicz three-valued logic. and by assuming that the following formulas which are in fact tautologies in two-valued logic define, respectively, negation, disjunction, conjunction, and equivalence3 : negation: de! -,p = p -+ (1) 0 (not p means the same as p implies falsehood) disjunction: pVq de! = (p-+q ) -+q (2) (p or q means the same as (p implies q) implies q) conjunction: p 1\ q de! = -,( -,p V -,q) (3) (p and q means the same as not (not p or not q)) equivalence: p == q de! = (p -+ q) 1\ (q -+ p) (4) (p and q are equivalent means the same as p implies q and q implies p) It can be easily checked that these definitions together with Table 1 yield the following truth tables for negation, disjunction, conjunction, and equivalence in Lukasiewicz three-valued logic: 3 Lukasiewicz did not consider three-valued equivalence in [4]. For the first time he did it in the realm of many-valued logics in 1922 [8] but the formula (4) is a standard one in two-valued logic and it was often used in many of Lukasiewicz papers published both before and after 1920. NON-CLASSICAL LOGICS o 1 1 1 "2 1 "2 0 Table 2. Truth values of negation....,p v 0 2 0 0 1 1 "2 1 "2 1 "2 1 71 "2 1 1 1 1 1 1 1 (not p). Table 3. Truth values of disjunction p V q (p or q). A 0 "2 1 1 0 0 0 0 0 0 1 "2 1 1 1 "2 "2 "2 1 1 Table 4. Truth values of conjunction p A q (p and q). o "2 o 1 1 1 "2 1 "2 o 1 1 "2 1 o 1 "2 1 "2 1 1 == q (p is equivalent to q or p if and only if q). Table 5. Truth values of equivalence p When the first break-through was made, further generalization to nvalued logics and infinite-valued logics was not so difficult and Lukasiewicz actually did it soon afterwards [8, 9]. Of course except for the case of n-valued logics with n being relatively small number, truth values of logical functors cannot be presented in the form of tables. Fortunately, Lukasiewicz found algebraic expressions, the same for all systems of many-valued logics (both finite and infinite-valued), which yield truth 72 JAROSLAW PYKACZ values of compound propositions as functions of their constituents, i.e., all Lukasiewicz many-valued logics are truth-functional. The basic logical functor for Lukasiewicz was implication and the following expression allows to calculate truth value of implication t(p -7 q) when the truth values t(p) and t(q) of the antecedent p and the consequent q are known [8,9]: t(p -7 q) = min[1- t(p) + t(q), 1]. (5) It can be easily checked that this formula applied to definitions (1)-(4) yields the following formulas for truth values of the remaining logical connectives: = 1 - t(p) (6) t(p V q) = max[t(p) , t(q)] (7) t(p 1\ q) = min [t(p) , t(q)] (8) t(-.p) t(p == q) = 1-lt(p) - t(q)l. (9) Lukasiewicz assumed that the set of truth values of an n-valued logic with 0 ~ k ~ n - 1. It is consists of all fractions of the form n~l straightforward to check that the formulas (5)-(9) yield Tables 1-5 for n=3. Lukasiewicz was not so much concerned with the interpretation of his subsequently developed n-valued (for n > 3) or infinite-valued logics as with the interpretation of the third truth value of his original threevalued logic, but he mentioned in [8] that "... 0 is interpreted as falsehood, 1 as truth, and the other numbers in the interval 0-1 as the degrees of probability corresponding to various possibilities ... ". Therefore, it is clear that at least in the year 1922 he still maintained his idea, expressed for the first time in the year 1913 [3], of interpreting non-classical truth values as degrees of probability. Contrary to Jan Lukasiewicz, the second founding father of the modern theory of many-valued logics: Emil Leon Post does not seem to be very much concerned with interpretation of non-classical truth values. His investigations were not so much founded on philosophical considerations but were rather of formal algebraic nature. Loosely speaking we can say that he studied algebraic aspects of n-valued logics without bothering to express them linguistically and in this respect his papers [5, 10] are more close in their style to modern treatises on many-valued logics (see, e.g., [11]) than contemporary papers by Lukasiewicz. Post based his n-valued propositional calculi on the linearly ordered set of truth values {tl' t2, ... t n } where the extreme elements express ''full NON-CLASSICAL LOGICS 73 truth" and ''full falsehood" and he followed Whitehead and Russell's Principia Mathematica [12] in choosing negation and disjunction as basic connectives. However, although his disjunction was the same as that of Lukasiewicz, Le., its truth value was the bigger of truth values of its constituents, Post's basic negation, which could be called cyclical was quite different 4 : p -,p Table 6. Truth values of Post's "cyclical" negation. Because of so defined negation if other connectives (conjunction, implication, and equivalence) are defined with the aid of tautologies known from two-valued logic, then they exhibit rather unexpected and counterintuitive features. In spite of this fact in modern times Post logics find their application in studying electronic networks and in Computer Science [13, 14]. It should be also mentioned that due to this particular negation n-valued propositional calculi of Post, contrary to these of Lukasiewicz, are functionally complete: any conceivable connective can be defined by basic connectives of negation and disjunction. Of course no intuitive interpretation can be given to the vast majority of connectives obtained in such a way. After the first break-through made by Lukasiewicz and Post many other systems of three-valued and n-valued propositional calculi were proposed. I can mention here three-valued calculi of Kleene [15]' [16]' Bochvar [17], and Finn [18] motivated by epistemological considerations concerning lack of meaning of some statements, a similarly motivated group of papers dealing with so-called nonsense-logics [19, 20, 21, 22], attempts at describing intuitionistic propositional calculus in terms of many-valued logics [23, 24], or papers motivated by considerations concerning peculiarities of quantum physis [25, 26, 27, 28, 29, 30, 31, 32J. Mathematically experienced reader can find a recent survey of most of the above-mentioned three-valued logics and some n-valued logics [33, 4 Besides of this "cyclical" negation Post considered also the other negation, identical with that of Lukasiewicz. However, it seems that he treated "cyclical" negation as more important. 74 JAROS LAW PYKACZ 34] in Chapters 3 and 4 of a recent book [11] by Bole and Borowik. Some examples of many-valued logics motivated by physical considerations are described in the Chapter 8 of Jammer's book [35]. It should be mentioned that many-valued Lukasiewicz logic endowed with negation (1), (6), disjunction (2), (7), and conjunction (3), (8) was criticized by Gonseth [36] in 1938 since it satisfies neither the law of excluded middle t(p V --,p) = 1 (it is true that p or not p) (10) nor the law of contradiction t(p A --,p) = 0 (it is false that p and not p). (11) Actually, neither of these formulas is satisfied for t(p) =I- 0,1, for example they both assume truth value! for t(p) = !. Most probably Gonseth did not know Polish so he could not have read the paper [37] published by Zawirski already in 19345 in which Zawirski noticed that if we replace the right-hand side of the formula (2) by which Lukasiewicz defined disjunction in [4] by the other (in two-valued logic equivalent) expression: --'p --7 q, then the disjunction obtained in this way: pUq t(p U q) de! = --,p --7 q = min[t(p) + t(q), 1] (12) and the conjunction adjoint to it by De Morgan's Law (3): pnq t(p n q) de! = --,( --,p U --,q) max[t(p) + t(q) - 1,0] (13) satisfy both the law of excluded middle: t(p U --,p) = min[t(p) + 1 - t(p), 1] = 1 (14) and the law of contradiction: t(p n --,p) = max[t(p) + 1 - t(p) - 1,0] = O. 5 The same idea was published in English by Orrin Frink Jr. in 1938 in [38]. (15) NON-CLASSICAL LOGICS 75 Therefore, Gonseth's critique cannot be applied to Lukasiewicz manyvalued logic endowed with his original implication (5), negation (1), (6), and Zawirski disjunction (12) and conjunction (13). As we shall see in the sequel this set of connectives seems also to be better suited for the description of behaviour of quantum physical systems than the set of connectives originally defined and studied by Lukasiewicz. 3. NON - C LAS SIC A L SET S The classical, two-valued logic is a basis of traditional mathematics and, in particular, of the traditional set theory. Although there exist wellelaborated systems of axioms for the classical set theory for all practical purposes it is enough to distinguish a set that we are interested in by a predicate which, according to two-valued logic, allows to divide unambiguously all considered objects into two disjoint classes: objects that belong to a set and objects that do not belong to a set and form its complement. For example, let U be a set consisting of speakers at the "Einstein meets Magritte" Conference. This predicate is precise enough to define this set as soon as the Conference is finished (otherwise we could fall into the Aristotelian trap of future contingents). All propositions of the form: x belongs to the set U where x denotes a name of an individual person are, as soon as the Conference is finished, either true or false, i.e., they belong to the domain of classical two-valued logic. Although every traditional set is defined by a "sharp" predicate, not every predicate is good enough to define a traditional set in an unambiguous way. Let us try to distinguish a subset A of the above-mentioned set of speakers U consisting of speakers whose talks were interesting. Even if we choose only one umpire in order not to deal with various opinions we are likely to get, besides "sharp" judgements of the form: "the talk of Prof. X was not interesting", "the talk of Dr. Y was interesting" also a lot of statements of the form: "the talk of ... was ... a little bit/only pariially/not so much/quite/in most of its paris/nearly almost ... interesting". Therefore, we see that besides the speakers who, like Prof. X surely do not belong to the set A and like Dr. Y surely belong to it, both membership and non-membership of other speakers to the set A is doubtful. It also would not be good to group all these other speakers into one category since from various judgements of our umpire we infer that different talks were intersting to him to different extent. The best solution would be to evaluate numerically degrees to which talks of various speakers were interesting and to say that "degrees of membership" of various speakers to the set A are proportional to these numbers. 76 JAROSLAW PYKACZ This is exactly the idea of a fuzzy set: If a is a fuzzy subset of the universe of discourse U (in our case the set U consists of all speakers), then some elements of U surely belong to A, some surely do not belong to it, but also all intermediate cases of "partial membership" are allowed. Moreover, membership is "graded": according to the original idea of Lotfi A. Zadeh [41], who is generally recognized as a founding father of the fuzzy set theory6, membership of an element x to a fuzzy set A, denoted /-LA(X) or simply A(x) can vary from 0 (full nonmembership) to 1 (full membership), i.e., it can assume all values in the interval [0,1]. Therefore, a membership function /-LA : x I--t /-LA(X) E [0,1] completely characterizes the fuzzy set A and it is an obvious generalization of a characteristic function XA(X) of a traditional set: 0 for x tf. A XA (x) = { 1 for x E A (16) Fuzzy subsets of a plane can be easily visualized as areas which, contrary to traditional sets (usually called crisp sets in the fuzzy set theory) have no sharp boundaries and vanish gradually (Figs. 1 and 2). They are smeared, blurred or simply fuzzy. Our everyday language provides us with numerous examples of "nonsharp" predicates which can define only non-crisp sets, e.g., young (man), ripe (apple), old (painting), fast (car), famous (artist), etc. In all these cases we can easily distinguish elements which surely belong to a set of objects defined by a given predicate, elements which surely do not belong to it, and elements whose membership is more or less doubtful. Actually, I dare say that in everyday communication "sharp" predicates which defne crisp sets are rather an exception than a rule. Of course in some cases it is possible to draw a borderline in a more or less arbitrary way to recover sharp discrimination between members and non-members of a set. For example we could state that a car Xl which can go faster than 150 km/h belongs to the set of fast cars which, according to twovalued logic, implies that a car X2 which can go at most 149, 999 km/h is, by the very definition, not fast so it does not belong to the set of fast cars. However, we feel that the car X2 "almost belongs" to the set of fast cars and should not be treated in the same way as a car X3 which can go at most 50 km/h. It is more natural to state that the grade of membership of the car X2 to the set of fast cars is very close to 1 while the grade of membership of the car X3 to this set is close to o. Thus, the 6 It seems that C. C. Chang [39] and D. Klaua [40] elaborated similar ideas independently of [41] and published them even slightly before [41]. It is a problem for historians of science to explain why their papers, contrary to [41] are almost neglected. 77 NON-CLASSICAL LOGICS I 1 '--_ _ _-'---_----' ____ 1 ___ -'-_ _ _-'- ______ _ Figure 1. Crisp set and its characteristic function along the axis X. Point Xl belongs to A, therefore XA(XI) = 1; point X2 does not belong to A, therefore XA(X2) = o. Each point either entirely belongs or does not belong to A . .: , I··.· .:1 0'0 1 - _____ l_--- e' ",I _____ -~l_ ~- I 1 "2 _ _ _ _ 1_ _ _ _ _ _ _ _ _ _ _ I _ _ _ _ _ _ _ _ _ _ _ ..J _ _ _ _ _ I Figure 2. Fuzzy set and its membership function along axis X. Point Xl belongs to A in 100%, therefore JLA(Xt) = 1; point X2 belongs to A in 0% (does not belong to A in 100%), therefore JLA(X2) = O. Other points belong to A to the extent expressed by respective values of the membership function JLA, e.g., points X3, X4, X5 and Xa half-belong to A, therefore, JLA(x3-a) = ~. idea of representing the collection of fast cars in a form of a fuzzy set is very appealing, although it should be mentioned that in general it is not 78 JAROSLAW PYKACZ at all obvious what a membership function of a specific fuzzy set should precisely look like7 . As soon as membership functions of fuzzy sets are established, these sets are characterized to the full extent and we can define on them all relations and operations known from the traditional set theory8. This is much expected since classical sets are actually special cases of fuzzy sets: they are fuzzy sets whose membership functions assume only two values: 0 and 1, i.e., these membership functions are in fact characteristic functions (16), and because all set-theoretic relations and operations on classical sets can be expressed in terms of these characteristic functions. The basic relations and operations on fuzzy sets were defined already by Zadeh in his historic paper [41] and they are till now the most frequently used in all contributions to and applications of the fuzzy set theory. We shall see in the sequel that Zadeh's intuitive choice was so natural because these operations follow from the connectives of Lukasiewicz many-valued logic exactly in the same way as operations on classical sets follow from the connectives of classical logic. Zadeh basic relations and operations are defined with the aid of membership functions as follows (we assume, as it is usually done in the fuzzy set theory, that all considered fuzzy sets are in fact fuzzy subsets of a fixed universe of discourse U; we shall represent fuzzy sets on pictures by their membership functions as it was done on the lower part of Figure 2): Equality of fuzzy sets (Fig. 3): A = B iff for all elements x in the universe U (17) Inclusion of fuzzy sets (Fig. 4): A universe U ~ B iff for all elements x in the (18) Complement (negation) of a fuzzy set (Fig. 5): A' is a complement of A iff for all elements x in the universe U (19) 7 This observation gave rise to the notion of probabilistic fuzzy sets introduced by Hirota [42] whose membership functions are themselves "fuzzy". Of course this procedure can be continued, but objects obtained in this way are less and less convenient to deal with. 8 There are also operations which can be defined on fuzzy sets but not on crisp sets, e.g., an operation of "sharpening" of a set which makes it "less fuzzy". NON-CLASSICAL LOGICS 79 1 ---------/ c A=B Figure 3. Equality and non-equality of fuzzy sets. 1 ?-~ J.l.B (x) ;;2 J.l.A (x) Figure 4. Inclusion of fuzzy sets. Union (sum) of fuzzy sets (Fig. 6): Au B is a union of A and B iff for all elements x in the universe U (20) 1 ---------, , \ A' \\ A ------------------7--/ / / _ - - / J.l.A' (x) = 1 - J.l.A (x) 1 ---------------:;. :2 / / J.l.A (x) / \ / \ Figure 5. Complement (negation) of fuzzy sets. 80 JAROSLAW PYKACZ Figure 6. Union (sum) of fuzzy sets. 1 Figure 7. Intersection (product) of fuzzy sets. Intersection (product) of fuzzy sets (Fig. 7): A n B is an intersection of A and B iff for all elements x of the universe U (21) The fuzzy set theory is by no means only a mathematical game. Although in the beginning it was treated with a kind of reservation by traditionally oriented "crisp" mathematicians it finds numerous practical applications which vary from earthquake forecasting, computer medical diagnoses, decision making and pattern recognition to the production of control systems for the underground and more efficient vacuum cleaners [43]. Moreover, it seems to be a very natural tool for all "soft" sciences which deal with vagueness or imprecision caused either, like in meteorology, by the excess of data or, like in economics, sociology, psychology, etc., by the human factor. Actually, the "applicational" aspect of fuzzy sets is maybe even better known than their theoretical aspects which still seem to be undervalued by the society of "crisp" mathematicians. NON-CLASSICAL LOGICS 81 4. FU Z ZY SET SAND INFINI TEVALUED LUKASIEWICZ LOGIC In order to explain why obvious links between fuzzy sets and manyvalued Lukasiewicz logics were not studied9 during the whole first decade of vivid development of the fuzzy set theory one should take into account two possible reasons: On the one hand Lotfi Zadeh and his followers seemed to be interested mostly in applications of the newly established theory and were not so much occupied with clarification of its foundations. On the other hand "pure" mathematicians of that time did not bother a lot about a theory which probably was seen by them as too simple in comparison with sophisticated problems emerging on very frontiers of contemporary "crisp" mathematics. Actually, the relation of classical logic to classical set theory, in particular definitions of settheoretic operations in terms of connectives of classical logic are taught in the beginning of the secondary school lO • They remain the same when classical logic is replaced by infinite-valued Lukasiewicz logic and classical sets are replaced by fuzzy sets, but this observation was published by Robin Giles [44]10 years after l l the successful launching of the idea of fuzzy sets by Zadeh in 1965. Let us remind relations between propositions and sets known from school and see how they work when classical logic and classical sets are replaced by Lukasiewicz infinite-valued logic and fuzzy sets. The notion of a set is adopted at school as a primitive notion and it is tacitly assumed that we know a set when we know all its elements. Therefore, any set A can be described as a collection of objects whose names turn a propositional function "x belongs to A" ("x E A") into a true proposition. Symbolically: A = {x : t("x E A") = 1} (22) Because of the equality in the bracket the set A defined by the formula (22) is unavoidably crisp even if we replace classical two-valued logic by infinite-valued logic, However, if we rewrite this formula into the following (in two-valued logic equivalent) form: A = {x : t("x E A") i= O} (23) then, since in infinite-valued logic a truth value of a non-false proposition can assume, besides 1, any value between 0 and 1, the set A occurs to 9 except in almost neglected papers [39] and [40] mentioned in footnote 6 10 at least in Poland 11 but cf. footnotes 9 and 6 82 J AROSLA W PYKACZ be a fuzzy set with membership function defined by truth values of propositions of the form "x belongs to A": f.LA(X) = t("x E A") (24) (the degree of membership of an object x to the fuzzy set A is equal to the truth value of the proposition "x belongs to A"). This equality allows to use well-known school definitions of a complement, union, and intersection of classical sets together with Lukasiewicz formulas for truth values of negation(6), disjunction (7), and conjunction (8) to justify Zadeh's intuitive choice of basic operations on fuzzy sets (19), (20), and (21): Membership function of a complement (negation) of a fuzzy set: f.LAI(X) = t("x E A''') = t("x rf. A") = t(-,"x E A") = 1- t("x E A") = 1 -/-tA(X). (25) Membership function of a union (sum) of fuzzy sets: f.LAUB(X) = t("x E A" V "x E B") max[t("x E A"), t("x E B")J = max[f.LA(x), f.LB)X )J. (26) Membership function of an intersection (product) of fuzzy sets: f.LAnB(X) = t("x E A" /\ "x E B") min[t("x E A"), t("x E B")J = min[f.LA(x) , f.LB)X)J. = (27) However, as it was mentioned at the end of Section 1, the original Lukasiewicz disjunction and conjunction are not the only conceivable connectives of this type. Therefore, Zawirski disjunction (12) and conjunction (13) placed inside (26) and (27) yield other union and intersection of fuzzy sets, usually called bold operations after Giles [44J (other names : Giles, truncated, bounded, arithmetic, Prink, Lukasiewicz operations) 12: t( "x f.LAUB(X) 12 E A" u "x = min[t("x E A") = min[f.LA(x) E B") + t("x E B"), 1J + f.LB(X), 1J (28) It is sure that Giles [44] was not aware of Zawirski 1934 paper [37] where these operations appeared for the first time (in the domain of many-valued logics), and most probably he was also not aware of Frink 1938 paper [38]. It seems also that these operations were rediscovered many times by various authors which explains multiplicity of their names. NON-CLASSICAL LOGICS J.LAnB(X) = = = t("X E A" n "x E B") max[t{"x E A") + t{"x E B") - 1,0] max[ILA(x) + ILB(X) - 1,0] 83 (29) It is obvious that other disjunction-like and conjunction-like connectives of infinite-valued logic13 define in the same way other operations of fuzzy set union and intersection and vice versa: All operations on fuzzy sets interpretable as fuzzy set union and intersection 14 yield disjunctionlike and conjunction-like connectives of infinite-valued Lukasiewicz logic. 5. NON - C LAS SIC ALP H Y SIC S Richard Feynman, the Nobel Prize winner in physics, undoubtedly one of those who knew quantum mechanics best said [47]: "I think I can safely say that nobody understands quantum mechanics". Actually, the behaviour of microobjects is sometimes so peculiar that, although it can be very well described and predicted with the aid of well-elaborated mathematical formalism, it cannot be comprehended to the full extent by our brain experienced by macroscopic world and, I dare say, twovalued logic. Let us consider a standard example of a double-slit experiment (Figs. 8 and 9) which illustrates famous wave-particle duality of quantum objects and which, again according to Richard Feynman [48] u... has in it the heart of quantum mechanics. In reality, it contains the only mystery". The double-slit experiment is an old experiment designed by Thomas Young in the beginning of the 19th century in order to show interference of light. Waves emitted by a source S pass through two narrow slits A and B made in a non-transparent barrier and interfere in the area behind this barrier where they overlap (Fig. 8). The interference is visible either directly (in the case of water waves) or after placing a screen or a system of detectors in the area of interference, as it is usually done in the case of light or acoustic waves. 13 2 There are 2(2 ) = 16 conceivable two-argument connectives in 2-valued logic, 3(3 2 ) = 19.683 two-argument connectives in 3-valued logic, n(n 2 ) two-argument connectives in n-valued logic and obviously infinity of two-argument connectives in infinite-valued logic. Of course not all of them could be in a reasonable way interpreted as disjunction or conjunction. Some of them could be interpreted as implication or equivalence, but the overwhelming majority of them have no counterparts in two-valued logic. 14 Even the whole families of such operations, parametrized by real numbers, have been already studied (see, e.g., [45, 46, 43]) 84 JAROSLAW PYKACZ barrier screen Figure 8. Double-slit experiment in the configuration W ("waves"). Wave-like aspects of light are shown. It should be noticed that interference is a typical wave phenomenon: When two waves of the same amplitude meet their "movements" sum up so the amplitude of the emerging wave is doubled in areas where both constituing waves have the same phase (totally constructive interference), disappears in areas where constituing waves have opposite phase (totally destructive interference) and is intermediate in other places. Specially the case of totally destructive interference shows the crucial difference between waves and particles of any kind: We cannot expect that in a place where two streams of particles (e.g., bullets from two machine guns) meet they could annihilate leaving no trace of their existence at all, like destructively interfering waves do. Interference experiments, relatively easy to be performed with light, convinced for some time the majority of physicists that light is a typical wave. However, the study of photoelectric effect by Albert Einstein led him to the conclusion that energy of light spreads in indivisible packets ("quanta", which gave name to quantum physics). A slight modification of a double-slit experiment (Fig. 9) shows that light quanta, usually called photons, are well-localized: It is enough to place a detector behind each slit and use very weak source of light, in which case photons "leak" one by one, to state that: (i) each of the detectors either registers the "whole" photon or nothing at all, nothing like a "fragment of a photon" is ever registered, (ii) detectors never register a photon simultaneously which implies that each photon passes through one slit only, i.e., photons NON-CLASSICAL LOGICS very weak source 85 no coincidence! Figure 9. Double-slit experiment in the configuration P ("particles"). Particle-like aspects of light are shown. behave in this situation like well-defined particles 15 . The double-slit experiment and its modifications reveal to us, macroscopic creatures, a lot of surprising mysteries of the microworld: MYSTERY 1. Interference pattern in the configuration W is built after enough long time even when the source of light is so weak that photons "leak one by one" 16. This shows that interference is not caused by collective motion or mutual influence of many photons but happens even for a single photon which "interferes with itself' . MYSTERY 2. Interference experiments can be performed not only with light but also with "solid" objects the particle-like character of which seems to be beyond any question. This hypothesis was formulated by Louis de Broglie in his famous doctoral dissertation [51] in 1923 and confirmed experimentally for electrons in 1927 by Davisson and Germer [52]. Nowadays wave-like aspects of electrons or even quite heavy atoms are utilized for practical purposes in every hospital or laboratory where electron or ion microscope is used. MYSTERY 3. If someone, in spite of the results of a double-slit experiment performed in the configuration P maintains that photons, electrons, etc., are "extended particles" which travel simultaneously through both slits, he should take into account that in some already performed versions of this experiment the two allowed paths were separated by several me15 The described experiment is a "gedanken" one but similar experiments were actually performed (see, e.g., [49]) and the results were just as described above. 16 Such experiments were done already in 1909 by G. I. Taylor [50). 86 JAROS LAW PYKACZ possible paths of a photon gra vitational lens made by remote galaxy Figure 10. Cosmic-scale double-slit experiment. ters. Moreover, the ingenious proposal of "cosmic-scale double-slit experiment" (Fig. 10) made by Wheeler [53J would require micro(!)-objects to extend over a distance of millions of light-years. MYSTERY 4. It seems that the very nature of microobjects is so mischievous that they never show us both their faces: wave-like and particlelike at the same time. Actually, any attempt to establish through which slit a photon passes inevitably leads either to catching it in a detector or to such disturbance of its movement (e.g., if it bounces against another microobject which in turn is registered in a detector) that the interference pattern is destroyed anyway. This is the essence of the famous Complementarity Principle: Microobjects always reveal to us only one of their complementary faces, in particular, we can never experience both their wave-like (e.g., interference) and particle-like (e.g., defined trajectory) features in the same experiment. However, the last statement belongs to the original Copenhagen paradigm which nowadays has to be slightly modified. In 1979 Wooters and Zurek [54J demonstrated that mathematical formalism of quantum mechanics allows to obtain in a double-slit experiment both non-perfect knowledge about a path of a quantum object together with non-perfect interference pattern. These predictions were experimentally confirmed in an experiment performed by Mittelstaedt, Prieur, and Schieder [55J in 1987 and actually in the course of this experiment partial interference pattern was obtained together with partial knowledge of a trajectory of a photon which allowed the authors to conclude that: "A photon possesses simultaneously particle properties and wave properties". In this way we encounter for the first time the situation in which many-valued logic seems to be better-suited for the description of quantum phenomena than classical two-valued logic: According to the orthodox Copenhagen formulation of the Complementarity Principle wave-like and particle-like properties of microobjects are mutually exclusive in the NON-CLASSICAL LOGICS 87 "absolute" sense. Therefore, propositions like PA,W (PA,P) = "an experiment A reveals wave-like (resp. particle-like) properties of photons" are supposed to be either entirely false or entirely true, i.e., it is believed that for any conceivable experiment they belong to the domain of classical two-valued logic. However, Mittelstaedt, Prieur, and Schieder (MPS) found in the course of their experiment that: "Even for a very high value of the particle property, there is still a nonvanishing amount of the wave property" and they for example wrote about "... a photon with 98,2% particle property and ... 1,8% wave property", i.e., they were not only able to state simultaneous "partial" or "non-perfect" existence of these properties, but also to express numerically the "degree" to which they hold. These percentages, after dividing by 100%, can be in a straightforward way identified with truth values t(PA,W), t(PA,P) of propositions PA,W andpA,p and in the course of the MPS-type experiment these truth values are usually different from 0 or 1. Of course it is still possible to avoid using non-classical truth values t(PA,W), t(PA,P) i- 0,1 and describe MPS-type experiments entirely in terms of classical two-valued logic using instead of propositions like PA,W or PA,r> propositions of the form "the amount of wave-like properties of photons in an experiment is x%" whose truth-values (at least in the idealized case when we do not allow any imperfections) are again either o or 117. However, there are also other indications for using non-classical logics in the description of quantum systems. We shall study them in a more detailed way in the section that follows. 6. THREE-VAL UED LO GICS IN FOUNDATIONS OF QU ANTUM MECHANICS In the years 1925-1926 the development of quantum physics experienced itself a "quantum jump": Under the influential works by Heisenberg, Schrodinger, Born, Jordan and Dirac [56, 57, 58, 59, 60, 61] physicists abandoned so called "older quantum theory" which was merely an amalgamate of ideas and models taken from classical physics with ad hoc added "quantum conditions" and developed quantum mechanics as internally consistent, although mathematically highly sophisticated theory which, at least in its nonrelativistic part, persists without drastic changes till today. However, some implications of the new theory were so bizarre that there were scientists who claimed that quantum theory cannot be comprehended on the ground of classical two-valued logic. 17 This is a standard procedure allowed by the fact that classical two-valued logic is a metalogic for many-valued logics. 88 JAROSLAW PYKACZ The first one who expressed such claims was a Polish logician Zygmunt Zawirski who was looking for the possible fields of application of Lukasiewicz many-valued logic. In the papers published in 1931 [25] and in 1932 [26] Zawirski argued that the equivalence of "complementary theories", e.g., wave and particle pictures in the description of microobjects is possible only on the ground of (at least) three-valued logic since in two-valued logic a statement like "light is a wave AND light consists of particles" is a statement which is a conjunction of two propositions which cannot be simultaneously true. Therefore, according to the laws of classical two-valued logic such conjunction is necessarily a false proposition. According to Zawirski complementarity, typical to quantum mechanics, can be comprehended only on the ground of (at least) three-valued logic when we ascribe to two mutually exclusive theories a third truth value interpreted as "possibility" or "equal probability". Indeed, if a truth value of two propositions p and q equals ~, then according to formula (8) t(p 1\ q) = min[!, ~J = ~ (30) so the conjunction of two "possible" statements is again "possible". Zawirski's papers: [25J published in Polish in a local journal, [26J published in French but in a journal not popular among physicists, as well as [37J published again in Polish, passed relatively unnoticed 18. More fortunate in spreading out his ideas was Fritz Zwicky [62J but the bestknown attempts at basing quantum mechanics on three-valued logics were elaborated in the fourties and early fifties by Paulette DestouchesFevrier [27, 28J and Hans Reichenbach [29, 30, 31, 32J. Specially the Reichenbach's book [29J in which he consequently tried to explain quantum mysteries on the ground of three-valued logic was, for some time, widely discussed. Reichenbach's ideas were pursued after his death by Hilary Putnam [63J but all attempts at using three-valued logics in order to describe quantum phenomena never gained such popularity as attempts based on quite other type of non-classical logic introduced in 1936 by Birkhoff and von Neumann which are described in the next section. 18 even in Poland: When in 1991 in the Polish National Library in Warsaw I got to my hands a copy of Zawirski's paper [37] it occured that the pages of a booklet were still not cut apart, i.e., most probably no one had read this copy during the whole 60 years! The homage should be paid to Max Jammer who mentions Zawirski papers in his famous book [35] on philosophy of quantum mechanics. The interested reader will find in Chapter 8 of this book more detailed historical survey of applications of many-valued logics in foundations of quantum mechanics up to the early seventies. NON-CLASSICAL LOGICS 89 In the author's opinion the attempts at founding quantum mechanics on any version of three-valued logics were bound to fail also because three-valued logics are not enough "rich in truth values" and, therefore, are not enough "flexible": One should not hope to be able to describe within them the whole variety of quantum phenomena, specially when truth values are supposed to be connected with numerical results of experiments, if there is only one truth value (besides the classical 0 and 1) at hand. 7. STILL OTHER NON-CLASSICAL LOGIC FOR QUANTUM MECHANICS Garrett Birkhoff and John von Neumann published in 1936 a seminal paper [64] whose title "The logic of quantum mechanics" gave name to the vast and vivid field of activity that stemmed from this paper and which was later on called the theory of quantum logics 19 . At that time quantum mechanics already achieved its mature form 2o whose basic ingredients were linear operators acting on some abstract vector space called Hilbert space after great mathematician David Hilbert who was the first to study such spaces in the beginning of the century. Some of these operators (so called projection operators) were regarded by Birkhoff and von Neumann as representing dichotomic experimental propositions, i.e., such propositions, that can be unambiguously classified as either true or false when a suitable experiment is completed. Please note that this resembles very much Aristotle's statement "There will be a sea battle tomorrow" which, when the experiment consisting in placing two navies in close vicinity is completed is also bound to be either true or false, but before it is completed it can be regarded as belonging to the domain of many-valued logic. However, the possibility of interpreting Birkhoffvon Neumann experimental propositions as many-valued propositions was for the long time overlooked and the opinion prevailed that these propositions are two-valued (cf., p. 346 of Jammer's book [35]). The non-classical character of Birkhoff-von Neumann experimental propositions was supposed to lay not in the number of their truth values but in the algebraic structure of the set they form. This structure is obviously the same as the structure of the set of projection opera19 The most recent trend is to speak about quantum structures rather than about quantum logics (vide: International quantum structures Association) since in the last years the mathematical structures studied within this field considerably transgressed the boundaries encircled by Birkhoff and von Neumann's paper (64). 20 von Neumann's own contribution to this form was one of the most influential. 90 JAROSLAW PYKACZ tors acting on an underlying Hilbert space and it is nowadays called an orthomodular lattice. Orthomodular lattices are slightly more general structures than Boolean algebras 21 which in turn are algebraic models (so called Lindenbaum algebras) of families of propositions which obey laws of classical two-valued logic. The law valid in Boolean algebras that is not secured in orthomodular lattices is the distributivity law which, written in logical notation, looks as follows: pl\(qVr) pV(ql\r) = (pl\q) V (pl\r) (pVq) 1\ (pVr). (31) (32) Therefore, the hypothetical quantum logic, i.e., a system of experimentally verifiable propositions that can be legitimately used in the description of properties and behaviour of quantum objects was judged to be non-distributive although two-valued. We shall argue in the next section that it should be rather judged as non-distributive and many-valued. 8. THE UNIFICATION OF TWO APPROACHES TO QUANTUM LOGIC: BIRKHOFF-VON NEUMANN QUANTUM LOGIC AS PARTIAL INFINITE-VALUED LUKASIEWICZ LOGIC As it was already mentioned, a Birkhoff-von Neumann experimental proposition p assumes one of two classical truth values 0 or 1 when suitable experiment is completed. However, before this is done, the mathematical formalism of quantum mechanics allows only to calculate probability that a proposition will be true or false. It is worth stressing that in the case of quantum physical systems, contrary to classical ones, this probability is "ontological", not "epistemological", i.e., it does not result from a lack of knowledge about actual values of some parameters that could have characterized a studied system and which, had they been known, would have allowed to predict the results of experiments without any uncertainty22. 21 The most typical example of a Boolean algebra is a family of (crisp) subsets of a fixed set which is closed with respect to the standard set-theoretic operations of union, intersection, and complement. Actually, due to the famous Stone Theorem [65] any Boolean algebra is isomorphic to a Boolean algebra of this kind. 22 The idea of supplementing quantum theory by such parameters, usually called hidden variables, was considered by various physicists since the very birth of quantum theory. However, some recent experiments [66] indicate that this idea can be maintained only if hidden variables act on quantum objects in a non-local way and, therefore, is rejected by the majority of contemporary physicists. NON-CLASSICAL LOGICS 91 In general the same physical system can be in different states (which "operationally" manifests itself by using different procedures to prepare it for an experiment) and probabilities that an experimental proposition p will turn out to be true are, in general, different for different states of a physical system23 • Therefore, we see that with any experimental proposition p and any state a of a physical system the mathematical formalism of quantum mechanics associates in an unambiguous way a number a(p) E [0, 1] interpreted as a probability of stating experimentally that the proposition p is true. In the mathematical parlance it is said that states of a physical system are represented by probability measures defined on a logic of a system, i.e., on a family of experimentally verifiable propositions pertaining to the studied physical system. This is in a full agreement with mathematical description of quantum systems in Hilbert spaces where actually, due to the famous Gleason Theorem [67], states of physical systems generate probability measures on families of projection operators. However, there is also the other possibility in which a number a(p) E [0,1] can be interpreted. It agrees with an old idea launched by Lukasiewicz already in 1913 [3] and elaborated further by Zawirski [37], according to which numerical value of a probability of a random event c coincides with a truth value of a (many-valued) proposition: "The event c will happen,,24. This possibility was pursued by the present author in a series of papers [68, 69, 70] and consists in assuming that a number a(p) E [0,1] is a truth value of an experimental proposition p(a) which can be linguistically expressed, e.g., in the form: "If we perform a dichotomic (yes-no) experiment designed to check an experimentally verifiable proposition p on a physical system that is in a state a, then p will turn out to be true", or simply: "If a system is in a state a, then p will be true", i.e., that a(p) = t(p(a)). (33) Let us stress once more that the Future Tense used in these linguistic statements clearly shows that at least in the case of quantum physical 23 If they are not, there is no reason to claim that these states of a system are actually different. 24 Actually, neither of these Authors paid much attention to distinguish this statement expressed in the Future Tense from the statement: "The event & happened" . In my opinion such distinction helps to maintain clarity: The first statement belongs to the domain of many-valued logic while the second one to the classical two-valued logic (provided that we exclude "ill-defined" events and "ill-performed" experiments which, after an experiment is completed, leave doubts whether an event actually happened or not). 92 JAROSLAW PYKACZ systems these statements do not, in general, belong to the domain of the classical two-valued logic which only rarely admits statements about future events. Since we cannot attach any truth value to palone (which means that it is not a logical proposition) but only to p(a), where a represents a state of a system, a statement p regarded after Birkhoff and von Neumann as a two-valued proposition actually turns out to be a many-valued propositional function defined on a set of states of a physical system. Of course there can exist also constant propositional functions of this form, i.e., statements which assume the same truth value on each state of a system. Two of them play particular role and obviously can be added to any family of propositional functions that describes any physical system: The always-false propositional function F and the always-true propositional function T which assume, respectively, truth values 0 and 1 on any state a of a physical system25 . It was proved by the author [68] that a set of propositional functions that possesses an algebraic structure of an orthomodular lattice (e.g., that is generated by a set of projection operators acting on a Hilbert space) is "minimal" in the sense that it does not contain any constant propositional function except F and T. However, many-valued models of recently studied more general quantum structures (orthoalgebras, effect algebras) do admit constant propositional functions other than F and T. Let us now consider what logical operations can be performed on a set of many-valued propositional functions associated to a given quantum system. The most obvious of them (and also causing least troubles) is negation. According to the original ideas of Birkhoff and von Neumann it is always possible to negate an experimental proposition p by interchanging meanings attached to the two possible outcomes of a dichotomic experiment designed to check p. The same procedure can be maintained to obtain negation -,p(.) of a propositional function p(.) also in our approach. It follows from our reinterpretation of Birkhoff-von Neumann probability a(p) as a truth value of a many-valued proposition p(a) that t(-,p(·)) = 1- t(p(·)). (34) Actually, whatever is a state a of a physical system, probability a( -,p) of getting the outcome "no" in any dichotomic experiment with random outcomes obviously equals 1 - a(p) where a(p) is the probability of 25 F can be linguistically expressed, for example, as: "If a system is in a state ex, then it does not exist" and T as: "If a system is in a state ex, then it exists", but there are also other linguistic expressions for F and T possible. 93 NON-CLASSICAL LOGICS getting the outcome "yes". Therefore, according to (33) we get for any state a of a physical system t(...,p(a)) = a(...,p) = 1- a(p) = 1- t(p(a)) (35) from which the general formula (34), which is in full accordance with the original Lukasiewicz formula (6), follows. The problem with a proper expressions for conjunctions and disjunctions of our propositional functions is much less trivial. The first difficulty consists in the abundance of possibilities mentioned at the end of Section 3. Secondly, the proposals made originally by Birkhoff and von Neumann [64] according to which conjunctions and disjunctions should be modelled, respectively, by order-theoretic operations of the greatest lower bound (g.l.b.) and the least upper bound (l.u.b.) with respect to the partial order relation introduced in the set of experimentally verifiable propositions, are so far from being satisfactory that they cannot guide us as they did in the case of negation. Actually, one of the features by which quantum systems differ from classical systems the most is the phenomenon expressed in famous Heisenberg Uncertainty Principle which says that there exist physical quantities that cannot be simultaneously measured with arbitrary accuracy. The standard example of such quantities is position x and momentum p of a microobject for which uncertainty relation takes form ~xp> h - 471" (36) where ~x and ~p denote, respectively, unavoidable uncertainties in finding the values of x and p in any conceivable experiment. It follows from (36) that if we manage to measure position of a microobject with great close to zero), then we do not know practically anything accuracy (~x about simultaneously measured value of momentum (~p very big) and vice versa. Therefore, although truth values of two experimentally verifiable propositions a = "the value of position is Xo ± ~x" and b = "the value of momentum is Po ± ~p" can be established in two independently performed experiments whatever and ~p, the conjunction "a AND b" is are the assumed uncertainties ~x not an experimentally verifiable proposition if ~xp < 4~ since there is 94 J AROSLA W PYKACZ no experimental possibility to measure simultaneously the position and the momentum of a microobject with arbitrary accuracy. Birkhoff and von Neumann were well aware that this fact makes the interpretation of g.l.b. and l.u.b. of two experimentally verifiable propositions as conjunction and disjunction questionable for a very simple reason: The family of all projection operators on a Hilbert space, which is an archetype of all "quantum logics" is a lattice, i.e., in this family g.l.b. and l.u.b. of any two elements exist, which should not be the case in view of just studied example. In other words, a "good" quantum logic should be a partial logic in a sense that it should be possible to form conjunctions and disjunctions only within some classes of propositions which are usually called compatible or simultaneously measurable. The other difficulty in attempts at treating g.l.b. and l.u.b. of two Birkhoff-von Neumann propositions as their conjunction and disjunction follows from the fact that, according to the logical parlance, these operations are not truth functional, which means that the knowledge of truth values of two propositions a and b is not sufficient to establish the truth values of their g.l.b. and l.u.b. This is not a very serious obstacle but, anyway, it would be better if we could calculate truth values of the conjunction "a AND b" and the disjunction "a OR b" from the truth values of a and b alone. Thus, we are left with the infinity of possible candidates for manyvalued conjunctions and disjunctions to be used for our propositional functions that describe probabilities of getting various results of future experiments performed on quantum systems. The task of choosing among them the right ones seems to be almost impossible. Of course, we can proceed a little bit by elimination: For example, the original Lukasiewicz disjunction (7) and conjunction (8) do not seem to be the proper ones since, as it was already mentioned in Section 1, they do not satisfy the law of excluded middle (10) and the law of contradiction (11), while algebraic counterparts of these laws hold true in Hilbert-space-based Birkhoff-von Neumann quantum logic. Happily, it occurs that Zawirski disjunction (12) and conjunction (13) do not only satisfy these laws, but it can be also shown [71, 72] that after suitable translation of Birkhoff-von Neumann logic, via families of fuzzy sets, into the language of infinite-valued Lukasiewicz logic [68, 69] these operations satisfy all the previously listed requirements best: They are only partially defined on families of propositional functions associated with a studied quantum system which, in view of previously listed arguments, is rather a virtue than a drawback. Moreover, it can be shown [72] that if they are defined on a given pair of propositional functions p(.), q(.), then Birkhoff-von Neumann counterparts of these NON-CLASSICAL LOGICS 95 propositional functions are compatible (i.e., simultaneously measurable) propositions (as they should be, of course) and that in this case Zawirski operations coincide with order-theoretic operations of g.l.b. and 1. u. b. that can be independently introduced on a given family of propositional functions. This finally explains why order-theoretic operations of g.1.b. and 1.u.b. were for such a long time regarded as proper models of quantum-logical conjunction and disjunction in spite of well-known interpretational difficulties encountered when they are applied to noncompatible propositions which already annoyed the founding fathers of the quantum logic theory. To sum up, according to the author, the present state-of-the-art is the following: It is a misunderstanding to speak about any "global" quantum logic, understood as a new kind of propositional calculus or a new way of reasoning, that could replace the old, good, two-valued Aristotelian logic in its role of being the base of scientific method of comprehending physical phenomena. However, there do exist "local" quantum logics consisting of propositional functions that predict results of not-yet-performed experiments. The "locality" of these logics follows from the fact that each of them consists of propositional functions that pertain to a specific physical system. These propositional functions are defined on a set of states of a studied physical system (which once more stresses the fact that the logic is "local": it consists only of statements about this physical system), and become propositions when a free variable in a propositional function is replaced by a definite state of a system. Since propositions obtained in this way speak about future events (results of experiments that will be performed on a physical system but, in the considered moment of time are not yet performed), these "local logics" are unavoidably manyvalued logics, otherwise all future events would be strictly determined. The meaning of truth values of these many-valued propositions is probabilistic, for example, t(p(a)) = 0,8 means that the probability that any experiment designed to check a proposition p when a physical system is in a state a will show that p is true equals 0,8. The mathematical results obtained in the course of "translating" Birkhoff-von Neumann quantum logics into the language of Lukasiewicz infinite-valued logic show that a family C consisting of all propositional functions of the form p( .) associated with any physical system has to fulfill the following conditions [69, 71]: (i) The always-false propositional function F and the always-true propositional function T belong to C. (ii) If a propositional function pO belongs to C, then also its negation .p(.) belongs to C, and for any state a, t(.p(a)) = 1- t(p(a)). 96 JAROSLAW PYKACZ (iii) For any (finite or countable) sequence of propositional functions {Pi (. )} that are pairwisely exclusive in the Zawirski sense26 there exists in ,c a propositional function that is a Zawirski disjunction (12) of all Pi(')' (iv) The always-false propositional function F is the only one propositional function in ,c that is exclusive27 with itself. It was proved by the author [69] that every quantum logic in the Birkhoff-von Neumann sense with so called ordering family of probability measures (which, in the case of logics associated with physical systems represent states of these systems) can be uniquely expressed in the form of such a family of propositional functions defined on its set of probability measures, and vice versa: Any family of many-valued propositional functions that satisfies conditions (i)-(iv) possess an algebraic structure that characterizes Birkhoff-von Neumann quantum logics. Therefore, Birkhoff-von Neumann quantum logic can be seen as being both non-distributive (which follows from its algebraic structure "inherited" from a Hilbert space) and many-valued logic, which finally unifies these two lines of thought existing in the quantum logic theory since its very beginning. ACKNOWLEDGMENTS Writing of this paper was financially supported by the joint PolishFlemish Project No. 007 (Poland) / VLW 12 (Flanders) and by the University of Gdansk Research Grant BW/5100-5-0276-7. The Author is very grateful to the whole staff of the Center "Leo Apostel" (Brussels) where the main part of this paper was written for the technical assistance and warm hospitality extended to him during his frequent stayings at CLEA, and to Mr. Wojciech Mostowski for producing computer drawings that illustrate the paper. 26 This means that Zawirski conjunctions (13) of all possible pairs of propositions Pi(a), pj(a) obtained from Pi(-) and pj(-) for i =1= j and for all possible states a of a physical system are false propositions, i.e., Vi#-/Vat(pi(a) npj(a» = 0, which can be equivalently denoted as Pi(') n pj(-) = F for all i =1= j. 27 In the Zawirski sense explained in the previous footnote. NON-CLASSICAL LOGICS 97 AFFILIA TION Jaroslaw Pykacz Instytut Matematyki Uniwersytet Gdanski 80-952 Gdansk, Poland pykacz@ksinet. univ. gda. pl REFERENCES [1] Lukasiewicz, J., "Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkiils", Comptes rendus des seances de ia Societe des Sciences et des Lettres de Varsovie, Cl. 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[52] Davisson, C.J. and Germer, L.H., "Diffraction of electrons by a crystal of nickel", Physical Review, 30, 1927, p. 705. [53] Wheeler, J.A., "Law without law", in: Wheeler, J.A. and Zurek, W.H. (eds.), Quantum Theory and measurement, Princeton University Press, Princeton, NJ., 1983, pp. 182-219. [54] Wooters, W.K. and Zurek, W.H., "Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle", Physical Review D, 19, 1979, pp. 473-484. [55] Mittelstaedt, P., Prieur, A., and Schieder, R., "Unsharp particle-wave duality in a photon split beam experiment", Foundations of Physics, 17, 1987, pp. 891-903. [56] Heisenberg, W., "Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen", Zeitschrijt fur Physik. 33, pp. 879-893. NON-CLASSICAL LOGICS 101 [57] Schrodinger, E., "Quantisierung als Eigenwertproblem", Annalen der Physik, 79, 1926, pp. 361-376; 499-507; 80, 1926, pp. 437-490; 81, 1926, pp. 109-139. [58] Born, M. and Jordan, P., "Zur Quantenmechanik", Zeitschrijt fur Physik, 34, pp. 858-888. [59] Born, M., Heisenberg, W. and Jordan, P., "Zur Quantenmechanik II", Zeitschrijt fUr Physik, 35, pp. 557-615. [60] Born, M., "Zur Quantenmechanik der Stossvorgange", Zeitschrijt fur Physik, 37, pp. 863-867. [61] Dirac, P.A.M., "The fundamental equations of quantum mechanics", Proc. Royal Society of London (AJ, 109, 1925, pp. 642-653. [62] Zwicky, F., "On a new type of reasoning and some of its possible consequences", Physical Review, 43, 1933, pp. 1031-1033. [63] Putnam, H., "Three-valued logic", Philosophical Studies, 8, 1957, pp.73-80. [64] Birkhoff, G. and von Neumann, J., "The logic of quantum mechanics", Annals of Mathematics, 37, 1936, pp. 823-843. [65] Stone, M.H., "The theory of representations for a Boolean algebra", Transactions of the American Mathematical Society, 40, 1936, pp. 37-111. [66] Aspect, A., Grangier, P., and Roger, G., "Experimental realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A new violation of Bell's inequality", Physical Review Letters, 49, 1982, pp. 9194. [67] Gleason, A., "Measures on the closed subspaces of a Hilbert space", Journal of Mathematics and Mechanics, 6, 1957, pp. 885-894. [68] Pykacz, J., "Fuzzy set ideas in quantum logics", International Journal of Theoretical Physics, 31,1992, pp. 1767-1783. [69] Pykacz, J., "Fuzzy quantum logics and infinite-valued Lukasiewicz logic". International Journal of Theoretical Physics, 33, 1994, pp. 14031416. [70] Pykacz, J., "Fuzzy quantum logics as a basis for quantum probability theory", International Journal of Theoretical Physics, 37, 1998, pp. 281-290. [71] Pykacz, J., "Attempts at the logical explanation of the waveparticle duality", in: Dalla Chiara, M.L., Giuntini, R., and Laudisa, F. (eds.), Language, Quantum, Music, Kluwer, Dordrecht, 1999, pp. 265-282. [72] Pykacz, J., manuscript in preparation. CLAUDIO GAROLA AGAINST "PARADOXES": A NEW QUANTUM PHILOSOPHY FOR QUANTUM MECHANICS l. INTRODUCTION It is a commonplace that XXth century physics has produced powerful new theories, such as Relativity and quantum mechanics, that upset the world view provided by XIXth century physics. But every physicist knows how difficult it may be to explain the basic aspects of these theories to people having a non-physical professional training. The main reason of this is that both Relativity and quantum mechanics are based on fundamental ideas that are not hard to grasp in themselves, but deeply contrast the primary categories on which our everyday thinking it; based, so that it is impossible to place relativistic and quantum results within the framework suggested by ordinary intuition and common sense. Yet, despite this similarity, there are some relevant differences between the difficulties arising in Relativity and in quantum mechanics. In order to understand this point better, let us focus our attention on Special Relativity first (analogous arguments can be forwarded by considering General Relativity). Here, the strange links between space and time following from the even more strange assumption that the velocity of light is independent of the motion of the observer conflict with the very simple conception of space and time implicit in our daily practice (and explicitly stated in classical Physics, think of Newton's "absolute space" and "absolute time"): but this conflict regards geometrical space-time models, not the very roots of our language, hence our thought. Then, let us consider quantum mechanics. Here it is a basic notion that properties of physical systems are nonobjective, in the sense that a property cannot be thought of as existing if a measurement of it is not performed. As Mermin [30] writes, "it is a fundamental quantum doctrine that a measurement does not, in general, reveal a preexisting value of the measured property" . This doctrine is expressed, commented on and elaborated in a huge number of books and papers, and it has been adopted more or less explicitly for seventy years in most arguments in quantum mechanics. Yet, one can easily realize that it challenges the basic procedures of our reasoning within a natural language. Indeed, a primary function of any language is attributing properties to things and deducing, via general 103 © 1999 Kluwer Academic Publishers. 104 CLAUDIO GAROLA laws, further properties. If the above quantum doctrine is accepted, this function must be completely reconsidered. The stronger impact of quantum mechanics on the deep structures of our reasoning is clearly witnessed by the fact that no physicist nowadays seriously challenges Special Relativity, while the debate on the interpretation and on the "paradoxes" of quantum mechanics is still lively. It is apparent that even experienced physicists may be in trouble when facing all consequences of the orthodox quantum philosophy. One could wonder whether it is worth maintaining the aforesaid quantum doctrine if it produces problems and paradoxes. But it is well known that [1] and the Bell-Kochen-Specker theorem [26], [30] that seem to prove some strange features of quantum systems from which this doctrine directly follows: thus, it appears to be imposed by technical results inside quantum mechanics itself rather than being an a priori philosophical choice, which implies that one cannot renounce it without rejecting all the heritage of knowledge provided by quantum physics. Therefore, most scholars concerned with the foundations of quantum mechanics accept the above doctrine, even when propounding an alternative to the orthodox interpretation or a modification of the theory, and some of them try to develop a consistent world view that embodies and "explains" the strange features of quantum systems mentioned above (these are mainly two: nonlocality, which means that a measurement on a particle 1 may instantaneously influence the properties of a particle 2, even if 1 and 2 are far away, if 1 and 2 have interacted in the past; contextuality, which means that a property of a given physical system in a given state may be asserted, or denied, or be meaningless, depending on the measurement context). But up to now no proposal has achieved the general consent of the scientific community, so that the situation is still unsatisfactory from several viewpoints. However, there are still scarcely explored ways of coping with quantum paradoxes. In particular, the belief that nonlocality and contextuality are technical results that cannot be avoided without messing up quantum mechanics can be criticized. This point is crucial, and in order to make it clear a somewhat detailed discussion is needed. Consider first any physical theory. According to a largely accepted epistemological conception (received viewpoint, e.g., Braithwaite, [6], or Hempel, [25]) this theory can be schematically described by saying that it consists of: (i) a mathematical (more generally, formal) apparatus; (ii) a domain of facts (the observative domain); (iii) a partial interpretation, which makes some (not generally all) terms of the mathematical apparatus correspond to observed, or ob- A NEW QU ANTUM PHILOSOPHY FOR QM 105 servable, facts and relations among facts in the observative domain; (iv) a space-time model, which is very important in order to grasp intuitively the content of the theory and to conceive new developments of it (the existence of a model is not retained however a necessary requisite of any physical theory). Among these constitutive elements, the interpretation, in particular, strongly depends on the (implicit or explicit) epistemological choices underlying the theory (for instance, if one retains, with Galileo, that "nature is written in mathematical terms", he will be inclined to interpret all terms of the mathematical apparatus as describing some actually existing, even if sometimes elusive, physical entity, to consider the mathematical relations in the theory as expressions of objective laws of nature and, finally, to retain that the space-time model is a faithful, though possibly approximate, representation of the actual world). Hence, the way in which a mathematical result is interpreted may change when the epistemological premises are changed. If this conclusion is applied to the special case of quantum mechanics, one realizes that the deduction of nonlocality and contextuality, hence of the fundamental quantum doctrine reported above, from the Bell and Bell-Kochen-Specker theorems, is based on an interpretation that follows from the standard philosophy of quantum mechanics. But this philosophy contains, in particular, the fundamental quantum doctrine itself, so that the claim that this follows from the Bell and Bell-Kochen-Specker theorems creates a conceptual loop. Therefore, one can still imagine that some changes in the standard philosophy might lead one to conceive a new interpretation of quantum mechanics which does not imply, on one side, a modification of its formal apparatus, thus saving the core of the quantum description of the world, and which, on the other side, allows one to read the Bell and Bell-Kochen-Specker results in a non standard way, thus avoiding "paradoxes" as nonlocality and contextuality and reconciling quantum physics with the basic procedures of everyday physicist's and mathematician's reasoning (it must be stressed that this does not mean to propound again classical models for physical reality, since even a classically structured language can be used in order to describe an infinity of non-classical models). However, the idea of introducing philosophical changes in order to solve some problems in quantum mechanics would probably be regarded with annoyance by many pragmatically oriented physicists, who consider philosophical reasoning "practically irrelevant", or even harmful, for physicists, and tolerate it only if devoted to elucidate and legitimate the autonomous progress of science. The arguments offered above should already convince the reader of the naIvety of this position. In addition, 106 CLAUDIO GAROLA one can remind that the birth of quantum mechanics was accompanied by a lively epistemological debate on the aims of physics, the basic requirements that a physical theory must fulfil, the kind of sentences that can be accepted as meaningful in its language, and so on. The famous paper written by Einstein, Podolski and Rosen in 1935 [9] (see also Bohm, [3]), in which the authors intended to show that quantum mechanics was not a complete physical theory, and Bohr's answer to it [5], defending the opposite thesis, can be taken as typical examples of this debate. Therefore, it is apparent that the founders of quantum mechanics were well aware of the relevance of the epistemological background on which physical theories are constructed, even from a strictly technical viewpoint (indeed, the way in which physicists collect, select and organize physical data is largely affected by their underlying convictions about physics, methodology and knowledge, which also guide the choice of mathematical tools and theoretical entities). On the other hand their convictions have not to be taken as definitive and eternal truths, even because the links between the epistemological premises and the body of a physical theory are not one-to-one, which makes every theory (hence, in particular, quantum mechanics) liable to different interpretations, or modifiable so as to fit in with different conceptual frameworks. The bias against the philosophical reflection on the roots of quantum mechanics, on the contrary, leads, on the one hand, to underestimation of the relevance of the epistemological premises in this theory, and, on the other hand, to implicit acceptance (sometimes with incongruity and misunderstanding) of the viewpoint inherited from the tradition, making it unquestionable. The above remarks being taken into consideration, the rest of this paper will deal with a recent attempt at developing the idea of coping with the paradoxes of quantum mechanics by analyzing, criticizing and then changing the epistemological premises more than the formal apparatus of this theory. If looked at without prejudices, a solution of this kind should be rather appealing for physicists, since it could lead to avoiding paradoxes without renouncing all the patrimony of acknowledged achievements of quantum mechanics; furthermore, it will be seen that the change of the basic paradigma of this theory provides some interesting suggestions on possible ways of going beyond it that have never been explored before, even by the authors who tried to produce alternative theories or interpretations (see, e.g., Garola, [15], [16], [17]; Garola and Solombrino, [22], [23]). A NEW QUA N TUM PHI LOS 0 P H Y FOR Q M 107 2. QU ANTUM MECHANICS AND TRUTH To follow the program described at the end of the Introduction, it is convenient to begin with some general considerations that will constitute a standpoint for a critical analysis of the standard quantum philosophy. Whenever one starts reasoning (which is one of the basic functions of the language), one adopts, consciously or not, some general procedures, which can be summarily described as follows. Firstly, he explicitly asserts or implicitly accepts some preliminary statements (or premises), part of which are factual statements ("Socrates is a man" , in a famous syllogism of Aristotelian logic), part are general statements ("all men are mortal"). Then, he deduces some consequences from these premises ("Socrates is mortal"). This deduction cannot be arbitrary (a reasoning that deduces from the above premises that "Socrates is Chinese" would not be accepted), hence it follows some rules (deduction rules) that can be analyzed and made explicit. In order to understand the deep nature of these rules, note that when asserting or accepting some preliminary statements we mean that these statements are true; furthermore, our conclusions are also retained to be true if the deduction procedure is correct. Thus, one can maintain that the deduction rules, whatever they may be, are used in order to recognize that some new statements are true, being based on the knowledge that some initial statements (the premises) are true. The explicit form of the deduction rules has been studied for a long time, and this research has generated a huge body of knowledge and theories, classical Logic included. In the context of logical studies, the classical concept of truth supported by classical Logic plays a crucial role, and it is well known that it has been elucidated and formalized by the Tarski theory of truth (see, e.g., Tarski, [37], [38]). It is then important to point out some fundamental features of this classical concept that are relevant to our aims in this paper. (i) The definition of a truth value (true/false) for the statements of a language is given by means of a set theoretical model. Whenever the language is such that this model applies, any statement in it has a truth value, independently of any assumption on the actual existence of the entities mentioned in the statement itself ("a unicorn is a feline" is false in the context of a story, even if the actual existence of unicorns can be seriously questioned). But the fact that a truth value is defined does not mean that it is known, or that it can be obtained by means of some (empirical or logical) procedures: truth and access to truth are different concepts. This is a crucial point that must be carefully pondered, since it frequently occurs in the everyday life that we sometimes separate and sometimes identify these concepts. In order to grasp it better, one can 108 CLAUDIO GAROLA resort to a mathematical analogy: one can well prove that the solution of a given differential equation exists, but he can have no means for actually producing that solution. Furthermore, think of the statement "this match is wooden and completely combustible". If one decides to verify first whether the second statement is true, he will not be able to verify, after burning the match, whether the first statement is true: nevertheless, he is however sure that the first statement has a truth value, that he could easily check if he decided to consider it first. It follows from the above remarks that one can plainly talk in classical Logic of the truth value of a statement without implying that he possesses a procedure for getting to know that value. (ii) Intuitively, the set theoretical model provided by Tarski for defining the concept of truth can be summarily described as follows. The statements of a language can be divided into elementary (or atomic) statements and complex statements. Elementary statements state a property of an individual object (or a relation among some individual objects, but this more general case will not be considered here for the sake of simplicity), e.g., "sun is bright". Complex statements are obtained by using elementary statements and connectives as "not", "and", "or", "implies" , or quantifiers as "for every", "exists". Then, in the set of all individual objects considered in the language, or universe of the language, a property is represented by a subset, and an elementary statement which attributes a property to a given object is true if the object belongs to the subset representing the property, false otherwise. Furthermore, the negation of the same statement, obtained by using the connective "not", is true if the object belongs to the complement in the universe of the set representing the property, false otherwise; the conjunction of two statements, obtained by using the connective "and", each attributing a property to the same individual object, is true if the object belongs to the intersection of the two sets representing the two properties, false otherwise; the disjunction of the same two statements, obtained by using the connective "or", is true if the object belongs to the union of the two sets representing the two properties, false otherwise; a complex statement which states that all individual objects having a given property P also have another property Q is true if the subset representing P is contained in the subset representing Q, false otherwise; and so on. By means of the above definitions one can obtain rules for attributing truth values to complex statements whenever the truth values of all elementary statements that appear in them are known (a simple example of rules of this kind is provided by the famous truth tables of the part of classical Logic called Propositional Logic). A NEW QU ANTUM PHILOSOPHY FOR QM 109 (iii) By combining the rules considered at the end of (ii), one can obtain deduction rules. These can be considered procedures for attaining knowledge of the truth of some given statements, whose truth values are defined but unknown, on the basis of the knowledge of the truth of some other statements (the premises), consistently. with our initial remarks on the deduction process in natural languages. Furthermore, one can introduce a logical order (actually, a pre-order) on the set of all statements, defined as follows: a given statement is smaller than another given statement if, whenever the former is true, then the latter is true. (iv) As it has already been stressed, the premises on which one applies a deduction procedure usually are not homogeneous: for, some describe factual situations, some state general laws. These laws usually are part of a theory, and it is apparent that this theory can be complete, i.e., such that its laws, together with suitable sets of factual premises, are sufficient for determining the unknown truth value of any statement of the language, or incomplete if this does not occur. The above features of classical Logic provide an intuitive insight of the way in which this discipline explains and systematizes the inference procedures of natural languages. However, these procedures are so complex and sometimes seemingly contradictory that the theoretical settlement supplied by classical Logic is not uncontroversial. It is well known, for instance, that the verificationist conceptions of truth deny the possibility itself of defining truth independently of the procedures that lead one to determine the truth values of the statements of a language: in other words, a truth value is defined for a given statement only if a procedure exists which allows one to determine effectively whether the statement itself is true or false. A relevant example of a formal logic that adopts a verificationist concept of truth is provided by Intuitionistic Logic, the position of which will be called logical verificationism in the following, since it collapses the concept of truth into the concept of logical verification (at least in the case of complex statements: elementary statements must have empirical proofs, see, e.g., Dalla Pozza and Garola, [8]). It is then important to note that logical verificationism does not simply reduce the set of statements that can be thought of as having a truth value, but implies deep changes in the logical rules themselves that enter the deductive process. By imitating this illustrious example, the standard interpretation of quantum mechanics adopts a similar attitude requiring an empirical proof in every case. Indeed, the standard philosophy of quantum mechanics regards as "metaphysical" every attempt of introducing physical properties or entities which, in principle, cannot be observed, and deprives of meaning all statements in a theory whose truth or falsity 110 CLAUDIO GAROLA cannot be checked by means of suitable measurements. To be precise, it maintains that no statement which cannot be verified by means of a suitable measurement procedure has a truth value and can be accepted as meaningful in the language of physics. The above quantum philosophy, that will be called empirical verificationism in the following since it collapses the concept of truth into the concept of empirical verification, both for elementary and for complex statements, may seem at first glance to express only a natural physicist's refusal of statements that cannot be justified on an experimental ground. But a deeper insight shows that it has a number of traumatic consequences that make it far more problematical than the logical verificationism of Intuitionistic Logic. Thus, we dedicate the next section to a brief discussion of some of these consequences that are relevant for our purposes. 3. THE SN ARES OF EMPIRICAL VERIFICATIONISM As we have anticipated at the end of Section 2, empirical verificationism has a number of consequences that are epistemologically problematical. Let us discuss some of them. (i) The fact that a statement has or not a truth value depends, according to empirical verificationism, on its verifiableness (or testability), hence it depends on the measurement theory that is adopted, which is a physical theory. But eliminating all statements that are meaningless since they are not verifiable entails radical changes in the structure of our language, hence, in particular, it modifies the deduction rules, which thus turn out to depend on the physical theory that one wants to express by means of the language itself. This entails that one cannot state any a priori rationality criterion which is independent of the theory. One can summarily say that the collapse of truth and empirical access to truth produces the collapse oflogic into physics (see, e.g., Putnam, [33]; Finkelstein, [10], [11]; we insist again on the fact that this goes well beyond the position of Intuitionistic Logic, where the distinction between the logic of a theory and the theory itself can be preserved: this is witnessed by the fact that the "logic of quantum mechanics" , or Quantum Logic, has a mathematical structure which is different from that of Intuitionistic Logic). This collapse can be seen as a primary source of quantum paradoxes: for, many of these follow from the attempt at applying physical rules to logical arguments. (ii) The specific theory that is being considered, i.e., quantum mechanics, produces a disconcerting situation whenever the above verificationist A NEW QUANTUM PHILOSOPHY FOR QM 111 criterion of meaning is applied to elementary statements that attribute properties (equivalently, measurement values) to individual samples of physical systems. Indeed, these statements cannot be neatly divided once and for all into meaningful and meaningless. On the contrary, there are physical contexts (that can be chosen by the observer himself) in which some statements are meaningful, other meaningless, and different physical contexts in which the former statements are meaningless, the latter meaningful. This can be easily understood if one reminds the famous Heisenberg uncertainty principle: indeed, loosely speaking, it follows from this principle that, if the position of a given particle is completely known, then it is meaningless to say that it also has a (even unknown) velocity (to be precise, a momentum), and, conversely, if the velocity of the same particle is completely known, then it is meaningless to say that it also has a (even unknown) position. But in this way subjectivity enters into physics, at least in the sense that, by choosing the physical verification context, one chooses which properties are meaningful and which are meaningless for the inquired object. From this viewpoint, such strange features of quantum mechanics as nonlocality and contextuality (see Section 1) can be guessed even without the aid of the deep theorems of Bell and Bell-Kochen-Specker. (iii) Consider a molecular complex statement, i.e., a statement which is obtained by connecting elementary statements attributing properties to a given individual sample of a physical system by means of connectives as "and", "or", etc .. The general principle of empirical verificationism establishes that this statement is meaningful if and only if it can be verified, that is, if and only if there exists an apparatus (possibly compounded of many sub-apparatuses, suitably linked by means of electronic or mechanical connections) which yields one of the outcomes true andfalse whenever applied to the sample that is being inquired. But, if this occurs, the measuring apparatus can be looked at as testing a physical property: hence, the complex statement that we are considering is logically equivalent to an elementary statement that attributes this property to the given sample. This implies that the complex statement itself is meaningful if and only if it is equivalent to an elementary statement, which constitutes a particular but relevant aspect of the collapse of the logical structure induced by empirical verificationism and discussed in (i). The above consequences of the basic philosophy of quantum mechanics have deeply influenced the epistemological thought of our century, and many speculations can be classified as attempts of coexisting with them. But one can also legitimately wonder whether empirical verificationism is epistemologically sound and, more important, if it is actually essential 112 CLAUDIO GAROLA in physics; in particular, one can wonder whether it is so inherent to quantum mechanics that it could not be renounced without losing all physical knowledge provided by this theory. Now, the answer to all these questions is negative in our opinion. Let us discuss this decisive point in some details. Firstly, consider the problem of philosophical soundness. It is well known that a number of logicians and epistemologists (see, e.g., Russell, [34]; Popper, [32]) have argued against the verificationist concept of truth, observing that the concept of verification itself presupposes that something is verified, which is just the truth of the statement that is being considered: thus, verification and truth cannot be identified. This rather abstract objection (which however is culturally relevant, since collapsing concepts that have been disentangled after centuries of philosophical efforts can hardly be considered a progress) applies to every form of verificationism (in particular, to logical and empirical verificationism). But in the case of empirical verification ism one can add some further criticism: for instance, one can observe that the standard justification for adopting this position, that is, freeing physics from old metaphysical hindrances, actually reveals the incapability of distinguishing between the semantic concept of truth (which can be defined by means of a set-theoretical model, as in classical Logic) and the ontological concept, according to which asserting that a statement is true means to postulate the actual existence of the objects that are mentioned in it. Secondly, consider the possibility of avoiding a verificationist position in quantum mechanics. This appears to be precluded if one accepts the fundamental quantum doctrine of nonobjectivity of physical properties reported in the Introduction, which seems unavoidable since it is supported by some technical results (the Bell and Bell-Kochen-Specker theorems) that are internal to quantum mechanics itself. But it has already been observed that the support to the fundamental quantum doctrine provided by these results depends on an interpretation that contains the quantum doctrine itself. If this loop is broken, a number of possible alternatives can be conceived to the adoption of empirical verificationism as the basic philosophy of quantum mechanics, one of which will be discussed in detail in the next Sections. 4. A L T ERN A T I V EST 0 EM P I RIC A L V E R I F I CAT ION ISM The difficulties inherent in the orthodox interpretation of quantum mechanics and in all interpretations that basically accept empirical verificationism are largely discussed in scientific literature. But, usually, they A NEW QUA N TUM PHIL 0 SOP H Y FOR Q M 113 are not explicitly ascribed to the adoption of empirical verificationism itself. Furthermore, the alternative interpretations that could occur if this doctrine is abandoned are not seriously taken into account, since the identification between the semantic and the ontological concept of truth mentioned at the end of Section 2 leads most scholars to retain that they are necessarily "metaphysical" or ingenuously realistic. These attitudes are maintained even in very sophisticated and advanced studies in quantum mechanics. As an instance, let us consider the recent book on the quantum theory of measurement by Busch, Lahti and Mittelstaedt [7]. In the first Chapter the Authors discuss the difficulties encountered by every interpretation of quantum mechanics which is based on the following assumptions: (i) quantum mechanics deals with properties of individual systems and not only with measurement outcomes; (ii) quantum mechanics is a complete theory (see Section 2, (iv); the notion of completeness can be extended beyond the framework of classical Logic, and intuitively means that quantum mechanics says all what is meaningful to say about the physical systems that it describes); (iii) the properties of physical systems are non-objective in the sense specified by the fundamental quantum doctrine illustrated in the Introduction. The authors point out that these assumptions entail, in particular, that one has to face the objectification problem, that is "the question of how definite measurement outcomes are obtained", since the properties are non-objective. Then, they observe that some interpretations of quantum mechanics, as the many-worlds, witnessing and modal interpretations, as well as those stressing the decisive status of the observer, "can be read as various attempts to live with the insolubility of the objectification problem" , and that it may finally turn out that only a kind of unsharp objectification is possible if the above assumptions are maintained: but they do not recognize explicitly that the objectification problem follows from adopting empirical verificationism, which implies a number of consequences that go beyond the objectification problem itself. In addition, Busch, Lahti and Mittelstaedt take into account the possible alternatives that keep the above-mentioned fundamental assumption (i), while abandoning (ii) and (iii), that is, the objectivity/incompleteness interpretations that renounce the requisite of completeness of quantum mechanics but repropose its "objectivity": yet, the different possible meanings of the latter term are ncit analyzed, and this may lead the reader to interpret it in the ontological sense, hence to reject immediately these interpretations as "metaphysical". 114 CLA UDIO GAROLA It must be also noted that a further bias against objectivity/incompleteness interpretations arises because these interpretations are often identified with hidden variables theories (i.e., theories that attempt to explain quantum mechanics in terms of underlying variables that have a classical behaviour but are impossible to observe), which have been for a long time a taboo for every well-bred physicist. Owing to the above-mentioned reasons most scholars conclude that no proper alternative exists for accepting assumptions (ii) and (iii), even if they lead to still unsolved problems. However, this conclusion can be rejected if the arguments on which it is based are confuted. Let us discuss this confutation in detail. First, in the objectivity/incompleteness interpretations the term "objectivity" may have different meanings. To be precise, it can be intended in an ontological sense, as seen above, which means in particular that the theoretical entities of quantum mechanics represent actually existing entities. But it can also be interpreted in a semantic sense, which means that one is allowed to state meaningful complex statements about the properties of physical systems in quantum mechanics independently of any direct measurement process, and even independently of any ontological assumption. This alternative follows from the possibility of defining the truth value of a given complex statement by means of a settheoretical model or by means of logical procedures, as we have seen in Section 2, but in any case independently of the existence of an empirical procedure that allows one to determine the value itself by means of a measurement. Second, the objectivity/incompleteness interpretations cannot be identified with hidden variables theories. Indeed, Kochen and Specker [26] have proved that a hidden variables model exists even in the case of orthodox quantum mechanics if the conditions imposed on hidden variables for retaining them physically acceptable are sufficiently weak. More generally, one can conceive contextual and/or nonlocal hidden variables theories (see, e.g., Mermin, [30]) whose "objectivity" is problematical (the measuring apparatus testing a physical object which has interacted with other objects in the past may instantaneously modify the properties of these objects, even if they are far away). On the other hand, we shall see that objectivity/incompleteness interpretations exist that have no hidden variables models which satisfy the more restrictive conditions required by Kochen and Specker "for the successful introduction of hidden variables" in the paper quoted above. Our confutation opens the way in which the objectivity /incompleteness interpretations that escape the standard criticism can be introduced. The desirability of an approach of this sort to quantum me- A NEW QU ANTUM PHILOSOPHY FOR QM 115 chanics clearly follows from the paradoxical consequences of empirical verificationism pointed out in Section 3, (i), (ii), (iii), and from the epistemological and logical arguments that can be offered against it (see again Section 3). Therefore, we purpose to present in the next Sections the general lines of a recent approach to physical theories, called Semantic Realism, in which empirical verificationism is given up and the term "objectivity" is intended in a purely semantic and non-ontological sense. We shall see that, within this approach, quantum mechanics turns out to be incomplete, so that the interpretation suggested by Semantic Realism actually belongs to the class of objectivity/incompleteness interpretations. 5. THE BASIC IDEAS OF SEMANTIC REALISM The main aim of Semantic Realism is to provide a general .logical and linguistic framework suitable for expressing a wide set of physical theories (classical and quantum mechanics in particular). Its basic choice is the rejection of empirical verificationism. But, of course, this choice is not sufficient for characterizing the philosophy of Semantic Realism completely, since, in particular, it does not specify the logic that one has to adopt when constructing a language for physics. However, the criticism at the end of Section 3, which applies to every form of verificationism, suggests that one should avoid all logics based on a verificationist concept of truth (hence, in particular, to avoid Intuitionistic Logic). On the other hand, the convenience of making quantum reasoning closer to everyday physicist's and mathematician's reasoning (see Section 1) strongly recommends the adoption of a classical theory of truth, hence of classical Logic. Thus, this adoption is explicitly done by Semantic Realism. The natural subsequent step consists in constructing a language for expressing physical theories that has a classical logical structure (this language must prove to be able, in particular, to provide a new interpretation and to express properly quantum mechanics in order to get over the incorrect but unavoidable objection that adopting classical Logic constitutes a step backward, which reintroduces classical models and/or ingenuous realism). This could be done informally, as usual in physics, by choosing a suitable part (or fragment) of a natural language and introducing into it new technical terms, whose meaning is defined in a precise and accurate way: the mathematical apparatus of a specific physical theory would then be formulated and formalized by means of this enriched part of the natural language, and the interpretation of the old and new 116 CLAUDIO GAROLA terms of the language would provide the core of the partial interpretation of the theory into the observative domain (see the schematic description of physical theories supplied in the Introduction). However, this informal procedure would not prevent all semantic ambiguities inherent in the use of a natural language, which might be decisive when dealing with paradoxes that can be avoided only by introducing subtle distinctions requiring semantic exactness. Thus, one could think it convenient to provide a complete formalization of the language of physics (not only of its mathematical apparatus): but this would be exceedingly complicated, and produce a cumbersome formal structure that would hinder intuitive reasoning. Therefore, Semantic Realism adopts a compromise, accepting the standard language of physics as its general language and formalizing mainly that part of it which is interpreted on the observative domain (the observative language: to be precise, Semantic Realism aims to construct a classical formal language in which the observative languages of a class of physical theories that contains classical and quantum mechanics can be regimented). Let us try to give an intuitive insight into the basic ideas inspiring this formalization and its use for coping with old and new quantum paradoxes. In order to do this, let us begin with some primitive notions that refer to the observative domain on which the formalized language must be interpreted. (i) First, one introduces the notion of laboratory. This simply is a space-time domain in the actual world, i.e., a portion of space (e.g., a given room) associated with an interval of time (e.g., one day). The convenience of introducing this notion is made apparent by some obvious remarks. Indeed, all physical experiments are performed in a certain place at a given time, and they are are often repeated (in the same place, at different times) in order to control their results again and again. They must be intersubjective (i.e., different experimenters in different places and at different times must be able to reproduce them). Therefore, one is naturally led to concentrate his attention on those portions of space and time in which physical experiments are performed, introducing the above notion of laboratory (but it must be noted that this is somewhat different from the non-technical notion used in natural languages, which usually does not make explicit reference to a time interval). (ii) Second, one introduces the notions of preparation and dichotomic registering device. In order to understand them, think of the simplest experiment (yes-no experiment) that can be conceived. Intuitively, this consists of a device (the preparation) that prepares an individual sample of a given physical system (which can be macroscopic, as a tennis ball, or microscopic, as a photon or an electron), and of a second device A NEW QUANTUM PHILOSOPHY FOR QM 117 (the dichotomic registering device) that performs a test on the sample, yielding one of two possible outcomes (yes/no, true/false, 1/0). Thus, the notions of preparation and dichotomic registering device come out in a natural way and their introduction is justified. The above scheme for a yes-no experiment (which is inspired by Ludwig's analysis of physical experiments [28]) needs however some further comments. Therefore, let us note that actual experiments, which usually are far more complicate, can be often thought of as composed of a number of elementary yes-no experiments that are performed at a given time, and then repeated at different times. It follows that the elementary experiment described above does not represent a particular case, but plays a central role in the conceptual construction of a theory. Furthermore, let us stress that the reference to an individual sample of a given physical system can be eliminated if one wants to avoid the slanderous accusation of introducing metaphysics into physics by assuming the actual existence of some microscopic objects. Indeed, it suffices to correlate directly the act of preparing with the outcome provided by the dichotomic registering device, avoiding the reference to an intermediate physical system. This choice would however be very expensive, both with concern to language and to conceptual economy. Therefore, Semantic Realism avoids it, and introduces the specific term physical object in place of the more cumbersome expression "individual sample of a given physical system": the reader may freely decide whether this new term is only a shortcut or actually denotes some existing entity. (iii) By using the basic notions discussed in (i) and (ii), one can introduce a number of derived notions that make the formalization process carried out by Semantic Realism intuitively clear. In order to do this, let us note first that in every physical theory there are preparations that are considered physically equivalent. Thus, preparations can be grouped into equivalence classes, and these classes are called states. Analogously, there are dichotomic registering devices that are considered physically equivalent. Thus, dichotomic registering devices can be grouped into equivalence classes, and these classes are called effects. The derived notions of states and effects are basic in Semantic Realism. Therefore, we note explicitly that they depend, through the concept of physical equivalence, on the theory whose observative language is being formalized, and that, in spite of this, the abstract structure of the language itself will not depend on the theory. Moreover, a general definition of physical equivalence for preparations and dichotomic registering devices will be provided in Section 6, (iii). (iv) Let us look at a given laboratory i and consider all physical objects that are prepared in it by repeating the same preparation at different 118 CLAUDIO GAROLA times or by activating different preparations. The (finite) set of all these objects is then called the domain Di of i. Inside this domain, one can select all objects that have been prepared by preparations belonging to a given state S: the subset of all these objects is called the extension of S in i, and if x is a physical object belonging to this extension, one briefly says that "x is in the state S (in i)". Analogously, one can select every object that would give answer yes if a test should be performed on it by means of a dichotomic registering device belonging to a given effect F: the subset of all these objects is called the extension of F in i, and if x is a physical object belonging to this extension, one briefly says that "x would induce answer yes (or true, or 1) in the effect F (in i)". It is important to observe that a physicist adhering to the standard quantum philosophy would object that the above definition of the extension of an effect F is not acceptable. Indeed, one can decide whether a given physical object x belongs to the extension ofF by performing a test on it by means of a dichotomic registering device that belongs to F. But it is well known that there are in quantum mechanics non-compatible (i.e., mutually exclusive) dichotomic registering devices: hence, if one has performed the above test, he can never know whether x belongs to the extension of another effect G made up by dichotomic registering devices that are non-compatible with those in F. It follows that the extension of G is not defined, for it has no meaning to refer to what "would have happened" if one had performed a test by means of a dichotomic registering device belonging to G. Since, of course, the argument holds even while interchanging F and G, the orthodox quantum physicist concludes that one cannot associate a definite extension to every effect. The above conclusion, however, clearly follows from the fundamental quantum doctrine illustrated in the Introduction, which ensues from adopting empirical verificationism. The contrary assumption that the extensions of effects are defined subtends the rejection of this philosophy, and is based on the fact that the extension of any effect F can be actually exhibited in a laboratory i by performing a test by means of a dichotomic registering device belonging to F on all physical objects in the domain D i . This is considered sufficient, according to Semantic Realism, for maintaining that the extensions of all effects are defined even if they cannot be exhibited conjointly. (v) From the viewpoint of logic, the definitions in (iv) allow one to construct a formal language L that has the set of states and the sets of effects as sets of predicates, standard connectives, quantifiers and formation rules, and which furthermore is endowed with a built in classical logical structure. Indeed, in the laboratory i the domain Di is the "universe" for this language in the sense specified in Section 2, (ii). Fur- A NEW QU ANTUM PHILOSOPHY FOR QM 119 thermore, states and effects are represented by their extensions, that are subsets of the domain Di, just as the "properties" in Section 2, (ii), are represented by subsets of the universe. Thus, one assumes that an elementary statement of the form: S(x): the physical object x is in the state S, or F(x): the physical object x would induce answer yes in the effect F, is true if and only if the object x belongs to the extension in i of S, or F, respectively. The truth of complex statements obtained by using elementary statements, connectives and quantifiers, is then defined by means of standard conventions in classical Logic (see again Section 2, (ii); it is important to remind that these conventions define the truth of complex statements independently of the existence of empirical or logical procedures that allow one to determine effectively the truth values of the statements themselves). The above remarks conclude the illustration of the basic ideas that inspire the attempt, carried out by Semantic Realism, at providing a classical formal language in which the observative languages of classical and quantum mechanics, in particular, can be regimented. However, one must go farther if he wants to use the language L for treating problems in these theories. More specifically, there are properties of states and effects, and relations among them, that hold both in classical and in quantum mechanics and that must be explicitly analyzed and possibly expressed by means of L. We intend to provide in the next Section an intuitive outline of these properties and relations. 6. S TAT E S, E F FEe T S, AND QUA N TUM LOG I C In order to realize the program presented at the end of Section 5, let us discuss some fundamental features of states and effects within Semantic Realism, also introducing a number of derived definitions that are useful in the following. We note that some items in this section are rather technical, especially (vii), (viii) and (ix), even if we try to discuss them in a non-technical way. We apologize for bothering the reader with a number of details: but, unfortunately, they are important, since the roots of some paradoxes in quantum mechanics can be found in some subtle consequences of the standard quantum philosophy criticized in Sections 2 and 3. However, we have tried to make the rest of this paper understandable even if some of the technical points discussed in this section are not completely clear to the reader. 120 CLAUDIO GAROLA (i) It follows from the definition of extension of a state S that the extensions of different states in a laboratory i have empty intersection. Indeed, no actual physical object in the domain Di of a laboratory i can be prepared by more than one preparation, hence no physical object can belong to more than one extension of state. Moreover, the union of all extensions of states in Di coincides with Di (mathematicians would say that the set of all extensions is a partition of D i ; it must be noted that this is a typical feature of Semantic Realism, which does not occur in other axiomatic approaches, as, for instance, Ludwig's). On the contrary, the extensions of two (or more) different effects may have non-empty intersections. Indeed, it may occur that a given physical object would induce answer yes in any dichotomic registering device belonging to one of the effects, and also in any dichotomic registering device belonging to the other. Taking into consideration the above remark, the extensions of effects can be used for introducing an order relation on the set of all effects. Indeed, one says that an effect F is smaller than an effect G if and only if the extension of F is contained in the extension of G in every laboratory. This order is partial, which intuitively means that not every pair of effects is such that one of its members is smaller than the other; thus, one obtains the partially ordered set (briefly poset) of all effects, which is (because of its interpretation and structure) an effect algebra in a sense made familiar to scholars through the literature on the foundations of quantum mechanics. It must be stressed that the order defined on the set of all effects is empirical, in the sense that it depends on the physical relations that occur among different effects. However, whenever F is smaller than G, in every laboratory a physical object that belongs to F also belongs to G: thus, if the statement F(x) is true, then also G(x) is true. A logician would say that F(x) is smaller than G(x) according to the standard definition of logical order in a formalized language (see Section 2, (iii)). This point should be carefully pondered. Indeed, the same logician would also say that the statement F(x) is smaller than F(x) or G(x), since, again, if F(x) is true, then F(x) or G(x) is true. But, now, this conclusion is independent of any empirical relation between F and G, since it depends only on the conventional definition of the connective or. Thus, one must be well aware that the standard "logical order" contains both an empirical and a strictly logical part, which is a relevant distinction in the following. (ii) The yes/no outcomes in every dichotomic registering device can be exchanged, thus obtaining a new dichotomic registering device that is obviously inequivalent to the original one. If the exchange of outcomes is A NEW QU ANTUM PHILOS OPHY FOR QM 121 done in all devices belonging to a given effect F, one obtains a new effect, say FC, that can be called the complement of F. In every laboratory i, the extension of FC obviously is the complement in Di of the extension of F. Moreover, a statement of the form FC(x): the physical object x would induce answer yes in FC is true if and only if the statement F(x): the physical object x would induce answer yes in F is false. This implies that the elementary statement FC(x) is logically equivalent to the complex statement not F(x) (or, symbolically, -,F(x)), which constitutes an important instance of logical equivalence between elementary and complex statements. (iii) It is convenient to select only preparations and dichotomic registering devices that yield definite frequencies when carrying out yes-no experiments. More precisely, let a given preparation be repeated a great number of times in a given laboratory, so as to produce an ensemble of identically prepared physical objects (hence, of physical objects that are in the same state), and let a dichotomic registering device be used in order to perform a test successively on all physical objects in the ensemble. The ratio between the number of yes answer that one has obtained and the number of elements in the ensemble provides the frequency of the yes answer. One can then repeat the whole experiment in another laboratory, obtaining a new frequency. There is no a priori reason for expecting that the two frequencies coincide. But one convenes now that preparations and dichotomic registering devices are chosen in such a way that this actually occurs whenever the ensembles contain "a great number" of physical objects; in other words, we convene that one can define a probability of the yes outcome as the limit (in the statistical sense) to which frequencies approach whenever the number of elements in any considered ensemble becomes larger and larger. The above convention allows one to introduce some assumptions that are physically quite natural. More specifically, let us consider two preparations that produce the same frequencies for every dichotomic registering device whenever an experiment of the kind described above is done: then, we assume that the two preparations are physically equivalent. Analogously, two dichotomic registering devices that produce the same frequencies for every preparation are assumed to be physically equivalent. Thus, one gets two explicit definitions of physical equivalence which allow one to decide whether some preparations or some dichotomic registering devices are physically equivalent. (iv) It may happen that the extension of a state S in a laboratory i is contained in the extension of an effect F. This means that all physical objects in the state S would induce, in i, answer yes in the effect F. 122 CLAUDIO GAROLA More generally, one can consider the set of all effects whose extensions contain the extension of a given state S in every laboratory. This set is called the certainly true domain of S: every effect in the certainly true domain of S is characterized by the fact that it yields answer yes in every laboratory when it is used in order to perform a test on a physical object in the state S (one briefly says that "the state S produces answer yes with certainty if tested by an effect which belongs to the certainly true domain of S itself"). In every laboratory i one can consider the intersection of the extensions of all effects that belong to the certainly true domain of a given state S. This intersection contains all physical objects that would induce answer yes in all effects that belong to the certainly yes domain of S, and has a crucial role in Semantic Realism. Therefore, we introduce the further symbol Pi(S) in order to denote it and briefly mention some of its more relevant uses. First, Pi(S) can be used in order to distinguish pure from mixed states, which is theoretically important from several viewpoints, in particular because the basic notions of many physical theories can be discussed by making reference to pure states only. According to Semantic Realism, a state S is said to be pure if and only if, in every laboratory i, its extension is the only extension of state which is completely included in the intersection Pi(S): intuitively, this means that S is pure if and only if it is the only state which produces answer yes with certainty if tested by an effect which belongs to the certainly true domain of S itself (it can then be proved that this definition coincides in particular theories, as classical and quantum mechanics, with the standard definitions of pure states provided by these theories). Second, Pi(S) can be used in order to introduce a binary preclusivity (nonreflexive and symmetric) relation on the set of pure states (for physicists: in quantum mechanics two pure states are in the preclusivity relation if and only if the two vectors representing them are orthogonal). Indeed, a pure state S can be said to be in this relation with another pure state S' whenever, in every laboratory i, the set Pi(S) has empty intersection with the extension of S'. Intuitively, this means that there is no physical object in the state S' which would activate answer yes on all effects in the certainly true domain of S. (v) The preclusivity relation introduced in (iv) can be used in order to pick out a special subset of the set of all effects: the set of all exact effects. We cannot enter here the technical procedures that allow one to make this selection (they mainly are based on introducing a suitable notion of closure for every subset of states, via the preclusivity relation). But it is important to grasp without ambiguity the intuitive interpretation A NEW QU ANTUM PHILOSOPHY FOR QM 123 of exact effects. An exact effect F is an effect which is associated with a physical property (as "being white", "being in the point P", "having velocity v"), and every dichotomic registering device in F can be used for testing whether this physical property holds for a physical object x. The set of all physical properties associated with exact effects is called the set of all testable properties (following a terminology known in the literature one could also say that exact effects correspond to sharp testable properties while non-exact effects correspond to unsharptestable properties). Since different exact effects obviously correspond to different testable properties, the correspondence between the set of exact effects and the set of testable properties is one-to-one, so that the two sets can be identified. Because of this identification, the statement F(x), with F an exact effect, is briefly interpreted as "the physical object x has the (testable) property F" , or "the (testable) property F is true for the physical object x", and every dichotomic registering device in F is said to test the property F on x. Let us analyze more carefully the concept of testable property. It is apparent from the definition of this concept that its introduction requires an idealization, since only ideal dichotomic registering devices (that can at most be "approached" by actual devices) can be used in order to test exactly whether a given property holds or not for a given physical object. More important, the concept of testable property must not be confused with the ambiguous concept of "physical property that can be tested in some way" that is rather usual in the language of physics. Indeed, a testable property can only be attributed to a physical object (i.e., to an individual sample of a physical system) and tested on it by means of a single act of testing: in logical terms, one would say that it is a first order property. On the other hand, there are in physics properties that refer to ensembles of physical objects (as the frequencies considered in (iii)), or even to ensembles of ensembles (for instance when one says that a given frequency is minimal in a given state): in logical terms, these are second or third order properties, respectively. Even these higher order properties can be tested: but their test actually consists of a (usually great) number of elementary tests of testable properties and of a successive comparison of the sets of results that have been obtained. It must be stressed that confusing physical properties of different logical orders may be a primary source of physical paradoxes. (vi) By making reference to testable properties, one can introduce two binary relations of compatibility (see [21], [22]) that have an intuitive interpretation, as follows. Semantic compatibility: the testable properties F l , F2 are semantically compatible iff they can be simultaneously true for a physical object x. 124 CLAUDIO GAROLA Pragmatic compatibility (or conjoint testability): the testable properties F}, F2 are pragmatically compatible (conjointly testable) if and only if they can be conjointly measured on a physical object x. The relation of semantic compatibility formalizes the intuitive idea that there are properties that can be simultaneously true for a given physical object (for instance, "being white" and "being round" in the case of a "white and round ball"). The relation of pragmatic compatibility formalizes the intuitive idea that there are properties that can be tested conjointly (for instance, "having x coordinate in the interval .6.x" and "having y coordinate in the interval .6.y" in quantum mechanics), and properties that cannot (for instance, "having x coordinate in the interval .6.x" and "having momentum along the x axis in the interval .6.px" if the product of the widths of the two intervals is less than 2~' again in quantum mechanics). These relations are obviously different (for instance, "having x coordinate in the interval .0.x" and "having x coordinate outside the interval .6.x" are not semantically compatible, since they can never be simultaneously true, but they are pragmatically compatible, since a test of the former also is a test of the latter, which is obviously true if and only if the former is false). Furthermore, pragmatic compatibility can be identified with the standard relation of compatibility in orthodox quantum mechanics. But if empirical verificationism is accepted, two testable properties can be semantically compatible (or not) only if they are pragmatically compatible, for it has no meaning to say that properties that cannot be measured conjointly can (or cannot) be "simultaneously true": thus, the relation of semantic compatibility is not defined on the set of all testable properties. On the contrary, according to Semantic Realism, the two compatibility relations are disentangled and both defined on the set of all testable properties: intuitively, this follows from the fact that the extensions of two testable properties are defined in every laboratory independently of the conjoint testability of the properties themselves. In particular, it may occur that two properties can be semantically compatible without being pragmatically compatible (e.g., "having spin along the x axis" and "having spin along the y axis" in quantum mechanics), which is an important novelty introduced by Semantic Realism. (vii) Let S be a pure state, i a laboratory, and let us come back to the intersection PieS) introduced in (iv). It is apparent that there is no a priori reason for maintaining that an effect exists which, whatever i may be, has just Pi (S) as extension. Whenever this occurs, we call testable support of the state S this effect, and denote it by F s. It is then easy to see that, if a testable support of S exists, it is the smallest element in the certainly true domain of S, according to the order defined in (i) A NEW QU ANTUM PHILOSOPHY FOR QM 125 (indeed, its extension in any laboratory is contained in the extensions of all effects that belong to the certainly true domain of S). The existence of a testable support F s for every pure state Scan be easily deduced both in classical and in quantum mechanics if one refers to the standard interpretation of these theories (for physicists: if I tp > is the vector representing the state S in quantum mechanics, Fs is the testable property represented by the projection I tp >< tp I); furthermore, Fs is an exact effect, i.e., a testable property). Therefore, pure states are often identified with the testable properties that are their testable supports in classical and in quantum mechanics (see, e.g., [31]). However, this identification is misleading in the latter theory, and originates some old quantum paradoxes (as Furry's, see [12], [13]' and [23]). Indeed, in classical Mechanics the extension ofF s in any laboratory i coincides with the extension of S, since there is in this theory at least one (possibly ideal) dichotomic registering device that yields answer yes if and only if a physical object is in the state S. But, on the contrary, the extension of F s in quantum mechanics is usually greater than the extension of S, since there are states in this theory that are different from S and induce answer yes in the effect F s with a nonzero frequency in every laboratory (for physicists: these are states that are represented by vectors which are not orthogonal to the vector representing S). This difference between the two theories outlines the probabilistic character of quantum mechanics and shows that, even if the correspondence between pure states and their testable supports is one-to one, it is not possible in this theory to identify pure states with their testable supports with regard to their extensions. Synthetically, one can say that states must not be confused with testable properties in quantum mechanics. (viii) In addition to the remarks in (vii), the assumption itself that a testable support exists for every pure state can be seriously questioned in quantum mechanics whenever compound physical systems are considered (a compound physical system can be intuitively defined as a system formed by various subsystems, which can obviously interact). Indeed, the states of such a system can be divided into two classes, the first type states and the second type (or entangled) states (this partition is based on technical reasons that will not be discussed here). If'S is a second type state, it can be proved that S may be such that one cannot find in quantum mechanics a class of ideal dichotomic registering devices·, i.e., an exact effect, which satisfies the condition of being minimal in the certainly true domain of S. Should this be the case, no exact effect, i.e., (first order) testable property, can be considered as the support of S. Basing on the above criticism, Semantic Realism adopts a new and rather radical position: indeed, it defines a first type state S as a state 126 CLA UDIO GAROLA which has a testable support F s, and assumes that F s then is an exact effect. The second type states (which may exist or not in a given theory: indeed, they do not exist in classical Mechanics) are then defined as those states which have no testable support. It is interesting to note that many quantum physicists would disagree with the above definitions and assumptions, since they would object that some experiments have been contrived in quantum mechanics which test "whether a physical object is in the state S" whenever S is a second type state. Yet, this disagreement is based, in our opinion, on the lack of awareness of the fact that all these experiments actually test only second order properties (see (v)), since they consist in testing some first order properties of the different subsystems that form the compound system, repeating the test a great number of times on different samples of the system and then correlating statistically the obtained results. These second order properties (that can be called correlation properties) cannot therefore be confused with testable supports of states of the second type. The characterization of the first and the second type states provided by Semantic Realism is theoretically relevant. For, the implicit assumption that a testable support exists for every state is the basis on which some arguments stand which aim to prove that compound quantum systems cannot be separated (nonseparability intuitively means that the properties of two or more physical systems that have interacted in the past remain interconnected in such a way that testing a property on one of the systems influences the properties of the other system). If one drops out this assumption, this strange conclusion is avoided. It is then apparent that the aforesaid characterization constitutes a new branching point in which the interpretation of quantum mechanics provided by Semantic Realism differentiates from the standard interpretation (for physicists: note that dropping the assumption that a testable support exists for every state implies renouncing the assumption that every Hermitean operator represents a physical apparatus, at least in the case of compound physical systems). (ix) The breakdown of the one-to-one correspondence between pure states and their testable supports implies in particular that the partially ordered set (or poset) of all exact effects (this set is indeed ordered, since it is embedded in the bigger poset of all effects, see (i)) is not endowed, in general, with the mathematical structure of a lattice. However, one can resort to well known mathematical tools (completionprocedures, see, e.g., [14]) that allow one to enlarge the poset of all exact effects by adjoining to it further elements that transform it into a lattice. These new elements, of course, have not an operational interpretation: but they can be considered as theoretical (first order) properties, which are upper A NEW QUA N TUM PHI LOS 0 P H Y FOR Q M 127 (or lower) bounds of sets of testable (first order) properties. Thus, one obtains an enlarged set of physical properties which is partitioned into a set of testable properties and a set of theoretical properties. In this new enlarged context, a one-to-one correspondence between pure states and minimal properties can be recovered. Indeed, one can associate to every pure state S a support, which is a property that is minimal in the certainly true domain of S (which now contains also theoretical properties): this property is testable (testable support) if S is a first kind state, theoretical ( theoretical support) if S is a second kind state. (x) The completion of the poset of exact effects and the ensuing distinction between testable and theoretical properties is a typical feature of Semantic Realism, which has long ranging consequences. Indeed, on the one hand, it allows one to recover a structure of (complete, orthacomplemented) lattice, which can be identified, in the case of quantum mechanics, with the lattices that appear in a number of axiomatic approaches to this theory (see, e.g., [29], [27], [31]), often collected under the name quantum logic approach to quantum mechanics (see, e.g., [7]). Thus "Quantum Logic" is recovered by Semantic Realism as an empirical structure, since the order on it is empirical in the sense specified in (i). But, on the other hand, this Quantum Logic obtained by completion does not contain, in the case of complex systems, only observative elements (testable properties), but even elements that are non-observative (theoretical properties), at least in the sense that they do not correspond to first order properties that can be directly tested on a physical object. This presence of non-observative elements in the lattice associated to a compound physical system is a further important ingredient for avoiding quantum paradoxes, as we intend to show in the following. 7. MEANING, TESTABILITY AND VALIDITY OF PHYSICAL LAWS The topics discussed in Section 6 may have annoyed the reader who is not interested in a detailed exposition of the concepts and methods of Semantic Realism. But they allow us now to introduce some new general epistemological perspectives that are typical of Semantic Realism and have a deep physical meaning. (i) Basing on the completion procedures introduced in Section 6, (ix) , one can construct a new language, the language of properties Le , which has the set of all states and the set of all testable and theoretical properties as set of predicates, and is endowed with the same (classical) logical structure of L. This language is actually different from L: indeed, the set 128 CLAUDIO GAROLA of its predicates does not contain non-exact effects, while it may contain a set of new predicates that are interpreted as theoretical properties. Furthermore, it contains, as L, elementary and complex statements, and its statements are ordered by the standard logical order (see Section 2, (iii)). (ii) The poset of all elementary statements of Le of the form E(x), with E a property, endowed with the logical order, is obviously isomorphic to the poset of all testable and theoretical properties (this means that a one-to-one mapping exists which maps the former set onto the latter, preserving the order): therefore, it can also be considered a Quantum Logic. It is then interesting to observe that this name is now intuitively appropriate, since the elements of our new poset are statements and its order is logical. However, this order still depends on the interpretation of the predicates, consistently with the remark that logical order also contains an empirical part (see Section 6, (i)), so that even this Quantum Logic must be considered an empirical structure, not a strictly logical one. (iii) The statements of Land Le (all of which have a truth value) can be divided into testable and nontestable statements: intuitively, a statement is testable if its truth value can be tested by means of a suitable measurement procedure, nontestable if it cannot. Let us consider 1. All elementary statements of this language are obviously testable (which is a reasonable operational feature, since L aims to formalize observative languages, see Section 5). On the contrary, there may be complex statements in L that are nontestable: indeed it may occur that there is no way of testing the truth value of a complex statement which contains testable properties that are pragmatically non-compatible. Let us consider Le. This language may contain also elementary statements that are non-testable (indeed, every elementary statement E(x) of L e , where E is a theoretical property, is non-testable). Thus if the set of all theoretical properties is non-void, one cannot assert that Le formalizes an observative language, and Le provides the first instance of how theoretical (i.e., non directly observative) statements naturally enter the language of physics. Furthermore, a complex statement of Le may be nontestable, if it contains theoretical properties and/or testable properties that are pragmatically non-compatible. The existence of meaningful testable and nontestable statements in Land Le shows that the fundamental epistemological distinction between truth and empirical access to truth (see Section 2) is maintained in these languages. Moreover, a general criterion of testability can be introduced both in L and in Le. Indeed, the reasoning used in Section 3 in order to show that empirical verificationism implies that a molecular A NEW QU ANTUM PHILOSOPHY FOR QM 129 complex statement is meaningful if and only if it logically equivalent to a testable elementary statement can now be used in order to show that a molecular complex statement of L or Le is testable if and only if it is logically equivalent to a testable elementary statement. One thus gets a general criterion of testability, which can be used in order to pick out sets of testable statements of L or Le (for instance, all molecular complex statements which contain only pragmatically compatible testable properties are testable). (iv) The distinction between testable and non-testable statements in Land Le does not mean that the latter cannot be legitimately used in physics as theoretical statements: indeed they have a role, in particular, in the inferential procedures. Hence, both testable and nontestable statements of Le can be used in order to state physical laws. Therefore, one must distinguish two classes of physical laws, as follows. Empirical laws: these are expressed by testable statements, hence can be empirically checked. Theoretical laws: these are expressed by nontestable statements, hence cannot be empirically checked (at least directly). The above distinction between empirical and theoretical laws leads one to wonder about the epistemological status of these laws. It is then evident that this question has many possible answers, depending on the philosophical position that one decides to adopt. Therefore, let us make clear here the standpoint of Semantic Realism: this approach adheres to the operational philosopy underlying quantum mechanics and assumes that it must be retained in every physical theory, but it also maintains that this philosophy requires avoiding ontologization of physical laws and knowledge rather than adopting empirical verificationism. By applying the above viewpoint to theoretical laws it follows that these must be considered as formal structures, whose role consists in producing, via logical deduction, empirical physical laws that can be directly tested. The case of empirical physical laws requires a more detailed treatment. Therefore, let us preliminarily observe that they are commonly used, together with some auxiliary assumptions (or premises), in order to obtain physical predictions, i.e., true statements attributing physical properties to given physical objects (as an intuitive instance of this procedure, consider the classical syllogism quoted at the beginning of Section 2: here, "all men are mortal", "Socrates is a man" and "Socrates is mortal" play the role of a physical law, an auxiliary assumption and a prediction, respectively). It is then important to observe that the premises specify the physical situation to which the empirical law is applied, attributing properties (as "being a man") to the physical objects (as "Socrates") 130 CLAUDIO GAROLA that one is considering: we say that they determine a context. Now, the relations of semantic and pragmatic compatibility introduced in Section 6, (vi), suggest that various types of contexts can be distinguished, as follows. Contradictory contexts, in which semantically non-compatible properties are assumed. These contexts are inconsistent and can never occur, hence they are irrelevant for physics. Non-accessible physical contexts, in which semantically compatible but pragmatically non-compatible properties are assumed. These contexts are consistent and can in principle occur, but are not accessible to physical investigation, since one can never identify them empirically (indeed, the pragmatically incompatible properties that define a non-accessible context can never be tested conjointly). Accessible physical contexts, in which semantically and pragmatically compatible properties are assumed. These contexts are consistent and can occur; furthermore, they are accessible to physical investigation, since one can identify them empirically. The appearance of non-accessible physical contexts is a typical feature of Semantic Realism, and occurs because of the existence of physical theories, as quantum mechanics, where a non trivial relation of pragmatic compatibility is defined (it is evident that these contexts do not appear in classical Physics, where all physical properties can be tested conjointly, at least in principle). If one adopts the epistemological position of Semantic Realism specified above, that is, avoiding ontologization of physical laws, one concludes that empirical laws cannot be assumed to be true (in the sense that they may be true as well as false) in nonaccessible contexts: indeed, the theory itself prohibits that an empirical law be verified in a context of this kind. It follows that, if one wants to avoid "metaphysics", the validity of empirical laws must be limited, at least in those theories which introduce a non trivial relation of pragmatic compatibility. This limitation can be introduced by stating a new epistemological principle (Metatheoretical Generalized Principle, or, briefly, MGP), that we express here in an informal way as follows. MGP. A sentence expressing an empirical physical law (deduced or not from a general theoretical law) is true in every physical context in which only semantically and pragmatically compatible pmperties are assumed for each physical object that is considered (accessible physical contexts). The significance of MGP should not be underestimated. Indeed, classical Physics has accustomed physicists to assume implicitly the unrestricted validity of physical laws, which can be expressed by stating a Metatheoretical classical Principle (briefly, MCP), as follows. A NEW QUANTUM PHILOSOPHY FOR QM 131 MCP. A sentence expressing an empirical physical law (deduced or not from a general theoretical law) is true in every (non-contradictory) physical context. This principle is clearly adequate to classical Physics, where it is equivalent to MGP since non-accessible physical contexts do not occur. But it is questionable, according to Semantic Realism, in every physical theory where these contexts occur. By introducing MOP in place of MOP, Semantic Realism proposes to shift the antimetaphysical issue of the operational viewpoint from accepting empirical verificationism to renouncing any extrapolation of our knowledge beyond its empirical limits. We show in the next section that this viewpoint greatly helps in reconciling quantum mechanics with everyday physicist's intuition. Before closing this point, we would like to defend MGP against the natural objection that it produces an idealistic dependence of physical laws on the choices of the observer, as it seems to be witnessed by the term "physical context" that appears in it. Indeed, MGP introduces a kind of pragmatic contextuality which is deeply different from the semantic contextuality introduced by the standard interpretation of quantum mechanics (see Section 1). For, the latter makes the properties themselves of a physical object depend, to some extent, on the observer's decision of performing some possible experiments; the former only restricts the possibility of deducing relations among observation-independent physical properties from physical laws, the restriction depending on the character of partial knowledge attributed to these laws and not on the observer's choices. (v) Whenever the perspective of Semantic Realism is applied to quantum mechanics, this theory turns out to be incomplete with respect to the language Le; this means (see Section 2, (iv)) that the laws of quantum mechanics are not sufficient for determining the truth values of all statements of L e , even if suitable premises are assumed. Intuitively, this result follows from the probabilistic character of quantum physical laws, which usually provide only probabilities of truth values, not the truth values themselves, for statements attributing properties to physical objects in given states [16]. Since all physical properties in the language Le are either true or false for a given physical object, the interpretation of quantum mechanics provided by Semantic Realism belongs to the class of the objectivity/incompleteness interpretations, as anticipated at the end of Section 4. 132 CLAUDIO GAROLA 8. ON THE BELL AND BELL-KOCHEN-SPECKER THEOREMS Let us come now to the Bell and Bell-Kochen-Specker theorems, the relevance of which with respect to our thesis in this paper has been illustrated in the Introduction. Indeed, if the orthodox interpretation of these theorems is accepted, one of the basic assumptions of Semantic Realism (the existence of an extension for every effect in every laboratory, which allows one to attribute properties to physical objects independently of measurement procedures, see Section 5, (v)) would be invalid in quantum mechanics. In order to avoid this criticism, various proofs of the above theorems have been analyzed in a number of papers [18J, [19J, [20]' [23J and it has been shown that all of them are based on assuming the validity of some empirical (correlation) laws deduced from the general theoretical laws of quantum mechanics outside the domain of validity established by MGP: hence, these proofs are not correct according to Semantic Realism. Let us look into the above subjects in more detail. We consider first the Bell theorem. This theorem was stated by Bell in 1964 [lJ. Subsequently, the Bell argument was refined by a number of authors, who supplied accurate analyses of the premises on which it is based, and a number of alternative statements and proofs. Nowadays most authors agree that the Bell theorem shows that quantum mechanics conflicts with one at least of the following assumptions (see Selleri, [36J; note that assumption R, though called reality for conforming to current literature, actually imposes a weak reality condition: indeed, it does not entail that the mathematical objects that appear in the formal apparatus of quantum mechanics correspond to ontologically existing entities, but only requires that the properties of a physical object, or of a preparation procedure, do not depend on the choices of the observer). R (reality). The results of all conceivable measurements are simultaneously prefixed (even in the case of non-compatible observables). LOC (locality). Whenever two subsystems, say 1 and 2, of a compound physical system are sufficiently far apart, a measurement on subsystem 1 (2) does not modify the values of the observables of subsystem 2 (1). However, it can be proved that non-R and quantum mechanics imply non-LOCo Thus, the Bell theorem is usually retained to prove the nonlocality of quantum mechanics. If one considers the more ancient proofs of this theorem, one sees that they essentially show that locality implies inequalities which are not consistent with quantum mechanics. In all these proofs, empirical physical laws are assumed as true for physical objects in non-accessible physical A NEW QU ANTUM PHILOSOPHY FOR QM 133 contexts (see Section 7, (iv)), which implies a violation of MGP. This is apparent, for instance, in the Wigner [39] and Sakurai [35] proofs, where many subsets of physical objects are considered, each set consisting of physical objects having the same properties (for physicists: spin components along different directions), and then an inequality is obtained by applying a quantum law (again for physicists: perfect correlation in the singlet state) to these objects. Indeed, the properties characterizing some sets are semantically compatible but pragmatically non-compatible, which prohibits the use of quantum laws according to MGP in these cases. In other proofs, as the original Bell's one, an inequality is obtained regarding sets of physical objects in non-accessible physical contexts, but the deduction does not introduce quantum laws (we reserve the name Bell inequality in the following to Bell-type inequalities obtained in this way). But quantum laws are then used whenever the Bell inequality is compared with an analogous inequality predicted by quantum mechanics. Thus, one can conclude that none of these proofs can be accepted according to Semantic Realism. More recent proofs do not make resort to inequalities, but directly show that, joining R, LOC and quantum mechanics, one obtains contradictory predictions (e.g., Greenberger et al., [24]). In these proofs, a number of empirical laws are deduced from a general theoretical quantum law, and it is implicitly assumed that they all are simultaneously true. But whenever one of these empirical laws is assumed to hold, the properties predicted by it are easily seen to be pragmatically noncompatible with the properties that one introduces further as premises in order to deduce predictions from another law of the set. Thus, one produces again a non accessible physical context, in which the validity of the latter law cannot be assured, because of MGP. One concludes that neither of these proofs can be accepted according to Semantic Realism. Let us come to the Bell-Kochen-Specker theorem. As we have seen in the Introduction, this theorem is usually retained to prove the contextuality of quantum mechanics, which can now be interpreted in the sense that the truth value of a statement attributing a property that belongs to a set of properties that are measured on a physical object depends on the whole set, not only on the property itself. It is evident that the fundamental quantum doctrine discussed in the Introduction follows at once from contextuality, and we remind that the relevance of the Bell-Kochen-Specker theorem consists, seemingly, in showing that this feature of quantum mechanics is imposed by the internal structure of quantum mechanics itself, so that it needs not be formulated as an a priori epistemological constraint. 134 CLA UDIO GAROLA The original proofs of the Bell-Kochen-Specker theorem [2], [26] were rather complicate, but there are some recent proofs (Mermin, [30]) that are quite simple and immediate. By considering these proofs, it has been shown (Garola and Solombrino, [23]) that they are invalidated, according to Semantic Realism, by the same argument used in order to reject the proofs of the Bell theorem that do not resort to inequalities. Let us discuss this subject more carefully. The core of the Mermin argument consists in stating a set of equations that express theoretical laws connecting quantum observables (i.e., mathematical operators that represent physical quantities that can be actually measured). In each equation the observables are compatible, i.e., they can be measured conjointly: hence, Mermin introduces the "plausible" condition that the measured values of the observables must satisfy the equation itself. If all values are simultaneously prefixed (assumption R) this condition can be applied repeatedly to all equations in the set, concluding that all these equations must be simultaneously satisfied by the measured values of the observables. But, then, Mermin shows that this is impossible. Thus, he deduces that the values of some observabIes may be different if different sets of measurements are performed, hence they cannot be prefixed and quantum mechanics turns out to be a contextual theory (note that this does not introduce contradictions between measurement outcomes and physical laws in standard quantum mechanics, since some of the observables that occur in different laws are non-compatible, which means that only restricted subsets of values can actually be measured). In order to invalidate this reasoning, let us note first that the laws used in the Mermin proofs are so general that they can neither be expressed by means of L nor by means of Le: indeed, they hold for every state of the physical system, so that stating them formally would require quantification on state variables, hence a complete formalization of the language of physics that has been avoided here for the sake of simplicity (see Section 5). However, this does not affect the general remarks on theoretical and empirical laws provided at the end of Section 7 and leading to state MGP. It follows that all empirical laws deduced from the theoretical laws used by Mermin are valid, according to Semantic Realism, within the limits estabilished by MGP, and cannot be applied unrestrictedly. Now, these limits prohibit that all empirical laws that can be deduced from the set of theoretical laws considered in the Mermin proofs be assumed as valid conjointly. Indeed, the fact that there are non-compatible observables in different laws implies that one can deduce empirical laws predicting properties that are not pragmatically compatible, so that assuming some of them as valid creates a non-accessible physical context A NEW QU ANTUM PHILOSOPHY FOR QM 135 in which one cannot maintain that the remaining empirical laws are also valid. Thus, one attains a not conventional conclusion: assuming a set of equations as theoretical laws does not imply, according to Semantic Realism, that the values of the observables that appear in them, which are prefixed, must all satisfy the equations themselves conjointly. Of course, this conclusion contradicts the conclusion obtained by Mermin by using repeatedly his "plausible" condition (which however rephrases a condition imposed by Kochen and Specker themselves in order to prove their theorem [26]), which is sufficient for invalidating the reasoning that leads to the Bell-Kochen-Specker theorem. The main aims of this section are thus attained. One can summarize our results by saying that nonlocality and contextuality of quantum mechanics, that are taken for granted by most physicists concerned with foundational research, do not necessarily hold if Semantic Realism is accepted. This conclusion eliminates, on one hand, a source of paradoxes, which unavoidably appear in the orthodox interpretation and in the alternatives that accept empirical verificationism; on the other hand, it prevents the charge of being inconsistent, that could otherwise be made against the interpretation of quantum mechanics provided by Semantic Realism because of the reasons illustrated at the beginning of this section. 9. C ONCL UDING REMARKS Judging from the results of Section 8, one could believe that the Bell and Bell-Kochen-Specker theorems simply are non-proved statements according to the interpretation of quantum mechanics provided by Semantic Realism. This is certainly true, but rather disconcerting for scholars concerned with the foundations of quantum mechanics, who usually consider these theorems as basic results. However, if one reconsiders the whole subject from an epistemological viewpoint, the aforesaid theorems still play an important role. Indeed, let us consider first the Bell-KochenSpecker theorem. Our invalidation above shows that Semantic Realism rejects, through MGP, the repeated use of Mermin's "plausible" condition. But this use can be seen as a particularization of the MCP principle to the special case under consideration. Therefore, according to Semantic Realism, the Bell-Kochen-Specker theorem can be reformulated as a general statement which asserts that assuming reality (R) and unrestricted validity of physical laws (Mep) together with quantum mechanics leads to inconsistencies. 136 CLA UDIO GAROLA It follows from the above reformulation of the Bell-Kochen-Specker theorem that one has some possible choices if he wants to avoid inconsistencies. For instance, if one considers only the choices that give up one of the three assumptions above, one has three possibilities: (i) modifying quantum mechanics, preserving Rand MCP; (ii) giving up R, preserving quantum mechanics and MCP; (iii) giving up MCP, preserving quantum mechanics and R. Choice (i) is made by those physicists who want to restore a basically classical conception of the world. Choice (ii) is made by the standard interpretation of quantum mechanics (note that this choice does not transgress the operational philosophy of quantum mechanics, since giving up R prohibits producing non-accessible physical contexts, hence the distinction between MCP and MGP becomes irrelevant). Choice (iii) is made by Semantic Realism (with some changes in the interpretation of quantum mechanics, see in particular Section 6, (viii)), which criticizes (i) as an impossible attempt of recovering old intuitive models in physics, and (ii) since abandoning R creates a number of conceptual difficulties and "paradoxes". However, we insist on the fact that the appearance of alternative (iii) is due to the Bell-Kochen-Specker theorem, if restated as above, and in this sense this theorem is still decisive (it is interesting to note that even recent attempts at modifying quantum mechanics in order to recover some forms of realism at the macroscopic level essentially deny R at the microscopic level, and possibility (iii) is ignored). Consider now the Bell theorem. We remind that physicists usually think that the Bell inequalities provide a method for testing experimentally whether quantum mechanics or locality (LOC) is correct. But the invalidation of the Bell theorem implies that quantum mechanics does not necessarily conflict with LOC. Then, a crucial question arises: what would happen if one should perform a suitable test of a given Bell inequality? Would the inequality be violated or not? The answer of Semantic Realism is simple but not trivial. In fact, Semantic Realism maintains that a Bell inequality is a correct theoretical formula which is not epistemically accessible in quantum mechanics. Any physical experiment actually tests something else (correlations among properties of physical objects in accessible contexts) and obviously yields the results predicted by quantum mechanics. No contradiction can actually occur, since the inequalities that can be tested in quantum mechanics could be identified with Bell inequalities only assuming the unrestricted validity of some physical laws, i.e., violating MGP. Thus, it is untrue that a Bell inequality provides a method for testing experimentally whether either quantum mechanics or LOC is correct. But the large number of experimental proofs that confirm the predictions of quantum mechanics proves A NEW QU ANTUM PHILOSOPHY FOR QM 137 indirectly that something must go wrong with quantum laws regarding compound systems within non-accessible physical contexts: otherwise, quantum mechanical inequalities would be reduced to Bell inequalities. The above remarks on the Bell and Bell-Kochen Specker theorems clearly suggest that a physical theory could exist which goes beyond quantum mechanics (which is an incomplete theory according to Semantic Realism, see Section 7, (v)) without contradicting it. Furthermore, Semantic Realism also supplies two interesting suggestions for a theory of this kind. First, the interpretation of the supports of the second kind states as theoretical properties (see Section 6, (ix)) shows that the quantum treatment of compound systems is semantically ambiguous. Indeed, in this treatment the support of any pure state is represented by a mathematical entity (for physicists: a one-dimensional projection) which should then correspond to a (first order) testable property in the case of a first kind state, to a (first order) theoretical property in the case of a second kind state. This suggests that a new theory should represent testable and theoretical properties by means of different mathematical entities, so that one can distinguish the former from the latter (or it should at least provide mathematical rules for distinguishing within the formalism the projections that represent physical apparatuses from the projections which do not). Second, the invalidation of the Bell theorem and the interpretation provided by Semantic Realism of the experimental results which confirm the quantum predictions suggest that, contrary to a widespread belief, local hidden variables models for quantum mechanics may exist (see Section 4): but these should be such to propose new laws that differ from quantum mechanical laws within non-accessible physical contexts, i.e., physical contexts which are only indirectly accessible by means of correlation measurements that essentially test physical properties of the second or third order (see Section 6, (v)). We stress that this does not mean that one must expect violations of quantum predictions whenever correlation measurements are performed: rather, one expects that quantum predictions be fulfilled, and that the new laws explain why quantum inequalities differ from Bell inequalities. To conclude with, it is interesting to note that Semantic Realism also proves to avoid the old Furry [12], [13] and Bohm-Aharonov [4] quantum paradoxes. These results are obtained by using the distinction between states and effects, and the distinction between semantic and pragmatic compatibility, respectively, and no use is required of the MGP principle (see Garola and Solombrino, [23]). 138 CLAUDIO GAROLA AFFILIATION Claudio Carola Dipartimento di Fisica dell' Universita Lecce, Italy [email protected] REFERENCES [1] Bell, J.S., "On the Einstein Podolsky Rosen paradox", Physics, 1, 1964, p. 195. [2] Bell, J .S., "On the Problem of hidden variables in quantum mechanics", Rev. Mod. Phys., 38, 1966, p. 447. [3] Bohm, D., Quantum Theory, Prentice Hall, Englewood Cliffs (N.J.), 1951. [4] Bohrn, D. and Aharonov, Y., "Discussion of experimental Proofs for the paradox of Einstein, Rosen, and Podolsky", Phys. Rev., 108, 1957, p. 1070. [5] Bohr, N., "Can Quantum Mechanical Description of Reality be Considered Complete?", Phys. Rev., 48, 1935, p. 696. [6] Braithwaite, R.B., Scientific Explanation, Cambridge University Press, Cambridge, 1953. [7] Busch, P., Lahti, P.J., and Mittelstaedt, P., The Quantum Theory of measurement, Springer, Berlin, 1991. [8] Dalla Pozza, C. and Garola, C., "A Pragmatic Interpretation of Intuitionistic Propositional logic" , Erkenntnis, 43, 1995, p. 8l. [9] Einstein, A., Podolsky, B., and Rosen, N., "Can quantum mechanical description of reality be considered complete?" , Phys. Rev., 47, 1935, p. 777. [10] Finkelstein, D., "Matter, space and logic", in: Hooker, C. A. (ed.), The Logico-Algebraic Approach to quantum mechanics, Vol II, Reidel, Dordrecht, 1979. [11] Finkelstein, D., "The physics of logic", in: Hooker, C. A. (ed.), The Logico-Algebraic Approach to quantum mechanics, Vol II, Reidel, Dordrecht, 1979. [12]Furry, W.H., "Note on the quantum mechanical theory of measurement", Phys. Rev., 49, 1936, p. 393. [13] Furry, W.H., "Remarks on measurements in quantum theory", Phys. Rev., 49, 1936, p. 476. A NEW QU ANTUM PHILOS OPHY FOR QM 139 [14] Garola, C., "Embedding of posets into lattices in quantum logic", Int. Journ. of Theor. Phys., 24, 1985, p. 423. [15] Garola, C., "classical foundations of quantum logic", Int. Journ. of Theor. Phys., 30, 1991, p. 1. [16] Garola, C., "Semantic incompleteness of quantum physics", Int. Journ. of Theor. Phys., 31, 1992, p. 809. [17] Garola, C., "Truth versus testability in quantum logic", Erkenntnis, 37, 1992, p. 197. [18] Garola, C., "Reconciling local realism and quantum physics: a critique to Bell", Teoreticheskaya i Matematicheskaya Fizika, 99, 1994, p.285. [19] Garola, C., "Criticizing Bell: Local realism and quantum physics reconciled", Int. Journ. of Theor. Phys., 34, 1995, p. 269. [20] Garola, C., "An operational Critique to Bell's Theorem" , in: Garola, C. and Rossi, A. (eds.), The Foundations of quantum mechanics. Historical Analysis and Open Questions, Kluwer Academic Publishers, Dordrecht, 1995. [21] Garola, C., "Pragmatic versus semantic contextuality in quantum physics", Int. Journ. of Theor. Phys., 34, 1995, p. 1383. [22] Garola, C. and Solombrino, L., "The theoretical apparatus of semantic realism: A new language for classical and quantum physics", Found. of Phys., 26, 1996, p. 1121. [23] Garola, C. and Solombrino, L., "Semantic realism versus EPRlike paradoxes: the Furry, Bohm-Aharonov and Bell paradoxes", Found. of Phys., 26, 1996, p. 1329. [24] Greenberger, D.M., Horne, M.A., Shimony A., and Zeilinger, A., "Bell's theorem without Inequalities", Am. Journ. of Phys., 58, 1990, p. 1131. [25] Hempel, C.C., Aspects of Scientific Explanation, Free Press, New York, 1965. [26] Kochen, S. and Specker, E.P., "The problem of hidden variables in quantum mechanics", Journ. of Math. Mech., 17, 1967, p. 59. [27] Jauch, J.M., Foundations of quantum mechanics, Addison-Wesley, Reading (Mass.), 1968. [28] Ludwig, G., Foundations of quantum mechanics I, Springer Verlag, New York, 1983. [29] Mackey, G.W., The Mathematical Foundations of quantum mechanics, Benjamin, New York, 1963. 140 CLAUDIO GAROLA [30] Mermin, N.D., "Hidden variables and the two theorems of John Bell", Reviews of Modern Physics, 65, 1993, p. 803. [31] Piron, C., Foundations of Quantum Physics, Benjamin, Reading, (Mass.), 1976. [32] Popper, K.R., Conjectures and Refutations, Routledge and Kegan Paul, London, 1969. [33] Putnam, H., "Is logic empirical?", in: Hooker, C.A. (ed.), The Logico-Algebraic Approach to quantum mechanics, Vol II, Reidel, Dordrecht, 1979. [34] Russell, B., An Inquiry into Meaning and Truth, Allen & Unwin, London, 1940. [35] Sakurai, J.J., Modern quantum mechanics, W.A. Benjamin, Reading (Mass.), 1985. [36] Selleri, F., "Even local probabilities lead to the paradox", in: Selleri, F. (ed.), quantum mechanics Versus Local Realism, Plenum Press, New York, 1988. [37] Tarski, A., Logic, Semantics, Metamathematics, Oxford University Press, Oxford, 1956. [38] Tarski, A., "The semantic conception of truth and the foundations of semantics", in: Linsky, L. (ed.), Semantics and the Philosophy of Language, University of Illinois Press, Urbana, 1952. [39] Wigner, E.P., "On hidden variables and quantum mechanical probabilities", Am. lourn. of Phys., 38, 1970, p. 1005. DIEDERIK AERTS QUANTUM MECHANICS: STRUCTURES, AXIOMS AND PARADOXES 1. INTRODUCTION In this article we present an analysis of quantum mechanics and its problems and paradoxes taking into account some of the results and insights that have been obtained during the last two decades by investigations that are commonly classified in the field of 'quantum structures research'. We will concentrate on these aspects of quantum mechanics that have been investigated in our group at Brussels Free Universityl. We try to be as clear and self contained as possible: firstly because the article is also aimed at scientists not specialized in quantum mechanics, and secondly because we believe that some of the results and insights that we have obtained present the deep problems of quantum mechanics in a simple way. The study of the structure of quantum mechanics is almost as old as quantum mechanics itself. The fact that the two early versions of quantum mechanics-the matrix mechanics of Werner Heisenberg and the wave mechanics of Erwin Schrodinger-were shown to be structurally equivalent to what is now called standard quantum mechanics, made it already clear in the early days that the study of the structure itself would be very important. The foundations of much of this structure are already present in the book of John von Neumann [1], and if we refer to standard quantum mechanics we mean the formulation of the theory as it was first presented there in a complete way. Standard quantum mechanics makes use of a sophisticated mathematical apparatus, and this is one of the reasons that it is not easy to explain it to a non specialist audience. Upon reflecting how we would resolve this 'presentation' problem for this paper we have chosen the following approach: most, if not all, deep quantum mechanical problems appear already in full, for the case of the 'most simple' of all quantum models, namely the model for the spin of a spin~ quantum particle. Therefore we have chosen to present the technical aspects of this paper as much as possible for the description of this most simple quantum 1 The actual members of our group are: Diederik Aerts, Bob Coecke, Thomas Durt, Sven Aerts, Frank Valckenborgh, Bart D'Hooghe and Bart Van Steirteghem. 141 © 1999 Kluwer Academic Publishers. 142 DIEDERIK AERTS model, and to expose the problems by making use of its quantum mechanical and quantum structural description. The advantage is that the structure needed to explain the spin model is simple and only requires a high school background in mathematics. The study of quantum structures has been motivated mainly by two types of shortcomings of standard quantum mechanics. (1) There is no straightforward physical justification for the use of the abstract mathematical apparatus of quantum mechanics. By introducing an axiomatic approach the mathematical apparatus of standard quantum mechanics can be derived from more general structures that can be based more easily on physical concepts (2) Almost none of the mathematical concepts used in standard quantum mechanics are operationally defined. As a consequence there has also been a great effort to elaborate an operational foundation. 2. QU ANTUM STR UCTURES AND QU ANTUM LOGIC Relativity theory, formulated in great part by one person, Albert Einstein, is founded on the concept of 'event', which is a concept that is physically well defined and understood. Within relativity theory itself, the events are represented by the points of a four dimensional space-time continuum. In this way, relativity theory has a well defined physical and mathematical base. quantum mechanics on the contrary was born in a very obscure way. Matrix mechanics was constructed by Werner Heisenberg in a mainly technical effort to explain and describe the energy spectrum of the atoms. Wave mechanics, elaborated by Erwin Schrodinger, seemed to have a more solid physical base: a general idea of wave-particle duality, in the spirit of Louis de Broglie or Niels Bohr. But then Paul Adrien Maurice Dirac and later John von Neumann proved that the matrix mechanics of Heisenberg and the wave mechanics of Schrodinger are equivalent: they can be constructed as two mathematical representations of one and the same vector space, the Hilbert space. This fundamental result indicated already that the 'de Broglie wave' and the 'Bohr wave' are not physical waves and that the state of a quantum entity is an abstract concept: a vector in an abstract vector space. Referring again to what we mentioned to be the two main reasons for studying quantum structures, we can state now more clearly: the study of quantum structures has as primary goal the elaboration of quantum mechanics with a physical and mathematical base that is as clear as the one that exists in relativity theory. We remark that the initial aim of quantum structures research was not to 'change' the theory-although STRUCTURES AND PARADOXES 143 it ultimately proposes a fundamental change of the standard theory as will be outlined in this paper-but to elaborate a clear and well defined base for it. For this purpose it is necessary to introduce clear and physically well defined basic concepts, like the events in the theory of relativity, and to identify the mathematiCal structure that these basic concepts have to form to be able to recover standard quantum mechanics. In 1936, Garret Birkhoff and John von Neumann, wrote an article entitled "The logic of quantum mechanics". They show that if one introduces the concept of 'operational proposition' and its representation in standard quantum mechanics by an orthogonal projection operator of the Hilbert space, it can be shown that the set of the 'experimental propositions' does not form a Boolean algebra, as it the case for the set of propositions of classical logic [2]. As a consequence of this article the field called 'quantum logic' came into existence: an investigation on the logic of quantum mechanics. An interesting idea was brought forward. Relativity theory is a theory based on the concept of 'event' and a mathematical structure of a four dimensional space-time continuum. This space-time continuum contains a non Euclidean geometry. Could it be that the article of Birkhoff and von Neumann indicates that quantum mechanics should be based on a non Boolean logic in the same sense as relativity theory is based on a non Euclidean geometry? This is a fascinating idea, because if quantum mechanics were based on a non Boolean logic, this would perhaps explain why paradoxes are so abundant in quantum mechanics: the paradoxes would then arise because classical Boolean logic is used to analyze a situation that intrinsically incorporates a non classical, non Boolean logic. Following this idea quantum logic was developed as a new logic and also as a detailed study of the logico-algebraic structures that are contained in the mathematical apparatus of quantum mechanics. The systematic study of the logico-algebraic structures related to quantum mechanics was very fruitful and we refer to the paper that David Foulis published in this book for a good historical account [3]. On the philosophical question of whether quantum logic constitutes a fundamental new logic for nature a debate started. A good overview of this discussion can be found in the book by Max Jammer [4]. We want to put forward our own personal opinion about this matter and explain why the word 'quantum logic' was not the best word to choose to indicate the scientific activity that has been taking place within this field. If 'logic', following the characterization of Boole, is the formalization of the 'process of our reflection', then quantum logic is not 144 DIEDERIK AERTS a new logic. Indeed, we obviously reflect following the same formal rules whether we reflect about classical parts of reality or whether we reflect about quantum parts of reality. Birkhoff and von Neumann, when they wrote their article in 1936, were already aware of this, and that is why they introduced the concept of 'experimental proposition'. It could indeed be that, even if we reason within the same formal structure about quantum entities as we do about classical entities, the structure of the 'experimental propositions' that we can use are different in both cases. With experimental proposition is meant a proposition that is connected in a well defined way with an experiment that can test this proposition. We will explicitly see later in this paper that there is some truth in this idea. Indeed, the set of experimental propositions connected to a quantum entity has a different structure than the set of experimental propositions connected to a classical entity. We believe however that this difference in structure of the sets of experimental propositions is only a little piece of the problem, and even not the most important one2 . It is our opinion that the difference between the logico-algebraic structures connected to a quantum entity and the logico-algebraic structures connected to a classical entity is due to the fact that the structures of our 'possibilities of active experimenting' with these entities is different. Not only the logical aspects of these possibilities of active experimenting but the profound nature of these possibilities of active experimenting is different. And this is not a subjective matter due to, for example, our incapacity of experimenting actively in the same way with a quantum entity as with a classical entity. It is the profound difference in nature of the quantum entity that is at the origin of the fact that the structure of our possibilities of active experimenting with this entity is different 3 . We could proceed now by trying to explain in great generality what we mean with this statement and we refer the reader to [6] for such a pre2 We can easily show for example that even the set of experimental propositions of a macroscopic entity does not necessarily have the structure of a Boolean algebra. This means that the only fact of limiting oneself to the description of the set of 'experimental' propositions already brings us out of the category of Boolean structures, whether the studied entities are microscopic or macroscopic [5] 3 We have to remark here that we do not believe that the set of quantum entities and the set of classical entities correspond respectively to the set of microscopic entities and the set of macroscopic entities as is usually thought. On the contrary, we believe, and this will become clear step by step in the paper that we present here, that a quantum entity should best be characterized by the nature of the structure of the possibilities of experimentation on it. In this sense classical entities show themselves to be special types of quantum entities, where this structure, due to the nature of the entity itself, takes a special form. But, as we will show in the paper, there exists macroscopic real physical entities with a quantum structure. STRUCTURES AND PARADOXES 145 sentation. In this paper we will explain what we mean mostly by means of a simple example. 3. THE E X AMP L E: THE QUA N TUM MAC H I N E As we have stated in the introduction, we will analyze the problems of quantum mechanics by means of simple models. The first model that we will introduce has been proposed at earlier occasions (see [5], [7], [8] and [9]) and we have named it the 'quantum machine'. It will turn out to be a real macroscopic mechanical model for the spin of a spin! quantum entity. We will only need the mathematics of high school level to introduce it. This means that alsothe readers that are not acquainted with the sophisticated mathematics of general quantum mechanics can follow all the calculations, only needing to refresh perhaps some of the old high school mathematics. We will introduce the example of the quantum machine model step by step, and before we do this we need to explain shortly how we represent our ordinary three dimensional space by means of a real three dimensional vector space. 3.1. The Mathematical Representation of Three Dimensional Space We can represent the three dimensional Euclidean space, that is also the space in which we live and in which our classical macroscopic reality exists, mathematically by means of a three dimensional real vector space denoted ]R3. We do this by choosing a fixed origin 0 of space and representing each point P of space as a vector v with begin-point 0 and end-point P (see Fig 1). Such a vector v has a direction, indicated by the arrow, and a length, which is the length of the distance from 0 to the point P. We denote the length of the vector v by Ivl. Fig 1 : A mathematical representation of the three dimensional Euclidean space by means of a three dimensional real vector space. We choose a fixed point 0 which is the origin. For an arbitrary point P we define the vector v that represents the point. i) The sum of vectors We introduce operations that can be performed with these vectors that indicate points of our space. For example the sum of two vectors v, W E ]R3 representing two points P and Q is defined by means of the parallelogram rule (see Fig 2). It is denoted by v + wand is again a vector of ]R3. 146 DIEDERIK AERTS P+Q Fig 2 : A representation of the sum of two vectors v and w, denoted by v+w. v o ii) Multiplication of a vector by a real number We can also define the multiplication of a vector v E lR 3 by a real number r E lR, denoted by rv. It is again a vector of lR 3 with the same direction and the same origin 0 and with length given by the original length of the vector multiplied by the real number r (see Fig 3). rP Fig 3 : A representation of the product of a vector v with a real number r, denoted by rv. rv P v o iii) The inproduct of two vectors We can also define what is called the inproduct of two vectors v and w, denoted by < v, w >, as shown in Fig 4. Q .. Ivlcosy ". w Fig 4 : A representation of the inproduct of two vectors v and w, denoted by <v,w>. P Y v 0 It is the real number that is given by the length of vector v multiplied by the length of vector w, multiplied by the cosine of the angle between the two vectors v and w. Hence < v, w > = Iv II w I cos r (1) where r is the angle between the vectors v and w (see Fig 4). By means of this inproduct it is possible to express some important geometric properties of space. For example: the inproduct < v, w > of STRUCTURES AND PARADOXES 147 two non-zero vectors equals zero iff the two vectors are orthogonal to each other. On the other hand there is also a straightforward relation between the inproduct of a vector with itself and the length of this vector < V,v >= Ivl 2 (2) iv) An orthonormal base of the vector space For each finite dimensional vector space with an inproduct it is possible to define an orthonormal base. For our case of the three dimensional real vector space that we use to describe the points of the three dimensional Euclidean space it is a set of three orthogonal vectors with the length of each vector equal to 1 (see Fig 5). V3 h3 h, ,·····················1 =,h,+"h,+"h, o V2 h2 ........... h2 ···!··········7 i ..."" .. Ih~: ..·. ~L. ..................·::::·..:~t2h .... v Fig 5 : An orthonormal base {hl,h2,h3} of the three dimen sional real vector space. We can write an arbitrary vector vasa sum of the three base vectors multiplied respectively by real numbers VI ,V2 and V3. These numbers are called the Cartesian coordinates of the vector v for the orthonormal base {hl,h 2,h3}. Hence the set {hI, h 2 , h3} is an orthonormal base of our vector space iff < hI, hI >= 1 < hl,h2 >= 0 < hI, h3 >= 0 < h 2 , hI >= 0 < h2, h2 >= 1 < h2, h3 >= 0 < h3,hl >= 0 < h3,h2 >= 0 < h 3, h3 >= 1 ]R3 (3) We can write each vector v E ]R3 as the sum of these base vectors respectively multiplied by real numbers VI, V2 and V3 (see Fig 5). Hence: (4) The numbers VI, V2 and V3 are called the Cartesian coordinates4 of the vector V for the orthonormal base {hI, h 2 , h3}. 4 It was Rene Descartes who introduced this mathematical representation of our three dimensional Euclidean space. 148 DIED ERIK AERTS v} The Cartesian representation of space As we have fixed the origin of our vector space we can also fix one specific orthonormal base, for example the base {hI,h2,h3}, and decide to express each vector v by means of the Cartesian coordinates with respect to this fixed base. We will refer to such a fixed base as a Cartesian base. Instead of writing v = vIh i + V2h2 + V3h3 it is common practice to write v = (VI, V2, V3), only denoting the three Cartesian coordinates and not the Cartesian base, which is fixed now anyway. As a logical consequence we denote hI = (1,0,0), h2 = (0,1,0) and h3 = (0,0,1). It is an easy exercise to show that the addition, multiplication with a real number and the inproduct of vectors are given by the following formulas. Suppose that we consider two vectors v = (VI,V2,V3) and W = ( WI, W2, W3) and a real number r, then we have: ° v + W = (VI + WI, V2 + W2, V3 + W3) rv = (rVI, rV2, rV3) < v, W >= VIWI + V2W2 + V3W3 (5) These are very simple mathematical formulas. The addition of vectors is just the addition of the Cartesian coordinates of these vectors, the multiplication with a real number is just the multiplication of the Cartesian coordinates with this number, and the inproduct of vectors is just the sum of one by one products of the Cartesian coordinates. This is one of the reasons why the Cartesian representation of the points of space is very powerful. vi} The representation of space by means of spherical coordinates It is possible to introduce many systems of coordination of space. We will in the following use one of these other systems: the spherical coordinate system. We show the spherical coordinates p, () and ¢ of a point P with Cartesian coordinates (VI, V2, V3) in Figure 6. We have the following well known and easy to verify relations between the two sets of coordinates (see Fig 6). VI = P sin () cos ¢ V2 = P sin () sin ¢ V3 = pcos() (6) STRUCTURES AND PARADOXES 149 Fig. 6 : A point P with Cartesian coordinates Vl,V2 and V3, and spherical coordinates p,O and ¢. 3.2. The States of the Quantum Machine Entity We have introduced in the foregoing section some elementary mathematics necessary to handle the quantum machine entity. First we will define the possible states of the entity and then the experiments we can perform on the entity. The quantum machine entity is a point particle P in three dimensional Euclidean space that we represent by a vector v. The states of the quantum machine entity are the different possible places where this point particle can be, namely inside or on the surface of a spherical ball (that we will denote by ball) with radius 1 and center 0 (Fig 7)5. Let us denote the set of states of the quantum machine entity by ~cq 6 . We will denote a specific state corresponding to the point particle being in the place indicated by the vector v by the symbol PV. SO we have: (7) 5 In the earlier presentations of the quantum machine [5], [7], [8] and [9], we only considered the points on the surface of the sphere to be the possible states of the quantum machine entity. If we want to find a model that is strictly equivalent to the spin model for the spin~ of a quantum entity, this is what we have to do. We will however see that it is fruitful, in relation with a possible solution of one of the quantum paradoxes, to introduce a slightly more general model for the quantum machine and also allow points of the interior of the sphere to represent states (see also [6] for the representation of this more general quantum machine) 6 The subscript cq stands for 'completed quantum mechanics'. We will see indeed in the following that the interior points of ball do not correspond to vector states of standard quantum mechanics. If we add them anyhow to the set of possible states, as we will do here, we present a completed version of standard quantum mechanics. We will come back to this point in detail in the following sections. 150 DIEDERIK AERTS Let us now explain in which way we interact, by means of experiments, with this quantum machine entity. As we have defined the states it would seem that we can 'know' these states just by localizing the point inside the sphere (by means of a camera and a picture for example, or even just by looking at the point). This however is not the case. We will define very specific experiments that are the 'only' ones at our disposal to find out 'where' the point is. In this sense it would have been more appropriate to define first the experiments and afterwards the states of the quantum machine entity. We will see that this is the way that we quantum entity 7. will proceed when we introduce the spin of a spin~ Fig. 7 : The quantum machine entity P with its vector representation v, its cartesian coordinates x,y and z, and its spherical coordinates p,() and <p. 3.3. The Experiments of the Quantum Machine Let us now introduce the experiments. To do this we consider the point u and the diametrically opposite point -u of the surface of the sphere ball. We install an elastic strip (e.g., a rubber band) of 2 units of length, such that it is fixed with one of its end-points in u and the other end-point in -u (Fig 8,a). As we have explained, the state Pv represents the point particle P located in the point v. We will limit ourselves in this first introduction of the quantum machine to states Pv where v = 1 and hence P is on the surface of the sphere. Later we will treat the general case. Let us now describe the experiment. Once the elastic is installed, the particle P falls from its original place v orthogonally onto the elastic, and sticks to it 7 It is essential for the reader to understand this point. In our model we have defined the states of the quantum machine entity, but actually there is no camera available to 'see' these states. The only experiments available are the ones that we will introduce now. STRUCTURES AND PARADOXES 151 (Fig 8,b). Then, the elastic breaks at some arbitrary point. Consequently the particle P, attached to one of the two pieces of the elastic (Fig 8,c), is pulled to one of the two end-points u or -u (Fig 8,d). Now, depending on whether the particle P arrives in u (as in Fig 8) or in -u, we give the outcome 01 or 02 to the experiment. We will denote this experiment by the symbol eu and the set of experiments connected to the quantum machine by Ceq. Hence we have: feq = {e u I u E ~3, u ~ (e) '-u lui = I} (8) Fig. 8 : A representation of the quantum machine. In(a) the particle P is in state pv, and the elastic corresponding to the experiment eu is installed between the two diametrically opposed points u and -u. In (b) the particle P falls orthogonally onto the elastic and sticks to it. In (c) elastic breaks and the particle P is pulled towards the point u, such that (d) it arrives at point u, and the experiment e u gets the outcome 01. Fig. 9 : A representation of the experimental process in the plane where it takes place. The elastic of length 2, corresponding to the experiment e u , is installed between u and -u. The probability, J.L(eu,pv,ol), that the particle P ends up in point u under influence of the experiment e" is given by the length of the piece of elastic L1 divided by the total length of the elastic. The probability, J.L(e u ,pv,02), that the particle P ends up in point -u is given by the length of the piece of elastic L2 divided by the total length of the elastic. 3.4. The Transition Probabilities of the Quantum Machine Let us now calculate the probabilities that are involved in these experiments such that we will be able to show later that they equal the quantum probabilities connected to the quantum experiments on a spin~ quantum particle. 152 DIED ERIK AERTS The probabilities are easily calculated. The probability, J-L(eu,pv, 01), that the particle P ends up in point u and hence experiment eu gives outcome 01, when the quantum machine entity is in state Pv, is given by the length of the piece of elastic L1 divided by the total length of the elastic (Fig 9). The probability, J-L(eu,pv, 02), that the particle P ends up in point -u, and hence experiment eu gives outcome 02, when the quantum machine entity is in state Pv, is given by the length of the piece of elastic L2 divided by the total length of the elastic. This gives us: 1 -L1 = -(1 + cos'Y) = cos2 -'Y 222 L2 1 . 2 'Y - = - (1 - cos 'Y) = sm 2 2 2 (9) (10) As we will see in next section, these are exactly the probabilities related to the spin experiments on the spin of a spin! quantum particle. 4. THE S PIN 0 F ASP I N ~ QUA N TUM PAR TIC L E Let us now describe the spin of a spin! quantum particle so that we can show that our quantum machine is equivalent to it. Here we will proceed the other way around and first describe in which way this spin manifests itself experimentally. 4.1. The Experimental Manifestation of the Spin The experiment showing the first time the property of spin for a quantum particle was the one by Stern and Gerlach [10], and the experimental apparatus involved is called a Stern-Gerlach apparatus. It essentially consists of a magnetic field with a strong gradient, oriented in a particular direction u of space (Fig 10). u A. direction of I,' the gradient I of the magnetic field outgoing beam arrival of the beam of quantum particles Fig 10: The Stern-Gerlach apparatus measuring the spin of a spin! quantum particle. The particle beam comes in from the left and passes through a magnetic field with a strong gradient in the u direction. The beam is split in two, one goes up, absorbed by a plate, and one down, passing through. STRUCTURES AND PARADOXES 153 A beam of quantum particles of spin ~ is directed through the magnet following a path orthogonal to the direction of the gradient of the magnetic field. The magnetic field splits the beam into two distinct beams which is a very unexpected phenomenon if the situation would be analysed by classical physics. One beam travels upwards in the direction of the gradient of the magnetic field and one downwards. The beam directed downwards is absorbed by a plate that covers this downwards part behind the magnetic field in the regions where the beams emerge. The upwards beam passes through and out of the Stern-Gerlach apparatus. We will denote such a Stern-Gerlach experiment by fu where u refers to the direction of space of the gradient of the magnetic field. If we consider this experiment being performed on one single quantum particle of spin ~, then it has two possible outcomes: (1) the particle is deflected upwards and passes through the apparatus, let us denote this outcome 01, (2) the particle is deflected downwards, and is absorbed by the plate, lets denote this outcome 02. The set of experiments that we will consider for this one quantum particle is the set of all possible SternGerlach experiments, for each direction u of space, which we denote by £89 8 . Hence we have: £89 = {Ju I u a direction in space} 4.2. The Spin States of a Spin~ (11) Quantum Particle Suppose that we have made a Stern-Gerlach experiment as explained and we follow the beam of particles that emerges. Suppose that we perform a second Stern-Gerlach experiment on this beam, with the gradient of the magnetic field in the same direction u as in the first preparing SternGerlach apparatus. We will then see that now the emerging beam is not divided into two beams. All particles of the beam are deviated upwards and none of them is absorbed by the plate. This means that the first experiment has 'prepared' the particles of the beam in such a way that it can be predicted that they will all be deviated upwards: in physics we say that the particles have been prepared in a spin state in direction u. Since we can make such a preparation for each direction of space u it follows that the spin of a spin ~ quantum particle can have a spin state connected to any direction of space. So we will have to work out a description of this spin state such that each direction of space corresponds to a spin state. This is exactly what quantum mechanics does and we will now expose this quantum description of the spin states. Since we have 8 The subscript sg stands for Stern-Gerlach. 154 DIED ERIK AERTS chosen to denote the Stern-Gerlach experiments by fu where u refers to the direction of the gradient of the magnetic field, we will use the vector v to indicate the direction of the spin of the quantum particle. Let us denote such a state by qv' This state qv is then connected with the preparation by means of a Stern-Gerlach apparatus that is put in a direction of space v. A particle that emerges from this Stern-Gerlach apparatus and is not absorbed by the plate 'is' in spin state qv' Let us denote the set of all spin states by L: sg . So we have: L:sg = {qv Iv a direction in space} (12) 4.3. The Transition Probabilities of the Spin If we now consider the experimental situation of two Stern-Gerlach apparatuses placed one after the other. The first Stern-Gerlach apparatus, with gradient of the magnetic field in direction v, and a plate that absorbs the particles that are deflected down, prepares particles in spin state qv. The second Stern-Gerlach apparatus is placed in direction of space u and performs the experiment fu. We can then measure the relative frequency of particles being deflected "up" for example. The statistical limit of this relative frequency is the probability that a particle will be deflected upwards. The laboratory results are such that, if we denote by /-L(fu, qv, od the probability that the experiment fu makes the particle deflect upwards-let us denote this as outcome ol-if the spin state is qv, and by /-L(fu, qv, 02) the probability that the experiment fu makes the particle be absorbed by the plate-let us denote this outcome by 02-if the spin state is qv, we have: (13) (14) where'Y is the angle between the two vectors u and v. 4.4. Equivalence of the Quantum Machine with the Spin It is obvious that the quantum machine is a model for the spin of a spin! quantum entity. Indeed we just have to identify the state Pv of the quantum machine entity with the state qv of the spin, and the experiment eu of the quantum machine with the experiment fu of the spin for all directions of space u and v. Then we just have to compare the transition STRUCTURES AND PARADOXES 155 probabilities for the quantum machine derived in formula's 9 and 10 with the ones measured in the laboratory for the Stern-Gerlach experiment 9 . Let us show in the next section that the standard quantum mechanical calculation leads to the same transition probabilities. 5. THE QU ANTUM DES CRIPTION OF THE SPIN We will now explicitly introduce the quantum mechanical description of the spin of a spin~ quantum particle. As mentioned in the introduction, quantum mechanics makes a very abstract use of the vector space structure for the description of the states of the quantum entities. Even for the case of the simplest quantum model, the one we are presenting, the quantum description is rather abstract as will become obvious in the following. As we have mentioned in the introduction, it is the abstract nature of the quantum description that is partly at the origin of the many conceptual difficulties concerning quantum mechanics. For this reason we will introduce again quantum mechanics step by step. 5.1. The Quantum Description of the Spin States quantum mechanics describes the different spin states qu for different directions of space u by the unit vectors of a two dimensional complex vector space, which we will denote by «::::2. This means that the spin states are not described by unit vectors in a three dimensional real space, as it is the case for our quantum machine model, but by unit vectors in a two dimensional complex space. This switch from a three dimensional real space to a two dimensional complex space lies at the origin of many of the mysterious aspects of quantum mechanics. It also touches some deep mathematical correspondences (the relation between SU(2) and SO(3) for example) of which the physical content has not yet been completely unraveled. Before we completely treat the spin states, let us give an elementary description of «::::2 itself. i) The vector space «::::2 «::::2 is a two-dimensional vector space over the field of the complex numbers. This means that each vector of «::::2-in its Cartesian representationis of the form (Zl' Z2) where Zl and Z2 are complex numbers. The addition 9 We do not consider here the states of the quantum machine corresponding to points of the inside of the sphere. They do not correspond to states that we have identified already by means of the Stern-Gerlach experiment. This problem will be analysed in detail further on. 156 DIEDERIK AERTS of vectors and the multiplication of a vector with a complex number are defined as follows: for (Zl, Z2), (tl, t2) E <e 2 and for Z E <e we have: (Zl, Z2) + (h, t2) = (Zl Z(Zl, Z2) = (ZZl, ZZ2) There exists an inproduct on + tl, Z2 + t2) (15) <e 2 : for (Zl, Z2), (tl, t2) E <e 2 we have: < (Zl,Z2),(tl,t2) >= Zrtl +Z2t2 (16) where * is the complex conjugation in <e. We remark that <, > is linear in the second variable and conjugate linear in the first variable. More explicitly this means that for (Zl, Z2), (tl, t2)' (rl, r2) E <e 2 and z, t E <e we have: < Z(Zl, Z2) + t(tl, t2)' (rl, r2) > < (Tl, T2), Z(Zl, Z2) + t(tl, t2) > z* < (Zl, Z2), (rl, r2) > +t* < (tl, t2)' (rl, r2) > (17) = Z < (Tl, r2), (Zl, Z2) > +t < (Tl, T2)' (tl, t2) > (18) = The inproduct defines the 'length' or better 'norm' of a vector (Zl, Z2) E <e 2 as: and it also defines an orthogonality relation on the vectors of <e 2 : we say that two vectors (Zl, Z2), (tl, t2) E <e 2 are orthogonal and write (Zl, Z2) ~ (tl,t2) iff: (20) A unit vector in <e 2 is a vector with norm equal to 1. Since the spin states are represented by these unit vectors we introduce a special symbol U(<e 2 ) to indicate the set of unit vectors of <e 2 • Hence: (21) The vector space has the Cartesian orthonormal base formed by the vectors (1,0) and (0,1) because we can obviously write each vector (Zl, Z2) in the following way: (22) STRUCTURES AND PARADOXES 157 ii) The quantum representation of the spin states Let us now make explicit the quantum mechanical representation of the spin states. The state qv of the spin of a spin~ quantum particle III direction v is represented by the unit vector Cv of U(C 2 ): Cv = (cos2e~,in-) e e·</> .</> = c(e,¢) (23) e where and ¢ are the spherical coordinates of the unit vector v. It is easy to verify that these are unit vectors in ((:2. Indeed, e . e Ic(e, ¢)I = cos2 '2 + sm2 '2 = 1 (24) It is also easy to verify that two quantum states that correspond to opposite vectors v = (1, e, ¢) and -v = (1, 7r - e, ¢ + 7r) are orthogonal. Indeed, a calculation shows (25) This gives us the complete quantum mechanical description of the spin states of the spin of a spin~ quantum particle. 5.2. The Quantum Description of the Spin Experiments We now have to explain how the spin experiments are described in the quantum formalism. As we explained already in detail, a spin experiment fu in the laboratory is executed by means of a Stern-Gerlach apparatus. To describe these experiments quantum mechanically we have to introduce somewhat more advanced mathematics, although still easy to master for those readers who have had no problems till now. So we encourage them to keep up with us. i) Projection operators The first concept that we have to introduce for the description of the experiments lO is that of an operator or matrix. We only have to explain the case we are interested in, namely the case of the two dimensional complex vector space that describes the states of the spin of a spin~ quantum particle. In this case an operator or matrix H consists of four complex numbers H =( Z11 Z21 Z12 ) Z22 (26) 10 We use 'experiment' to indicate the interaction with the physical entity. A measurement in this sense is a special type of experiment. 158 DIED ERIK AERTS that have a well defined operation on the vectors of C 2 : for (Zl' Z2) E C 2 we have H(Zl, Z2) = (81,82) where (27) It is easily verified that this operation is linear, which means that for (Zl' Z2), (h, t2) E C 2 and z, t E C we have: (28) Moreover it can be shown that every linear operation on the vectors of C 2 is represented by a matrix. There is a unit operator I and a zero operator 0 that respectively maps each vector onto itself and on the zero vector (0,0): I=(~ ~) o=(~ ~) (29) The sum of two operators HI and H2 is defined as the operator HI + H2 such that (30) and the product of these two operators is the operator HI . H2 such that (31) We need some additional concepts and also a special set of operators to be able to explain how experiments are described in quantum mechanics. We say that a vector (Zl' Z2) E C 2 is an 'eigenvector' of the operator H iff for some Z E C we have: (32) If (ZI' Z2) is different from (0,0) we say that Z is the eigenvalue of the operator H corresponding to the eigenvector (ZI' Z2)' Let us introduce now the special type of operators that we need to explain how experiments are described by quantum mechanics. A projection operator P is an operator such that p2 = P and such that for (ZI, Z2), (tl' t2) E C 2 we have: (33) We remark that a projection operator, as we have defined it here, is sometimes called an orthogonal projection operator. Let us denote the set of all projection operators by .C(C 2 ). STRUCTURES AND PARADOXES 159 If P E £((:2) then also 1 - P E £((:2). Indeed (I - P)(I - P) = 1 - PP + p2 = 1 - P - P + P = 1 - P. We can show that the eigenvalues of a projection operator are 0 or 1. Indeed, suppose that Z E (: is such an eigenvalue of P with eigenvector (Zl, Z2) #- (0,0). We then have P(Zl, Z2) = Z(Zl, Z2) = p 2(Zl, Z2) = Z2(Zl, Z2). From this follows that Z2 = Z and hence Z = 1 or Z = 0. Let us now investigate what the possible forms of projection operators are in our case ([:2. We consider the set {P, I - P} for an arbitrary P E £(([:2). i) Suppose that P(Zl,Z2) = (0,0) V (Zl,Z2) E ([:2. Then P = 0 and 1 - P = 1. This means that this situation is uniquely represented by the set {O,I}. ii) Suppose that there exists an element (Zl, Z2) E (:2 such that P(Zl, Z2) (0,0). Then P(P(Zl' Z2)) = P(Zl, Z2) which shows that P(Zl' Z2) is an eigenvector of P with eigenvalue 1. Further we also have that (I - P)(P(Zl' Z2)) = 0 which shows that P(Zl' Z2) is an eigenvector of 1 - P with eigenvalue O. If we further suppose that (I - P)(Y1, Y2) = o V (Yl, Y2) E (:2, then we have P = I and the situation is represented by the set {I,O}. i= iii) Let us now suppose that there exists a (Zl' Z2) as in ii) and an element (Y1, Y2) E (:2 such that (I - P)(Y1, Y2) =f. O. An analogous reasoning then shows that (I - P)(Y1, Y2) is an eigenvector of I - P with eigenvalue 1 and an eigenvector of P with eigenvalue O. Furthermore we have: < P(Zl' Z2), (I - P)(Y1, Y2) > < (Zl' Z2), P(I - P)(Y1, Y2) > = = (34) 0 which shows that P(Zl' Z2) -.l (I - P)(Yl, Y2). This also proves that P(Zl,Z2) and (I - P)(Y1,Y2) form an orthogonal base of <e 2 , and P is in fact the projection on the one dimensional subspace generated by P(Zl' Z2), while 1- P is the projection on the one dimensional subspace generated by (1 - P)(Y1, Y2). These three cases i), ii) and iii) cover all the possibilities, and we have now introduced all the elements necessary to explain how experiments are described in quantum mechanics. ii) The quantum representation of the measurements An experiment e in quantum mechanics, with two possible outcomes 01 and 02, in the case where the states are represented by the unit vectors 160 DIEDERIK AERTS of the two dimensional complex Hilbert space ((:2, is represented by such a set {P, I - P} of projection operators satisfying situation iii). This means that P i= 0 and I - P i= o. Let us state now the quantum rule that determines in which way the outcomes occur and what happens to the state under the influence of a measurement. If the state p of the quantum entity is represented by a unit vector (ZI, Z2) and the experiment e by the set of non zero projection operators {P, I - P}, then the outcome 01 of e occurs with a probability given by < (ZI,Z2),P(ZI,Z2) > and if 01 (35) occurs the unit vector (ZI, Z2) is changed into the unit vector P(ZI, Z2) IP(ZI, z2)1 (36) that represents the new state of the quantum entity after the experiment e has been performed. The outcome 02 of e occurs with a probability given by (37) and if 02 occurs the unit vector (ZI' Z2) is changed into the unit vector (I - P)(ZI,Z2) I(I - P)(ZI, z2)1 (38) that represents the new state of the quantum entity after the experiment e has been performed. We remark that we have: < (Zl' Z2), P(ZI, Z2) > + < (ZI, Z2), (I - P)(ZI, Z2) > =< (ZI' Z2), (ZI, Z2) >= 1 (39) such that these numbers can indeed serve as probabilities for the respective outcomes, i.e., they add up to 1 exhausting all other possibilities. iii) The quantum representation of the spin experiments Let us now make explicit how quantum mechanics describes the spin experiments with a Stern-Gerlach apparatus in direction u, hence the experiment f u· The projection operators that correspond to the spin measurement of the spin of a spin ~ quantum particle in the u direction are given by {Pu , 1- Pu } where Pu =~ ( 2 1 i:" cosa e+ 2{3 sin a e- i {3 sina ) 1 - cos a (40) STRUCTURES AND PARADOXES 161 where u = (1, a, (3) and hence a and (3 are the spherical coordinates angles of the vector u. We can easily verify that 1 ( 1- cosa . - e+~(J sin a 1- Pu = P u = 2 sin a ) 1 + cos a _e-i(J (41) Let us calculate in this case the quantum probabilities. Hence suppose that we have the spin in a state qv and a spin measurement f u in the direction u is performed. The quantum rule for the probabilities then gives us the following result. The probability /1(fu, qv, 01) that the experiment f u gives an outcome 01 if the quantum particle is in state qv is given by: (42) And the probability /1(fu, qv, 02) that the experiment fu gives an outcome 02 if the quantum particle is in state qv is given by: (43) This is not a complicated, but a somewhat long calculation. If we introduce the angle 'Y between the two directions u and v (see Fig. 11), it is possible to show that the probabilities can be expressed by means of this angle 'Y in a simple form. Fig. 11 : Two points v=(l,O,¢) and u=(l,a,(J) on the unit sphere and the angle 'Y between the two directions (O,¢) and (a,{3). We have: J..l(fu, qv, 01) = cos 2 ~ (44) J..l(fu, qv, 02) = sin2 ~ (45) 162 DIED ERIK AERTS These are indeed the transition probabilities for the spin measurement of the spin of a spin~ quantum particle found in the laboratory (see formulas 13, 14). To see this immediately we can, without loss of generality, choose the situation of the spin measurement for the z axis direction. Hence in this case we have u = (1, a, 13) with a = 0 and 13 = 0 for the experiment giving outcome 01 and a = 7r and 13 = 0 for the experiment giving the outcome 02. This gives us for fu the projection operators {Pu , P- u } given by: (46) For the probabilities we have: f-l(fu, qv, 01) =< cv , PuCv ) f-l ( fu,qv, 02 =< Cv,P-uCv 2 e >= cos "2 . 2 e >= SIn "2 (47) (48) We mentioned already that quantum mechanics describes the experiments in a very abstract way. This is one of the reasons why it is so difficult to understand many of the aspects of quantum mechanics. Since the quantum machine that we have presented also generates an isomorphic structure we have shown that, within our common reality, it is possible to realize this structure. The quantum machine is a model for the spin of a spin ~ quantum particle and it is also a model for the abstract quantum description in the two dimensional complex vector space. This result will make it possible for us to analyse the hard quantum problems and to propose new solutions to these problems. 6. QUANTUM PROBABILITY Probability, as it was first and formally introduced by Laplace and later axiomatised by Kolmogorov, is a description of our lack of knowledge about what really happens. Suppose that we say for a dice we throw, we have one chance out of six that the number 2 will be up when it lands on the floor. What is the meaning of this statement? It is very well possible that the motion of the dice through space after it has been thrown is completely deterministic and hence that if we knew all the details we could predict the outcome with certaintyll. So the emergence of the 11 We do not state here that complete determinism is the case, but just want to point out that it could be. STRUCTURES AND PARADOXES 163 probability ~ in the case of the diee does not refer to an indeterminism in the reality of what happens with the dice. It is just a description of our lack of knowledge about what precisely happens. Since we do not know what really happens during the trajectory of the dice and its landing and since we also do not know this for all repeated experiments we will make, there is an equal chance-for reasons of symmetry-for each side to be up after the diee stopped moving. Probability of ~ for each event expresses this type of lack of knowledge. Philosophers say in this case that the probability is 'epistemic'. Probability that finds its origin in nature itself and not in our lack of knowledge about what really happens is called 'ontologie'. Before the advent of quantum mechanics there was no important debate about thenature of the probabilities encountered in the physical world. It was commonly accepted that probabilities are epistemic and hence can be explained as due to our lack of knowledge about what really happens. It is this type of probability-the one encountered in classical physics-that was axiomatized by Kolmogorov and therefore it is commonly believed that Kolmogorovian probabilities are epistemic. It was rather a shock when physicists found out that the structure of the probability model that is encountered in quantum mechanics does not satisfy the axioms of Kolmogorov. Would this mean that quantum probabilities are ontologic? Anyhow it could only mean two things: (1) the quantum probabilities are ontologic, or (2) the axiomatic system of Kolmogorov does not describe all types of epistemic probabilities. In this second case quantum probabilities could still be epistemic. 6.1. Hidden Variable Theories Most of investigations that physicist, mathematicians and philosophers carried out regarding the nature of the quantum probabilities pointed in the direction of ontologie probabilities. Indeed, a whole area of research was born specifically with the aim of investigating this problem, it was called 'hidden variable theory research'. The reason for this naming is the following: if the quantum probabilities are merely epistemic, it must be possible to build models with 'hidden variables', their prior absence causing the probabilistic description. These models must be able to substitute the quantum models, i.e., they are equivalent, and they entail explicitly epistemic probabilities, in the sense that a randomisation over the hidden variables gives rise to the quantum probabilities. Actually physicists trying to construct such hidden variable models had thermodynamics in mind. The theory of thermodynamies is independent of classical mechanics, and has its own set of physical quantities 164 DIEDERIK AERTS such as pressure, volume, temperature, energy and entropy and its own set of states. It was, however, possible to introduce an underlying theory of classical mechanics. To do so, one assumes that every thermodynamic entity consists of a large number of molecules and the real pure state of the entity is determined by the positions and momenta of all these molecules. A thermodynamic state of the entity is then a mixture or mixed state of the underlying theory. The programme had been found feasible and it was a great success to derive the laws of thermodynamics in this way from Newtonian mechanics. Is it possible to do something similar for quantum mechanics? Is it possible to introduce extra variables into quantum mechanics such that these variables define new states, and the description of the entity based on these new states is classical? Moreover would quantum mechanics be the statistical theory that results by averaging over these variables? This is what scientists working on hidden variable theories were looking for. John von Neumann gave the first proof of the impossibility of hidden variables for quantum mechanics [1]. One of the assumptions that von Neumann made in his proof is that the expectation value of a linear combination of physical quantities is the linear combination of the expectation values of the physical quantities. As remarked by John Bell [11], this assumption is not justified for non compatible physical quantities, such that, indeed, von Neumann's proof cannot be considered conclusive. Bell constructs in the same reference a hidden variable model for the spin of a spin ~ quantum particle, and shows that indeed von Neumann's assumption is not satisfied in his model. Bell also criticizes in his paper two other proofs of the nonexistence of hidden variables, namely the proof by Jauch and Piron [12] and the proof by Gleason [13]. Bell correctly points out the danger of demanding extra assumptions to be satisfied without knowing exactly what these assumptions mean physically. The extra mathematical assumptions, criticized by Bell, were introduced in all these approaches to express the physical idea that it must be possible to find, in the hidden variable description, the original physical quantities and their basic algebra. This physical idea was most delicately expressed, without extra mathematical assumptions, and used in the impossibility proof of Kochen and Specker [14]. Gudder [15] gave an impossibility proof along the same lines as the one of Jauch and Piron, but now carefully avoiding the assumptions criticized by Bell. One could conclude by stating that everyone of these impossibility proofs consists of showing that a hidden variable theory gives rise to a certain mathematical structure for the physical quantities (see [1], [13J and [14]) or for the properties (see [12J and [15]) of the physical entity under consideration. The physical quantities and the properties of a STRUCTURES AND PARADOXES 165 quantum mechanical entity do not fit into this structure and therefore it is impossible to replace quantum mechanics by a hidden variable theory. More recently this structural difference between classical entities and quantum entities has been studied by Accardi within the category of the probability models itself [16], [17]. Accardi explicitly defines the concept of Kolmogorovian probability model starting from the concept of conditional probability. He identifies Bayes axiom as the one that, if it is satisfied, renders the probability model Kolmogorovian, i.e., classical. Again this approach of probability shows the fundamental difference between a classical theory and a quantum theory. A lot of physicists, once aware of this fundamental structural difference between classical and quantum theories, gave up hope that it would ever be possible to replace quantum mechanics by a hidden variable theory; and we admit to have been among them. Meanwhile it has become clear that the state of affairs is more complicated. 6.2. Hidden Measurement Theories Years ago we managed to built a macroscopic classical entity that violates the Bell inequalities [18], [19] and [20]. About the same time Accardi had shown that Bell inequalities are equivalent to his inequalities characterizing a Kolmogorovian probability model. This meant that the example we had constructed to violate Bell inequalities should also violate Accardi's inequalities characterizing a Kolmogorovian probability model, which indeed proved to be the case. But this meant that we had given an example of a macroscopic 'classical' entity having a non Kolmogorovian probability model. This was very amazing and the classification made by many physicists of a micro world described by quantum mechanics and a macro world described by classical physics was challenged. The macroscopic entity with a non Kolmogorovian probability model was first published in [7] and refined in [8], but essentially it is the quantum machine that we have presented again in this paper. We were able to show at that time the state of affairs to be the following. If we have a physical entity S and we have a lack of knowledge about the state p of the physical entity S, then a theory describing this situation is necessarily a classical statistical theory with a Kolmogorovian probability model. If we have a physical entity S and a lack of knowledge about the measurement e to be performed on the physical entity S, and to be changing the state of the entity S, then we cannot describe this situation by a classical statistical theory, because the probability model that arises is non Kolmogorovian. Hence, lack of knowledge about measurements, that change the state of the entity under study, gives rise 166 DIED ERIK AERTS to a non Kolmogorovian probability model. What do we mean by 'lack of knowledge about the measurement e'? Well, we mean that the measurement e is in fact not a 'pure' measurement, in the sense that there are hidden properties of the measurement, such that the performance of e introduces the performance of a 'hidden' measurement, denoted e A , with the same set of outcomes as the measurement e. The measurement e consists then in fact of choosing one way or another between one of the hidden measurements e A and then performing the chosen measurement eA. We can very easily see how these hidden measurements appear in the quantum machine. Indeed, if we call the measurement e~ the measurement that consists in performing eu and such that the elastic breaks in the point A for some A E [-u, u], then, each time eu is performed, it is actually one of the e~ that takes place. We do not control this, in the sense that the e~ are really 'hidden measurements' that we cannot choose to perform. The probability fL(eu,pv, or) that the experiment eu gives the outcome 01 if the entity is in state Pv is a randomisation over the different situations where the hidden measurements e~ gives the outcome 01 with the entity in state Pv. This is exactly the way we have calculated this probability in section 3.4. 6.3. Explaining the Quantum Probabilities First of all we have to mention that it is possible to construct a 'quantum machine' model for an arbitrary quantum mechanical entity. This means that the 'hidden measurement' explanation can be adopted generally, explaining also the origin of the quantum probabilities for a general quantum entity. We refer to [8], [21], [22], [23] and [24] for a demonstration of the fact that every quantum mechanical entity can be represented by a hidden measurement model. What is now the consequence of this for the explanation of the quantum probabilities? It proves that quantum probability is epistemic, and hence conceptually no ontologic probability has to be introduced to account for it. The quantum probability however finds its origin in the lack of knowledge that we have about the interaction between the measurement and the physical entity. In this sense it is a type of probability that is non-classical and does not appear in the classical statistical theories. Quantum mechanical states are pure states and descriptions of the reality of the quantum entity. This means that the 'hidden variable' theories which try to make quantum theory into a classical statistical theory are doomed to fail, as the mentioned no-go theorems for hidden variable theories had proven already without explaining why. Our approach allows to understand why. Indeed, STRUCTURES AND PARADOXES 167 there is another type of epistemic probability than the one identified in the classical statistical theories, namely the probability due to a lack of knowledge on the interaction between the measurement and the physical entity under study. It is natural that this new type of epistemic probability cannot be eliminated from a theory that describes the reality of the physical entity, because it appears when one incorporates the experiments related to the measurements of the properties of the physical entity in question. Therefore it is also natural that it remains present in the physical theory describing the reality of the physical entity, but it has no ontologic nature. This means that, with the explanation of the quantum probabilities put forward here, quantum mechanics does not contradict determinism for the whole of reality12. 6.4. Quantum, Classical and Intermediate If we demand for the quantum machine that the elastic can break at everyone of its points, and the breaking of a piece is such that it is proportional to the length of this piece, then this hypothesis fixes the possible 'amount' of lack of knowledge about the interaction between the experimental apparatus and the physical entity. Indeed, only certain type of elastic can be used to perform the experiments. On the other hand, we can easily imagine elastics that break according to different laws depending on their physical construction. Let us introduce the following different kinds of elastics: at one extremity we consider elastics that can break in everyone of its points and such that the breaking of a piece is proportional to the length of this piece. These are the ones we have already considered, and since they lead to a pure quantum structure, we call them quantum elastics. At the other extremity, we consider a type of elastic that can only break in one point, and let us suppose, for the sake of simplicity, that this point is the middle of the elastic (in [5], [25], [26] and [27] the general situation is treated). This kind of elastic is far from elastic, but since it is an extreme type of real elastics, we still give it that name. We shall show that if experiments are performed with this class of elastic, the resulting structures are classical, and therefore we will call them classical elastics. For the general case, we want to consider a class of elastics that can only break in a segment of length 2E around the center of the elastic. Let us call these E-elastics. 12 Our explanation does of course not prove that the whole of reality is deterministic. It shows that quantum mechanics does not give us an argument for the contrary. 168 DIEDERIK AERTS Fig 12 : An experiment with an E-elastic. The elastic can only break between the points -E and +E. L1 is the length of the interval where the elastic can break such that the point P finally arrives in u, and L2 is the length of the interval where the elastic can break such that the point P finally arrives at -u. The elastic with E = 0, hence the O-elastic, is the classical elastic, and the elastic with E = 1, hence the I-elastic, is the quantum elastic. In this way, the parameter E can be interpreted as representing the magnitude of the lack of knowledge about the interaction between the measuring apparatus and the physical entity. If E = 0, and for the experiment eu only classical elastics are used, there is no lack of knowledge, in the sense that all elastics will break at the same point and have the same influence on the changing of the state of the entity. The experiment eu is then a pure experiment. If E = 1, and for the experiment e u only quantum elastics are used, the lack of knowledge is maximal, because the chosen elastic can break at any of its points. In Fig 12 we have represented a typical situation of an experiment with an E-elastic, where the elastic can only break between the points -E and +E. Let us calculate the probabilities p( e u , Pv, 01) and p( e u , Pv, 02) for state-transitions from the state Pv of the particle P before the experiment e u to one of the states Pu or p-u. Different cases are possible: (1) If the projection of the point P lies between -u and -E (see Fig 12), then p(eu,pv, 01) = 0 (49) p(eu,pv, 02) = 1 (2) If the projection of the point P lies between +E and u, then p(eu,pv,od = 1 p(eu,pv, 02) = 0 (3) If the projection of the point P lies between 1 -(E - cos,),) 2E 1 2E (E + cos')') (50) -E and +E then (51) (52) The entity that we describe here is neither quantum, nor classical, but intermediate. If we introduce these intermediate entities, then it becomes STRUCTURES AND PARADOXES 169 possible to describe a continuous transition from quantum to classical (see [5], [25], [26] and [27] for details). It gives us a way to introduce a specific solution to 'classical limit problem'. 7. QUANTUM AXIOMATICS: THE OPERATIONAL PART In the foregoing example of the intermediate situation we have the feeling that we consider a situation that will not fit into standard quantum mechanics. However the situation is either not classical. But how could we prove this? This could only be done if we had an axiomatic formulation of quantum mechanics and classical mechanics, such that the axioms could be verified on real physical examples of entities to see whether a certain situation is quantum or classical or neither. This means that the axioms have to be formulated by means of concepts that can be identified properly if a real physical entity is given. This is certainly not the case for standard quantum mechanics, but within the quantum structures research large parts of such an axiomatic system has been realised through the years. 7.1. State Property Spaces: the Ontologie Part By lack of space we can not expose all the details of an operational axiomatic formulation, but we will consider the most important ingredients quantum particle or the in some detail and consider the spin of a spin~ quantum machine as an example. In the first place we have to formalize the basic concepts: states and properties of a physical entity. i) The states of the entity S With each entity S corresponds a well defined set of states E of the entity. This are the modes of being of the entity. This means that at each moment the entity S 'is' in a specific state pEE. ii) The properties of the entity S Historically quantum axiomatics has been elaborated mainly by considering the set of properties 13 . With each entity S corresponds a well defined set of properties .c. The entity S 'has' a certain property or does not have it. We will respectively say that the property a E .c is 'actual' or is 'potential' for the entity S. 13 We have to remark that in the original paper of Birkhoff and von Neumann [2], the concept of 'operational proposition' is introduced as the basic concept. An operational proposition is not the same as a property (see [28], [29]), but it points at the same structural part of the quantum axiomatic. 170 DIEDERIK AERTS To be able to present the axiomatisation of the set of states and the set of properties of an entity S in a mathematical way, we have to introduce some additional concepts. Suppose that the entity S is in a specific state p E ~. Then some of the properties of S are actual and some are not (potential). This means that with each state p E ~ corresponds a set of actual properties, subset of .c. Mathematically this defines a function ~ : ~ --+ P('c), which makes each state p E ~ correspond to the set ~(p) of properties that are actual in this state. With the notation P('c) we mean the 'powerset' of 'c, i.e., the set of all subsets of .c. From now on-and this is methodologically a step towards mathematical axiomatization-we can replace the statement 'property a E ,C is actual for the entity S in state p E ~' by 'a E ~ (p)', which is just an expression of set theory. Suppose now that for the entity S a specific property a E ,C is actual. Then this entity is in a certain state pEL: that makes a actual. With each property a E ,C we can associate the set of states that make this property actual, i.e., a subset of~. Mathematically this defines a function '" : ,C --+ P(~), which makes each property a E ,C correspond to the set of states ",(a) that make this property actual. This is a similar step to axiomatization. Indeed, this time we can replace the statement 'property a E ,C is actual if the entity S is in state p E ~' by the set theoretical expression 'p E ",(a)'. Summarising the foregoing we now have: property a E ,C is actual for the entity S in state p E ¢:? a E ~(p) ¢:? p E ",(a) ~ (53) This expresses a fundamental 'duality' among states and properties. We will introduce a specific mathematical structure to represent an entity S, its states and its properties, taking into account this duality. We need the set ~, the set 'c, and the two functions ~ and "'. Definition 1 (state property space). Consider two sets two functions ~:P('c) pf-7~() '" : ,C --+ P(~) If p E ~ a f-7 ",(a) ~ and ,C and (54) and a E ,C we have: aE ~(p) ¢:? P E ",(a) (55) then we say that (~,'c "') is a state property space. The elements of ~ are interpreted as states and the elements of ,C as properties of the STRUCTURES AND PARADOXES 171 entity S. The interpretation of (55) is 'property a is actual if S is in state p' 14 There are two natural 'implication relations' we can introduce on a state property space. If the situation is such that if 'a E [, is actual for S in state p E ~' implies that 'b E [, is actual for S in state p E ~' we say that the property a implies the property b. This 'property implication' relation is expressed by a mathematical relation on the set of properties (see following definition). If the situation is such that 'a E [, is actual for S in state q E ~' implies that 'a E is actual for S in state p E ~' we say that the state p implies the state q. Again we will express this 'state implication' by means of a mathematical relation on the set of states. Definition 2 {state implication and property implication}. Consider a state property space (~, £,~ ~). For a, bE £ we introduce: a -< b {:} and we say that a ~(a) (56) C ~(b) 'implies' b. For p, q p -< q {:} ~(q) E ~ we introduce: C ~(p) (57) and we say that p 'implies' q 15. We will introduce now the mathematical concept of a pre-order relation. Definition 3 {pre-order relation}. Suppose that we have a set Z. We say that -< is a pre-order relation on Z iff for x, y, z E Z we have: x-<x x -< y and y -< z =? x -< z For two elements x, y E Z such that x -< y and y -< x we denote x and we say that x is equivalent to y. (58) :::::! y 14 We remark that it is enough to give two sets L: and £ and a function ~ : L: --> P(£) to define a state property space. Indeed, if we define the function r;, : £ --> P(L:) such that r;,(a) = {p I pEL:, a E ~(p)} then (L:, £,~ r;,) is a state property space. This explains why we do not explicitly consider the function r;, in the formal approach outlined in [6], [30] and [31] in the definition of a state property system, which is a specific type of state property space. Similarly it would be enough to give L:, £ and r;, : £ --> P(L:). 15 Remark that the state implication and property implication are not defined in a completely analogous way. Indeed, then we should for example have written p -< q <=} ~(p) c ~(q). That we have chosen to define the state implication the other way around is because historically this is how intuitively is thought about states implying one another. 172 DIED ERIK AERTS It is easy to verify that the implication relations that we have introduced are pre-order relations. Theorem 1. Consider a state property space (2:, L, C K:), then 2:,-< and L, -< are pre-ordered sets. We can show the following for a state property space Theorem 2. Consider a state property space (2:, L, ~, K:). (1) Suppose that a, bEL and P E 2:. If a E ~(p) and a -< b, then b E ~(p). (2) Suppose that p, q E 2: and a E L. If q E K:(a) and p -< q then p E K:(a). Proof: (1) We have p E K:(a) and K:(a) C K:(b). This proves that p E K:(b) and hence b E ~(p). (2) We have a E ~(q) and ~(q) C ~(p) and hence a E ~(p). This shows that p E K:(a). The reader will now better understand why the original studies of the axiomatization of quantum mechanics have been called quantum logic. Indeed, we have also used the name 'implication'. We will see that we can also introduce concepts that are close to 'disjunction' and 'conjunction'. But we point out again that we are structuring more than just the logical aspects of entities. We aim at a formalization of the complete ontologic structure of physical entities. 7.2. Meet Properties and Join States If we have a structure with implications and we are inspired by logic, we are tempted to wonder about conjunctions and disjunctions. Here again it becomes clear that we are studying a quite different situation than the one analyzed by traditional logic. Suppose we consider a set of properties (aik It is very well possible that there exist states of the entity S in which all the properties ai are actual. This is in fact always the case if niK:(ai) "I- 0. Indeed, if we consider p E niK:( ai) and S in state p, then all the properties ai are actual. If it is such that the situation where all properties ai of a set (ai)i and no other are actual is again a property of the entity S, we will denote this new property by !\iai, and call it the 'meet property' of all ai. Clearly we have !\iai is actual for S in state p E 2: iff ai is actual for all i for S in state p. This means that we have !\iai E ~(p) iff ai E ~(p) Vi. This formulation of the 'meet property' gives us the clue how to introduce it formally in a state property space. Suppose now that we consider a set of states (pj) j of the entity S. It is very well possible that there exist properties of the entity such that these properties are actual if S is in anyone of the states Pj. This is in STRUCTURES AND PARADOXES 173 fact always the case if nj~(p) =I=- 0. Indeed suppose that a E nj~(p). Then we have that a E ~ (pj) for each one of the states Pj, which means that a is actual if S is in anyone of the states Pj. If it is such that the situation where S is in anyone of the states Pj is again a state of S, we will denote this new state by VjPj and call it the 'join state' of all Pj. Clearly we have that a property a E C is actual for S in state V jPj iff this property a is actual for S in any of the states Pj. This formulation of the 'join state' indicates again the way we have to introduce it formally in a state property space 16 . The existence of meet properties and join states will give additional structure to L; and C. Definition 4 (complete state property space). Consider a state propK:). We say that the state property space is 'property erty space (L;, C,~ complete' iff for an arbitrary set (ai)i, ai E C of properties there exists a property !\iai E C such that for an arbitrary state pEL;: (59) We say that a state property space is 'state complete' iff for an arbitrary set of states (pj) j, Pj E ~ there exists a state V jPj E ~ such that for an arbitrary property a E C: (60) If a state property space is property complete and state complete we call it a 'complete' state property space. The following definition and theorem explain why we have chosen to call such a state property space complete. Definition 5 (complete pre-ordered set). Suppose that Z, --< is a preordered set. We say that Z is a complete pre-ordered set iff for each subset (Xi)i, Xi E Z of elements of Z there exists an infimum and a supremum in Z17. 16 We remark that we could also try to introduce join properties and meet states. It is however a subtle, but deep, property of reality, that this cannot be done on the same level. We will understand this better when we introduce in the next section the operational aspects of the axiomatic approach. We will see there that only meet properties and join states can be operationally defined in the general situation. 17 An infimum of a subset (Xi)i of a pre-ordered set Z is an element of Z that is smaller than all the Xi and greater than any element that is smaller than all Xi. A supremum of a subset (Xi)i of a pre-ordered set Z is an element of Z that is greater than all the Xi and smaller than any element that is greater than all the Xi. 174 DIEDERIK AERTS Theorem 3. Consider a complete state property space Then ~,-< and C, --< are complete pre-ordered sets. (~, C,~ K,). Proof: Consider an arbitrary set (ai)i, ai E C. We will show that !\iai is an infimum. First we have to proof that !\iai --< ak V k. This follows immediately from (59) and the definition of --< given in (56). Indeed, from this definition follows that we have to prove that K,(!\iai) C K,(ak) V k. Consider p E K,(!\iai). From (55) follows that this implies that !\iai E ~(p). Through (59) this implies that ak E ~(p) V k. If we apply (55) again this proves that p E /'i,(ak) V k. So we have shown that K,(!\iad C K,(ak) V k. This shows already that !\iai is a lower bound for the set (aik Let us now show that it is a greatest lower bound. So consider another lower bound, a property b E C such that b --< ak V k. Let us show that b --< !\iai. Consider p E K,(b), then we have pEak V k since b is a lower bound. This gives us that ak E ~(p) V k, and as a consequence !\iai E ~(p). But this shows that p E /'i,(!\iai). So we have proven that b --< !\i ai and hence !\i ai is an infimum of the subset (ai k Let us now prove that V jPj is a supremum of the subset (pj k The proof is very similar, but we use (60) in stead of (59). Let us again first show that V jPj is an upper bound of the subset (pj k We have to show that Pl --< VjPj V l. This means that we have to prove that ~(VjP) C ~(pl) V l. Consider a E ~(VjP), then we have VjPj E /'i,(a). From (60) it follows that Pl E /'i,(a) V l. As a consequence, and applying (55), we have that a E ~(pl) V l. Let is now prove that it is a least upper bound. Hence consider another upper bound, meaning a state q, such that Pl --< q V l. This means that ~(q) C ~(Pl) V l. Consider now a E ~(q), then we have a E ~(pl) V l. Using again (55), we have PI E K,(a) V l. From (60) follows then that Vjpj E /'i,(a) and hence a E ~(Vja). We have shown now that !\iai is an infimum for the set (ai)i, ai E C, and that VjPj is a supremum for the set (Pj)j,Pj E ~. It is a mathematical consequence that for each subset (ai)i, ai E C, there exists also a supremum in C, let is denote it by Viai, and that for each subset (pj) j , Pj E ~, there exists also an infimum in ~, let us denote it by !\jPj. They are respectively given by Viai = !\xE£:',ai...;xVi x and !\jPj = VyE'E,Y"';PjVj y18. For both C and ~ it can be shown that this implies that there is at least one minimal and one maximal element. Indeed, an infimum of all elements of C is a minimal element of C and an infimum of the empty 18 We remark that the supremum for elements of £:. and the infimum for elements of E, although they exists, as we have proven here, have no simple operational meaning, as we will see in the next section. STRUCTURES AND PARADOXES 175 set is a maximal element of £'. In an analogous way a supremum of all elements of E is a maximal element of E and a supremum of the empty set is a minimal element of I;. Of course there can be more minimal and maximal elements. If a property a E £, is minimal we will express this by a ~ 0 and if a property bE£' is maximal we will express this by b ~ I. An analogous notation will be used for the maximal and minimal states. For a complete state property space we can specify the structure of the maps ~ and K, somewhat more after having introduced the concept of 'property state' and 'state property'. Theorem 4. Consider a complete state property space (~, £', ~, K,). For pEE we define the 'property state' corresponding to p as the property s(p) = /\aE~(p). For a E .c we define the 'state property' corresponding to a as the state t( a) = VpEK(a)P. We have two maps: t : .c ~ I; a H t(a) s :~ -+ £, P f---7 s(p) a -< b {:} t(a) -< t(b) p -< q {:} s(p) -< s(q) t(l\iai) ~ I\i t (ai) s(Vjpj) ~ Vjs(pj) (61) (62) Proof: Suppose that p -< q. Then we have ~(q) C ~(p). From this follows that s(p) = l\aEt;(p)a -< l\aEt;(q)a = s(q). Suppose now that s(p) -< s(q). then we have s(q) -< a. Hence also s(p) -< a. But this Take a E ~(q), Hence this shows that ~(q) C ~(p) and as a implies that a E ~(p). consequence we have p -< q. Because I\iai -< ak \:f k we have t(l\iai) -< t(ak) \:fk. This shows that t(l\iai) is a lower bound for the set (t(adk Let us show that it is a smallest lower bound. Suppose that p -< t(ak) \:f k. We remark that t(ak) E K,(ak). Then it follows that p E K,(ak) \:f k. As a \:f k. But then I\iai E ~(p) which shows consequence we have ak E ~(p) that p E K,(l\iai). This proves that p -< t(l\iai). So we have shown that t(l\iai) is a smallest lower bound and hence it is equivalent to I\it(ai). Theorem 5. Consider a complete state property space (~, I:,~ K,). For ~ we have ~(p) = [s(p), +00] = {a E I: I s(p) -< a}. For a E I: we have K,(a) = [-00, t(a)] = {p E ~ I p -< t(a)}. pE Proof: Consider b E [s(p) , +00]. This means that s(p) -< b, and hence ~(p). Consider now b E ~(p). Then s(p) -< b and hence bE [s(p) , +00]. bE 176 DIEDERIK AERTS If p is a state such that ~(p) = 0, this means that there is no property actual for the entity being in state p. We will call such states 'improper' states. Hence a 'proper' state is a state that makes at least one property actual. In an analogous way, if /';;(a) = 0, this means that there is no state that makes the property a actual. Such a property will be called an 'improper' property. A 'proper' property is a property that is actual for at least one state. Definition 6. Consider a state property space (~,.c /';;). We call p E a 'proper' state iff ~(p) =I=- 0. We call a E .c a 'proper' property iff /';;(a) =I=- 0. A state p E ~ such that ~(p) = 0 is called an 'improper' state, and a property a E .c such that /';;(a) = 0 is called an 'improper' property. ~ It easily follows from theorem 5 that a complete state property space has no improper states (J ~ 1\0 E ~(p) and no improper properties (0 ~ v0 E /';;(a)). 7.3. Tests and Preparations: the Operational Part Our contact with physical entities of the exterior world happens by means of experiments we can perform. A test is an experiment we perform on the physical entity in a certain state testing a certain hypothesis. States can often be prepared. A preparation is an experiment we perform on the physical entity such that as a result of the experiment the entity is in a certain state. We will not develop the algebra of experiments connected to a physical entity in a complete way in this paper, and refer to [6] for such an elaboration. Here we will only introduce the concepts that we need for our principal purpose: the presentation of quantum axiomatics. i) The tests on the entity S Tests are experiments that verify a certain hypothesis about the entity S. More specifically tests can test properties of the entity S in the following way. With a property a E .c corresponds a test a(a), which is an experiment with two possible outcomes yes and no. If the test a(a) has an outcome yes it does not yet prove that the property a is actual. It is only when we can predict with certainty that the test would have an outcome yes, without necessarily performing it, that the property a is actual. STRUCTURES AND PARADOXES 177 Definition 1 (testing a property). Suppose that we have an entity S with corresponding state property space (~, [" ~, /'i,). a( a) is a test of the property a E [, if we have a E ~(p) {:? yes can be predicted for a(a) S being in state p (63) ii) The preparations of the entity S Preparations are experiments that prepare a state of the entity S. More specifically, with a state p E ~ corresponds a preparation 7r(p) which is an experiment such that after the performance of the experiment the entity 'is' in state p. Definition 8 (preparing a state). Suppose that we have an entity S with corresponding state property space (~, [,~ /'i,). 7r(p) is a preparation of the state p E ~ p E /'i,(a) if we have {:? a is actual after the preparation 7r(p) (64) For a set of tests (ai)i and for a set of preparations (7rj)j we can now introduce in a very natural way a new test, that we call the product test, denoted by lIiai, and a new preparation, that we call the product preparation, denoted by IIp!"j, as follows: Definition 9 (product test and preparation). To execute lIiai we choose one of the ai, perform it and consider the outcome that we obtain. To execute lIj7rj we choose one of the 7rj, perform is and consider the state that we obtain. We want to show now that the product test tests an infimum of a set of properties, while the product preparation prepares a supremum of a set of states. Theorem 6. Suppose that we have an entity S with corresponding state property space (I::, 12,~ K.). Consider a set of properties (ai)i, ai E £ and a set of states (Pj)j, Pj E I::. Suppose that we have tests and preparations available for all properties and states. Then the product test lIia(ai) tests a meet property /\iai, where a(ak) tests ak V k, and the product preparation lIj7r(pj) prepares a join state VjPj, where lI(Pl) prepares PI V l. Proof: We have to show that 'yes can be predicted for lIia(ai) the entity S being in state P' is equivalent to 'ak E ~(p) V k'. This follows immediately from the definition of the product test. Indeed 'yes can be 178 DIED ERIK AERTS predicted for IIia(ai) the entity S being in state p' is equivalent to 'yes can be predicted for a(ak) \j k the entity S being in state p'. Consider now an arbitrary property a E 1: and suppose that (7r(Pj))j is a set of preparations that make a actual if the entity S is in state Pj. Consider now the preparation II j 7r(pj) , that consists of choosing one of the 7r(pj) and performing it. Then it is clear that a is actual after this preparation, since S will be in one of the states Pj. On the other hand, suppose now that IIj1r(pj) is a preparation that makes a E 1: actual. Consider an arbitrary one of the preparations 7r(Pk) of the product preparation. Then obviously also this preparation has to make a actual, since it could have been this one that was chosen by performing the product preparation. This shows that II j 7r(pj) prepares the state V jPj. This theorem shows that it is natural to introduce the meet property for a set of properties and the join state for a set of states, like we did in the foregoing section. It is time now that we expose the concepts that we have introduced here for the example of the spin of a spin! quantum particle. 7.4. The Example of the Spin Model In section 5 we have explained in detail the standard quantum description of the spin of a spin! quantum particle. For the case of the spin of a spin! quantum particle, the experiments fu have only two outcomes 01 and 02, and hence they are tests in the sense that if 01 is interpreted as yes then 02 is no. This means that we can represent the properties by means of the projection operators that we use to represent the experiments. For each direction u we have a property au that is represented by the projection operator given in formula (40) Pu = ~ ( 1 + cos a 2 e+ 2 {3 sin a e- i {3 sin a ) 1 - cos a (65) where u = (1, a, (3) and hence a and (3 are the spherical coordinates angles of the vector u. The set of properties 1:sPin~ is given by these properties au and the maximal and minimal property that we will respectively denote by I and 0 19 . Hence: 1: sPin ! = {au'!, 0 Iu E surface of the sphere ball} (66) 19 It will become clear in the next section why for the quantum case there is a unique minimal property and a unique maximal property 179 STRUCTURES AND PARADOXES We have stated in section 5 that a state qv of the spin of a spin! quantum particle in direction v is represented by means of the unit vector Cv in the two dimensional complex vector space (2. We have to elaborate a little bit more on the description of the states. Indeed, if we consider again our quantum machine, which is a model for the spin of a spin~ quantum particle, then we can see that there are more states than the ones represented by the unit vectors. If we consider a point w in the interior of the sphere, hence not on the surface of the sphere, then this is also a possible state of the quantum machine, not represented however by a unit vector of the vector space, but by a density operator 20 . Let us analyse this situation in detail. Let us calculate the probabilities for such a state Pw where w is a point inside the sphere. First we remark the following. Because ball is a convex set, each vector w E ball can be written as a convex linear combination of two vectors v and -von the surface of the sphere (see Fig 13). More concretely this means that we can write (referring to the wand v and -v in the Figure 13): w = Al . V - A2 . v, 0::; AI, A2 ::; 1, Al + A2 = 1 (67) Hence, if we introduce these convex combination coefficients AI, A2 we have w = (AI - A2) . v. Let us calculate now the transition probabilities in a general state Pw with w E ball and hence Ilwll ::; 1 (see Fig 13). Again the probability f.-l(eu,pw, 01), that the particle P ends up in point u and hence experiment eu gives outcome 01 is given by the length of the piece of elastic Ll divided by the total length of the elastic. The probability, f.-l(eu,pw, 02), that the particle P ends up in point -u, and hence experiment e u gives outcome 02 is given by the length of the piece of elastic L2 divided by the total length of the elastic. This means that we have: (68) (69) (70) (71) 20 A density operator is an operator W such that < c, We >=< We, c > VeE 1[2 (which means self-adjointness), and such that 0 ::; < W c, W c > V c E 1[2 (which means positiveness), and such that the trace equals to 1. 180 DIEDERIK AERTS v -v Fig 13 : A representation of the experimental process in the case of a state Pw where w is a point of the interior of the sphere. The elastic of length 2, corresponding to the experiment e u , is installed between u and -u. The probability, JL(e u ,pw,ol), that the particle P ends up in point u under influence of the experiment eu is given by the length of the piece of elastic Ll divided by the total length of the elastic. The probability, JL( eu ,pw ,02), that the particle P ends up in point -u is given by the length of the piece of elastic L2 divided by the total length of the elastic. These are new probabilities that will never be obtained if we limit the set of states to the unit vectors of the two dimensional complex space. The question is now the following: can we find a mathematical concept, connected in some way or another to the Hilbert space, that would allow us, with a new quantum rule for calculating probabilities, to recover these probabilities? The answer is yes. We will show that these new 'pure' states of the interior of the sphere can be represented using density operators, the same operators that are used within the standard quantum formalism to represent mixed states. And the standard quantum mechanical formula that is used to calculate probabilities connected to mixed states, represented by density operators, can also be used to calculate the probabilities that we have identified here. But of course the meaning will be different: in our case this standard formula will represent a transition probability from one pure state to another and not the probability connected to the change of a mixed state. Let us show all this explicitly and do this by constructing the density operators in question. The well known quantum formula for the calculation of transition probabilities related to an experiment e, represented by the projections {P, 1- P}, and where the quantum entity is in a mixed state p represented by the density operator W, is the following: f1( e, p, P) = tr(W . P) (72) where tr is the trace of the operator21. A standard quantum mechanical calculation shows that the density ().¢ ().¢ operator representing the ray state Cv = (cos 2ez"2, sin 2e-z"2) (see (23)) is given by: (73) 21 The trace of an operator is the sum of its diagonal elements. STRUCTURES AND PARADOXES 181 and the density operator representing the diametrically opposed ray state C v is given by: sin2 It. - sin fl.2 cos fl.e-ir/> ) 2 W(-v) = ( - sin fl. cos2 fieir/> 2 cos fi 2 2 2 (74) We will show now that the convex linear combination of these two density operators with convex weights Al and A2 represents the state Pw if we use the standard quantum mechanical formula (72) to calculate the transition probabilities. If, for w = Al V + A2 ( -v), we put: (75) we have: W( w) = \ Al COS ( (Al \ - 2 () 2 + A2 \ sm . 2 2,() \ ) . () () ~¢ A2 sm 2 cos 2 e (\ . () () -i¢ ) sm 2 cos 2 e (76) \ 2 13 Al sm 2 + A2 cos 2 Al - \ A2 ) \ . 2 () and it is easy to calculate now the transition probabilities using (72) and: p=(~ ~) (77) We have: W( W· ) P -_ ( ) (78) . 2() 2() = Al sm 2' + .\2 cos 2' (80) Al cos2 ~ + A2 sin2 ~ B B ~ (AI - A2) sin 2 cos '2e t ~ and hence, comparing with (68), we find: In an analogous way we find that: tr(W(w) . (1 - P)) = (..L(eu,pw, 02) So we have shown that we can represent each one of the new states Pw by the density operator W(w) if we use (72) for the calculation of the transition probabilities. We can also prove that each density operator in ([:2 is of this form. We show this easily by using the general properties of density operators. 182 DIEDERIK AERTS Let us identify now the set of states for the case of the spin of a spin~ quantum particle. Each state is of the form Pw with w a point of the sphere ball. There also exist a zero state 0 and a unit state J. = {Pw, 0, J I w L;sPin~ E (81) ball} The state property space corresponding to the spin of a spin~ particle is given by (L;sPin~' .csPin~' ~spin' i'\;sPn~) where quantum (82) We have for u and v belonging to the sphere ball: au E ~sPin We have, for ~spin(PV) Ivl = (Pw) ~ Pw E i'\;spn~ (au) ~ u =w (83) 1, and hence pertaining to the surface of ball: = {av,I} (84) For w < 1, and hence pertaining to the interior of ball we have: ~sPin(w) = {J} (85) 8. QUA N TUM A X 10M A TIC S: THE TEe H N I CAL PAR T In the foregoing section we have introduced the structure that can be operationally founded. To come to the full structure of standard quantum mechanics some additional axioms have to be introduced which are more of a technical nature. This section will be mathematically more sophisticated, but can be skipped for those readers that are mainly interested in the results that are presented in next section. In the introduction we mentioned that quantum axiomatics was developed to build up standard quantum mechanics and not to change it. Meanwhile it has become clear that some of the more technical axioms of standard quantum mechanics are probably not generally satisfied in nature. This finding will be our main comment regarding the standard axioms, and for this reason we anyhow have to introduce them. An analysis of the failing axioms and their consequences is presented in the next section. 8.1. State Property Systems Since we will introduce in this section the axioms that have very little operational meaning, we will enter much less in detail. 183 STRUCTURES AND PARADOXES (1) The identification of properties As we have seen in the example of the spin of a spin~ quantum par- ticle, a property of the physical entity is represented by a projection operator. This remains true for a general quantum entity. Suppose that we consider two properties a and b of a general quantum entity, and their corresponding projection operators Pa and Pb. The implication of properties, giving rise to a pre-order relation on is translated for a quantum entity as follows by means of the projection operators: .c, Theorem 7. Consider a state property space (~, 12, ~, r;,) corresponding to a quantum entity S, described by means of the standard quantum formalism in a Hilbert space H. For two properties a, b E 12, and corresponding projection operators Pa and Pb we have: (86) Definition 10. Consider a pre-ordered set Z, -<. The pre-order relation -< is a partial order relation iff we have for x, Y E Z x -< y and y -< x =? x =Y (87) r;,) corresponding Theorem 8. Consider a state property space (I:, 12,~ to a quantum entity S, described by means of the standard quantum formalism in a Hilbert space H. For two properties a, bE £ we have: a -< band b -< a =? a= b (88) and hence the pre-order relation on £ is a partial order relation. Proof: Suppose that a -< band b -< a. Then from theorem 7 follows that Pa = PaPb = PbPa = Pb. which means that a = b. (2) Completeness of the property lattice The second special property for quantum entities that we will identify has already been explained in the foregoing section. It is related to the existence of the meet property for a set of properties. Suppose that we consider again a quantum entity and a set of properties (ai)i with corresponding set of projection operators (Paik Then there exists a unique projection operator Pa that corresponds to the meet property !\iai· 184 DIEDERIK AERTS Theorem 9. Consider a state property space eE, £, (, K,) corresponding to a quantum entity S, described by means of the standard quantum formalism in a Hilbert space 11. For a set of properties (ai)i, ai E £, and corresponding set of projection operators (PaJi' there exists a projection operator Pa such that a E ((p) ¢:? ai E ((p) Vi (89) which means that a = !\iai and which means that the state property space is property complete (see definition 4) We will not prove this theorem because it will lead us into too many technical details. We only mention that this is a well known result about Hilbert space projectors. (3) Minimal property and maximal property We can remark now also that for the case of a quantum entity the maximal property I is always actual for any state of the entity and the minimal property 0 is never actual (both are unique since the set of properties is a partially ordered set). Indeed: Theorem 10. Consider a state property space (E, £, (, K,) corresponding to a quantum entity S, described by means of the standard quantum formalism in a Hilbert space 11. Consider an arbitrary state pw E E and let 0 be the minimal property and I be the maximal property of £, then o tI. ((p) and I E ((pw). Proof: The minimal property is represented in quantum mechanics by the zero projection 0, and the maximal property by the unit operator I. We have tr(W . 0) = 0 and tr(W . I) = tr(W) = 1 which shows that o tI. ((pw) and I E ((pw). The additional structure that we have observed for a quantum entity in (1), (2) and (3) will be our inspiration for the first axiom, and will make us introduce the structure of a state property system, that we have studied intensively in [30] and [31]. Definition 11 (state property system). Suppose that property space (E, £, (, K,). This state property space erty system iff (£, -<,!\, v) is a complete lattice22 , and maximal element and 0 the minimal element of £, and 22 A complete lattice is a complete partially ordered set. we have a state is a state propfor pEE, I the ai E £, we have: STRUCTURES AND PARADOXES 185 L, ~, /\,) correspondTheorem 11. Consider a state property space (~, ing to a quantum entity S, described by means of the standard quantum formalism in a Hilbert space 71, then (~, L, ~, /\,) is a state property system. The foregoing results make it possible to introduce the first axiom: Axiom 1 (state property system). Suppose that we have a state property space (L:,.c, corresponding to an entity S. We say that axiom 1 is satisfied iff the state property space is a state property system. e, /\,) If axiom 1 is satisfied we will call the set of properties the 'property lattice' corresponding to entity S. 8.2. Atomic States We have seen that the set of states ~ of a general physical entity has a natural pre-order relation that we have called the state implication. We have also explained that a state in standard quantum mechanics can be represented by a density operator. Some of the density operators represent vector states of the Hilbert space. The representation theorem that we will put forward in this section is inspired on the classical representation theorem formulated for the situation where all states are vector states. Therefore we wonder whether it is possible to characterize the vector states in a more general way. This is indeed possible by means of the mathematical concept of 'atom' that we will introduce now. Definition 12. Consider a pre-ordered set Z, -<. We say that x E Z is an 'atom' iff x is not a minimal element and for y -< x we have y ~ x or y is a minimal element of Z. Definition 13. Consider a state property space (L:,.c, t;, /\,) describing an entity S. We know that L:, -< is a pre-ordered set. We call pEL: an 'atomic state' of the entity S iff p is an atom for the pre-order relation on L:. We will denote the set of atomic states of a state property space by means of A. Let us first investigate which are the atomic states for our example of the quantum machine. Following (84) and (85) we have Pv -< Pw and Pv ¢ Pw for Ivl = 1 and w < 1. This shows that none of the states Pw with Iwl < 1 is an atomic state. From (83) follows that Pv -< Pw for Ivl = Iwl = 1 implies that v = wand hence Pv = Pw' This shows that all of the states Pv with Ivl = 1 are atomic states. Hence the atomic states for the quantum machine are exactly these states that correspond 186 DIEDERIK AERTS to points on the surface of ball. For the situation of a general quantum entity described in a Hilbert space 1{ a similar result can be shown: the atomic states are those states that are represented by density operators that correspond to vectors of the Hilbert space. Theorem 12. Consider a state property space eE, £,~ K,) corresponding to a quantum entity S described by a Hilbert space 1{. A state pEE is atomic iff it is represented by a density operator corresponding to a vector of the Hilbert space. Proof: Consider a state PW E E represented by a density operator W of 1{. This density operator W can always be written as a convex linear combination of density operators Wi corresponding to vectors of 1{ and representations of states PWi E E23. Hence we have W = Li Ai Wi. Consider a property a E £ such that a E ~(pw). This means that tr(W Pa) = 1 where Pa is the projection operator representing the property a. We have tr(W Pa ) = tr(Li Ai WiPa ) = Li Aitr(WiPa). Since a :=:; Ai :=:; 1 V i and Li Ai = 1 and a :=:; tr(WiPa) :=:; 1 V i this proves that tr(WiPa ) = 1 V i. As a consequence we have a E ~(pwJ. This shows that ~ (pw) c ~ (pw;) and hence pWi --< PW. This shows already that genuine density operators that are not corresponding to a vector of the Hilbert space are not atomic. Suppose now that we consider a state Pw E E where W corresponds to a vector c E 1{ and another state pv E E such that pv --< pw· This means that ~(pw) c ~(pv). Suppose the property represented by the projector operator on the vector c, let us denote it Pc, is contained in ~(pw) and hence also in ~(Pv). From this follows that V = Pc and hence pv = PW. This shows that states represented by density operators corresponding to vectors of the Hilbert space are atomic states. The reader has perhaps meanwhile understood in which way we will gradually arrive at a full axiomatization of standard quantum mechanics. We analyse step by step what are the requirements that are additionally satisfied for the state property space of a quantum entity described by standard quantum mechanics. To proceed along this line we will now first show that for an entity satisfying axiom 1, the atomic states can be identified unequivocally with atomic properties of the property lattice. This will make it possible to concentrate only on the structure of the property lattice. Theorem 13. Consider a state property space (E, £,~ K,) satisfying axiom 1 (hence a state property system) with set of atomic states A. 23 This is a well known property of density operators in a Hilbert space. 187 STRUCTURES AND PARADOXES Let us denote by A the set of atoms of L. If we consider the function s : L: --t C, then s(A) = A. Proof: Suppose that rEA and let is show that s(r) is a atom of L. Consider a E such that a -< s(r) and a =/::. O. We remark that in this case K;(a) =/::. 0. Indeed, suppose that K;(a) = 0 then a -< b V b E and hence a = O. If K;( a) =/::. 0 there exists apE 2: such that p E K;( a). We then have a E ~(p) and as a consequence s(p) -< a -< s(r). From this follows that p -< r, but since rEA we have p ~ r. This implies that s(p) = s(r) and hence a = s(r). This proves that s(r) is an atom of C. Consider now a E A. Let us show that there exists apE A such that s(p) = a. We have 11:( a) i= 0 since a i= o. This means that there exists pEL: such that p E l1:(a). Hence a E ~(p) and as a consequence we have s(p) -< a. From this follows that s(p) = 0 or s(p) = a. We remark that s(p) = 0 is not possible since this would imply that ~(p) = C and hence 0 E ~(p) which is forbidden. Hence we have s(p) = a. We must still show that p is an atom. Indeed suppose that r -< p, then s(r) -< s(p) = a. This shows that s(r) = a and hence ~(r) = ~(p). .c .c The foregoing theorem shows that we can represent an atomic state by means of the atom of the property lattice on which it is mapped by the map s. That is the reason we will concentrate on the property lattice c. It could well be possible that no or very few atomic states exist. For the case of standard quantum mechanics there are however many atomic states. Theorem 14. Consider a state property space (L:, C, ~, 11:) describing a standard quantum mechanical entity S in a Hilbert space H. Each property ap 1= 0 represented by a non zero projection operator P equals the supremum of the set of rays (one dimensional projections) contained in P. This is the inspiration for the next axiom. First we introduce the concept of a complete atomistic lattice. Definition 14. Consider a complete lattice C, -< and its set of atoms A. We say that C is 'atomistic' iff each a E C is equal to the supremum of its atoms, i.e., a = V cEA,c-<ac. Axiom 2 (atomicity). Consider a state property system (L:, C,~ 11:) describing an entity S. We say that the state property system satisfies axiom 2 iff its property lattice is atomistic. 188 DIED ERIK AERTS We know from the foregoing that the atoms of C represent the atomic states of the entity S. Apart from atomicity, the property lattice of an entity described by standard quantum mechanics satisfies an additional property, called the 'covering law'. It is the following: Axiom 3 (covering law). The property lattice C of a state property system (~, c, r;,) satisfies the 'covering law' iff for c E A and a, b E C such that a 1\ c = 0 and a --< b --< a V c we have a = b or a V c = b. e, The covering law demands that the supremum of a property and an atom 'covers' this property, in the sense that there does not exists a property in between. The first three axioms introduce the linearity of the set of states of the entity. Indeed it can be shown that a complete atomic lattice satisfying the covering law and containing sufficiently many atoms is isomorphic to a projective geometry. Making use of the fundamental theorem of projective geometry we can construct of vector space coordinating this geometry and also representing the original lattice of properties. We refer to [34] for a proof of this fundamental representation theorem for complete atomic lattices satisfying the covering law. 8.3. Orthogonality The next axiom is inspired by the specific and strong orthogonality structure that exists on a Hilbert space. If axiom 1 is satisfied the set of properties is a complete lattice. We give now the definition of the structure of an orthocomplementation, which will make it possible for us to introduce the next axiom. Definition 15 (orthocomplementation). Consider a complete lattice C. We say that I : C ~ C is an orthocomplementation iff for a, b E C we have: a --< b =} b' --< a' (a ' )' = a 1\ a' a (91) =0 e, Theorem 15. Consider a state property space (~, c, r;,) corresponding to a quantum entity S, described by means of the standard quantum formalism in a Hilbert space 1i. If we define for a property a E C, and corresponding projection operator Pa , the property a ' as corresponding to the projection operator 1- Pa , then I : C ~ C is an orthocomplementation. STRUCTURES AND PARADOXES 189 Proof: Suppose that we have a, bEL and their corresponding projection operators Pa and Pb. If a -< b then Pa = PaPb = PbPa. We have (I - Pa)(1 - H) = 1 - H - Pa + PaH = 1 - H = 1 - Pb - Pa + HPa = (1 - H)(1 - Pa). This shows that b' -< a ' . We have 1 - (1 - Pa) = Pa which proves that (a')' = a. Since Pa(I - Pa) = 0 we have a 1\ a' = O. This gives us the next axiom: Axiom 4 (orthocomplementation). A state property system (L;, L, ~,/1;) describing an entity S is called 'property orthocomplemented' and satisfies axiom 4 iff there exists an orthocomplementation on the complete lattice of properties. Apart from the orthocomplementation the property lattice of an entity described by standard quantum mechanics satisfies an additional property called 'weak modularity'. It is a purely technical axiom expressed as follows: Axiom 5 (weak modularity). The property lattice £ of a state property space eE, £,~ /1;) satisfying axiom 1 and 4 (hence a property orthocomplemented state property system) is 'weakly modular' iff for a, bE £ such that a -< b we have (a Vb') 1\ b = a. 8.4. Full Axiomatisation of Standard Quantum Mechanics We need more requirements in order to be able to prove that the obtained structure is isomorphic to standard quantum mechanics. We leave the proof for these requirements to be satisfied in standard quantum mechanics to the dedicated reader. The first requirement is called 'plane transitivity'. It has been identified only recently ([32] [33]). Axiom 6 (plane transitivity). The property lattice £ of a state property space (L:,£~/1;) is 'plane transitive' iff for p,q E L: there are r i= s E L: and an automorphism of £ that maps ponto q and leaves the 'plane' interval [0, r V s] invariant. Let us introduce the next axiom: Axiom 7 (irreducibility). The property lattice £ of a state property space (L:, £,~ /1;) satisfying axiom 1 and 2 (hence a property orthocomplemented state property system) is 'irreducible', i.e., whenever b E £ is such that b = (b 1\ a) V (b 1\ a') V a E £ then b = 0 or b = I. 190 DIEDERIK AERTS The standard representation theorem has been proven for the irreducible components of the property lattice. The foregoing axiom is in this sense not on the same level as the other ones. Indeed even if we do not require the property lattice to be irreducible, the representation theorem can be proven for each irreducible component. It can indeed be shown that a general property lattice is the direct product of its irreducible components. We refer to [38] and [41] and more specifically to [42] and [43] for a detailed analysis of this decomposition. When we mentioned the representation theorem derived from the fundamental theorem of projective geometry in section 8.2 we already pointed out that the property lattice has to contain enough states to be able to derive this theorem. For the full representation theorem of standard quantum mechanics we need infinitely many atoms. Axiom 8 (infinite length). The property lattice £. of a state property space (l:, £.,~ K,) is 'infinite' if it contains an infinite set of mutually orthogonal elements 24 . Theorem 16 (Representation theorem). Suppose that we have an entity S described by means of a state property space (l:, £.,~ K,) for which axioms 1 to 8 are satisfied. Then £. is isomorphic to the complete lattice of the projection operators of an infinite dimensional real, complex or quaternionic Hilbert space N. The atoms of £. and hence also the atomic states of S correspond to the rays of N. The orthocomplementation is induced by the orthogonality structure of N. We will not prove this theorem, but refer to [32, 33, 34, 35, 36, 37, 38, 39, 40,41,42,43,44,45,46,47,48,49] where pieces preparing the proof can be found. We refer to [32, 33] for a recent and more complete overview and the inclusion of the new axiom of plane transitivity. 9. PAR ADO XES AND F A I LIN G A X 10M S The aim of quantum axiomatics was to construct an operational foundation for standard quantum mechanics starting from basic concepts, states and properties, that are easy to identify physically. Once a full axiomatics has been constructed this gives of course a powerful tool to investigate the well known paradoxes that quantum mechanics entails. Let us investigate some aspects of this possibility. 24 Two elements are orthogonal iff they imply respectively a property and its orthocomplement. STRUCTURES AND PARADOXES 191 9.1. The Description of an Entity Consisting of Two Entities We will consider the description of two spins by using the quantum machine model for these spins. So we consider now two quantum machines. Let us call them Sl and S2, and the entity that just consists of these two quantum machines. In a general way, the entity 81 is described by a state property space (~l' £1, 6, fi:1) and the entity 82 is described by a state property space (~2, £2, 6, fi:2). Let us denote the sphere of the first quantum machine Sl by balh, the points of this sphere by Vl E balh, and the states, the experiments and the properties connected to this quantum machine by PVl' eUl and aUl . In an analogous way we denote the sphere of the second quantum machine S2 by ball2, and its states, experiments and properties by PV 2' eU2 and aU2 • As we have shown in (66) and (81) the sets of properties, the sets of states and the sets of experiments are given by: {a Ul {PVl I Ul E balh, lUll I Vi E balh} = 1} {euIIU1Ebalh,lull=1} {a U2 i U2 E ball2, iU2i = 1} {PV2 i V2 E ball2} {e U2 i U2 E ball2, IU2i = 1} (92) Let us call S the compound physical entity, consisting of the two quantum machines, and (~, L,~ fi:) the state property space describing this entity S. To see in which way the three state property spaces are connected we have to analyse the physical situation. The states Clearly a state P of the entity S completely determines a state Pi of Sl and a state P2 of S2-using the physical principle that when the entity S 'is' in state P then the entities Sl and S2 'are' in two corresponding states Pi and P2 25 . This defines two functions: ml:~-* m2 : ~ 25 p~ml() ----* ~2 P~ m2 (p) (93) If we take the ontologie meaning of the concept of state seriously, we can hardly ignore this physical principle. Although, as we will see, standard quantum mechanics gives rise to problems here. 192 DIEDERIK AERTS The properties Each experiment e1 on 51 is also an experiment on 5 and each experiment e2 on 52 is also an experiment on 5-following the physical principle that if we perform an experiment on one of the sub-entities we perform it also on the compound entity26. Since the properties are operationally defined by means of the experiments, from the same physical principle, we have that each property of a sub-entity is also a property of the compound entity. This defines again two functions: n1 : £1 n2 : £2 --+ --+ £ £ a1 a2 1---+ 1---+ n1(a1) n2 (a2) (94) A covariance principle If property a1 is actual for entity 8 1 in state m1(p), then property n1(a1) is actual for entity 5 in state p. An analogous covariance principle is satisfied between entity 52 and entity 5. This means that we have the following equations: a1 E 6(m1(p)) {:} n1(a1) E ~(p) a2 E 6(m2(p)) {:} n2(a1) E (p)~ (95) It has been shown in [50], [30], [31] and [33], that for the case of physical entities satisfying axiom 1 (hence the three state property spaces are state property systems) this covariance principle gives rise to a unique minimal structure for the state property system of the compound entity. It is the (co )product in the category of state property systems. Theorem 17. Suppose that we have two entities 51 and 52 with state property systems (L:1' £1, 6, /1;1) and (L:2' £2, 6, /1;2) that form a compound entity 5 with a state property system (L:, £,~ /1;) according to (93), (94) and (95). The minimal solution is as follows: L: = L:1 X L:2 (96) where L:1 x L:2 is the cartesian product of L:1 and L: 2. For (PI, P2) E L: we have:- (97) 26 Again, if we take the meaning of what an experiment is seriously, it is hard to ignore this principle. We have even no reason here to doubt it, because standard quantum mechanics agrees with it. STRUCTURES AND PARADOXES For PI, ql E ~1 and P2, q2 E ~2 193 we have: (98) £1 II £2 (99) I alE £ 1, a2 E £2, =f 01, a2 =f 02} U {O} (100) { (aI, a2) al where £1 U £2 is called the co-product of £1 and £2. For al E £1 and a2 E £2 we have: nl(al) nl(Ol) n2(a2) n2(02) = (a1,h) if a1"1 01 =0 = (h, a2) if a2 "1 02 = (101) ° (aI, a2) --< (b l , b2) {::} al --< bi and a2 --< b2 (102) 0--< (aI, a2) (103) Ai(aL a~) = (AiaL Aia~) ai (104) if Ai "1 01 and Ai a~ o if Ai = 01 or Ai a~ ai "1 02 = 02 (105) I a1 E 6 (PI), a2 E 6 (P2 )} I PI E K;(al),P2 E K;(a2)} (106) Further we have: .; (PI, P2) = {(a 1 , a2) K;(al' a2) = ((Pl,P2) (107) 9.2. The Covering Law and Compound Entities This structure of (co ) product is the simplest one that can be constructed for the description of the compound physical entity S. One would expect that it is 'the' structure to be used to describe the compound entity S. This is however not the case for quantum entities in standard quantum mechanics. The reason is that the co-product 'never' satisfies two of the axioms of standard quantum mechanics, namely axiom 3 (covering law) and axiom 4 (orthocomplementation). Let us prove this for the case of the covering law. 194 DIED ERIK AERTS Theorem 18. Suppose that axiom 1 and 2 are satisfied and consider two entities Sl and S2 described by state property systems (E1' £1, 6, ~1) and (E2' £2, 6, ~2) and the minimal compound entity S consisting of Sl and S2 and described by the state property system (E1 xE 2 , £1 U £2'~) as defined in theorem 17. Suppose that axiom 3 is satisfied for the entity S, then one of the two entities Sl or S2 has a trivial property lattice consisting only of the minimal and maximal element. Proof: Suppose that £1 has at least one element a1 different from It and 0 1. Since £1 is atomistic there exists an atom C1 -< aI, and at least one atom d 1 -I< a1. Hence C1 =f. d 1. Consider now two arbitrary atoms C2, d2 E £2. We have: (q, C2) -< (C1 V d 1, C2) -< (C1 V d 1, C2 V d2) (Cl V d 1 , C2 V d 2) = (q, C2) V (d 1, d 2) (108) (109) Since (d 1, d2) is an atom of £1 U £2, the property (q, c2)V(d l , d2) 'covers' (Cl' C2), because the covering law is satisfied for S. We therefore have: (q, C2) = (Cl V d1, C2) or (C1 V d 1, C2) = (q V d 1, C2 V d2) (110) This implies that (111) Since C1 =I d 1 we cannot have q = C1 V d 1. Hence we have C2 = d 2. Since C2 and d2 were arbitrary atoms of £2, this proves that £2 contains only one atom. From this follows that £2 = {02, h}. This theorem proves that for two non-trivial entities Sl and S2 the property lattice that normally should represent the compound entity S never satisfies the covering law. This same theorem also proves that, since we know that for an entity described by standard quantum mechanics the covering law is satisfied for its property lattice, in quantum mechanics the compound entity S is 'not' described by the minimal product structure. The covering law, as we remarked earlier already, is the axiom that introduces the linear structure for the state space. This means that for a property lattice that does not satisfy the covering law it will be impossible to find a vector space representation such that the superposition principle of standard quantum mechanics is available. It can be shown that this fact is at the origin of the Einstein Podolsky Rosen paradox as it is encountered in quantum mechanics (see [19] and [20]). It means indeed that the compound entity S, as it is described in standard quantum STRUCTURES AND PARADOXES 195 mechanics, will have additional elements in its state property structure, that are not contained in the minimal product structure that we have proposed here. As we will see, these additional elements are the so called 'non-product states'. 9.3. The Quantum Description of the Compound Entity For standard quantum mechanics the compound entity S consisting of two entities SI and S2 is described by means of the tensor product HI ® H2 of the Hilbert spaces HI and H2 that describe the two subentities 51 and 52. We have studied this situation in detail in earlier work [51]' and will here only expose the scheme. Let us consider an entity 5 described with a state property space (~, L, ~, "') corresponding to the Hilbert space H consisting of two entities 51 and 52 described by state property spaces (~1' L1, 6, "'1) and (~2' L2, 6, "'2) corresponding to Hilbert space HI and H2. Let us first identify the functions m and n that describe the situation where 5 is the joint entity of 51 and 52. For the function n this identification is straightforward. We have: n1 : L1 n2 : L2 --+ --+ L L ap1 ap2 f-> f-> ap1 rg, I2 = n1(apl) ahrg,P2 = n2(ap2) (112) This shows that for standard quantum mechanics, as in the case where we would describe the compound entity by means of the co-product, for each property al of 51 (a2 of 52) there is a unique property n1(a1) (n2(a2)) of 5. The requirement that with each state p of the compound entity 5 correspond unique states PI and P2 of the sub-entities gives rise to a special situation in the case that P corresponds to a non-product vector of the Hilbert space HI ® H2, i.e., a vector c = I:i ci ® c~ that cannot be reduced to a product of a vector in HI and a vector in H2. It can be shown that, taking into account the covariance requirement-this time also for the probabilities-there do correspond two unique states PWI and P W2 to such a state Pc, but when c is a non-product vector, WI and W 2 are density operators that do not correspond to vectors in HI and H2. This means (see theorem 12) that pW1 and PW 2 are non atomic states although Pc is an atomic state. We have not stated this too explicit till now, but in standard quantum mechanics there is a real physical difference between the atomic states-represented by density operators corresponding to a vector-and the non atomic states-represented by density operators not corresponding to a vector. The atomic states are 'pure states' and the non atomic states are 'mixed states'. This is in fact also the case in our operational definition of the join states in (see 196 DIED ERIK AER TS definition 9). The join state of a set of states, as defined there is a mixture of these states, which means that the entity is in the join state of this set iff it is in one of the states of this set, but we lack the knowledge about which one. So we repeat: a mixed state of a set of pure states describes our lack of knowledge about the pure state where the entity is in. If this is the meaning of a non atomic state, hence a mixed state over some set of atomic states, as it is the case in standard quantum mechanics, we can conclude that the entity is always in an atomic state. The non atomic states only describe our lack of knowledge about the atomic state the entity exactly is in. We can now see where the fundamental problem arises with the tensor product coupling procedure of quantum mechanics. If entities are always in atomic states, and since for an atomic state of the compound entity that corresponds to a non-product vector of the tensor product Hilbert space, the component states are strictly non atomic, it would indicate that the sub-entities are not in a state. This is of course very strange. Indeed, it seems even contradictory with the concept of state itself. An entity must always be in a state (and hence a quantum entity always in an atomic state), whether it is a sub-entity of another entity or not. As we have mentioned already, these non-product states also give rise to EPR type correlations between the two sub-entities 8 1 and 8 2. We remark that the presence of these correlations also indicates that the quantum description of the compound entity is not a description of 'separated' entities. So something really profoundly mysterious occurs here. We also mention that it is excluded that the non-product states of the quantum compound entity would be mathematical artifacts of the theory, since entities are without many problems prepared in these non-product states in the laboratory these days27. So the non-product states exists and are real states of the compound entity consisting of two quantum entities. Should we then decide that the sub-entities have disappeared as entities, and only some properties are left? We want to reflect more about this question and investigate what the possibilities are. Most of all we want to put forward an alternative possibility, that is however speculative, but should be worth further investigation. 27 The question about the reality of the non-product states was settled during the second half of the seventies and the first half of the eighties by means of the well known Einstein Podolsky Rosen correlation experiments. Meanwhile it has become common laboratory practice to prepare 'entangled' entities-that is what they are referred to now in the literature-in non-product states. STRUCTURES AND PARADOXES 197 9.4. About Mixtures, Pure States, Non Atomic Pure States We have to remark that the problem that we explained in the foregoing section was known from the early days of quantum mechanics but concealed more or less by the confusion that often exists between pure states and mixtures. Let us explain this first. The reality of a quantum entity in standard quantum mechanics is represented by a pure state, namely a ray of the corresponding Hilbert space. Mixed states are represented in standard quantum mechanics by density operators (positive self adjoint operators with trace equal to 1). But although a mixed state is also called a state, it does not represent the reality of the entity under consideration, but a lack of knowledge about this reality. This means that if the entity is in a mixed state, it is actually in a pure state, and the mixed state just describes the lack of knowledge that we have about the pure state it is in. We have remarked that the deep conceptual problem that we indicate here was noticed already in the early days of quantum mechanics, but disguised by the existence of the two types of states, pure states and mixed states. Indeed in most books on quantum mechanics it is mentioned that for the description of sub-entities by means of the tensor product procedure it is so that the compound entity can be in a pure state (and a non-product state is meant here) such that the sub-entities will be in mixed states and not in pure states (see for example [52]11-8 and [53] p 306). The fact that the sub-entities, although they are not in a pure state, are at least in a mixed state, seems at first sight to be some kind of a solution to the conceptual problem that we indicated in the foregoing section. Although a little further reflection shows that it is not: indeed, if a sub-entity is in a mixed state, it should anyhow be in a pure state, and this mixed state should just describe our lack of knowledge about this pure state. So the problem is not solved at all. Probably because quantum mechanics is anyhow entailed with a lot of paradoxes and mysteries, the problem was just added to the list by the majority of physicists. Way back, in a paper published in 1984, we have already shown that in a more general approach we can define pure states for the sub-entities, but they will not be "atoms' of the lattice of properties [50]. As we have shown already the ray states of quantum mechanics give rise to atoms of the property lattice, such that "pure states' in quantum mechanics correspond to "atomic' states of the state property space. This means that the non atomic pure states that we have identified in [50] can anyhow not be represented within the standard quantum mechanical formalism. We must admit that the finding of the existence of non atomic pure states in the 1984 paper, even from the point of view of generalized quantum 198 DIEDERIK AERTS formalisms, seemed also to us very far reaching and difficult to interpret physically. Indeed intuitively it seems that only atomic states should represent pure states. We know now that this is a wrong intuition. But to explain why we first have to present the other pieces of the puzzle. A second piece of the puzzle appeared when in 1990 we built a model of a mechanistic classical laboratory situation violating Bell inequalities with y2, exactly 'in the same way' as its violations by the EPR experiments [54J. With this model we tried to show that Bell inequalities can be violated in the macroscopic world with the same numerical value as the quantum violation. What is interesting for the problem of the description of sub-entities is that new 'pure' states were introduced in this model. We will see in a moment that the possibility of existence of these new states lead to a possible solution of the problem of the description of sub-entities within a Hilbert space setting, but different from standard quantum mechanics. More pieces of the puzzle appeared steadily during the elaboration of the general formalism presented in [6J. We started to work on this formalism during the first half of the eighties, reformulating and elaborating some of the concepts during these years. Then it became clear that the new states introduced in [54], although 'pure' states in the model, appear as non atomic states in the general formalism. This made us understand that the first intuition that classified non atomic states as bad candidates for pure states was a wrong intuition. Let us present now the total scheme of our possible solution. In the example that we proposed in [54J we used two spin models as the one presented here (the quantum machine) and introduced new states on both models with the aim of presenting a situation that violates the Bell inequalities exactly as in the case of the singlet spin state of two particles do. We indeed introduced a state for both spin coupled spin~ models that corresponds to the point in the center of each sphere, and connecting these two states by a rigid rod we could generate a violation of Bell's inequalities. We have shown in the last part of section 7.4 that the centre of ball is a non atomic state of the quantum machine. This means that we have 'identified' a possible 'non-mixture' state (meaning with 'non-mixture' that it really represents the reality of the entity and not a lack of knowledge about this reality) that is not an atom of the preordered set of states. Is this a candidate for the 'non-mixed' states that we identified in [50J and that were non-atoms? It is indeed, as we prove explicitly in [6J. The states Pw, where wE ball and Iwl < 1, as defined in 7.4, and that are certainly pure states for the quantum machine entity, are represented by the density operator W (w) if we use formula 72 for the calculation of the transition probabilities. STRUCTURES AND PARADOXES 199 9.5. Completing Quantum Mechanics? The idea that we want to put forward is the following: perhaps density operators just do represent pure states, also for a quantum entity. Such that the set of pure states would be represented by the set of density operators and not by the set of rays of the Hilbert space. If this would be the case, the conceptual problem of using the tensor product would partly be solved. We admit immediately that it is a very speculative idea that we put forward here. The problem is also that it will be difficult to test it experimentally on one physical entity. Indeed, the so called new pure states, corresponding to density operators of the Hilbert space, cannot experimentally be distinguished from mixtures of old pure states, corresponding to vectors, and represented also by these density operators. It is an easy mathematical result since in all probability calculations only the density operator appears. We can also see it explicitly on the quantum machine. Let us go back to the calculation that we made in the second part of section 7.4. The 'pure' state Pw corresponding to an interior point w = Al . V + A2 . (-v) of ball is represented by the density operator W( ) w - ( \ /\1 ( /\1 \ \ sm"2 . 20 cos 2 "20 + /\2 \ ) . 0 0 iq, - /\2 sm"2 cos "2 e (\ . 0 sm"2 cos "20 e-iq, ) \ 2 0 /\1 sm "2 + /\2 COS "2 /\1 - \. \ /\2 ) 2 0 But this density operator represents also the 'mixed' state describing the following situation of lack of knowledge: the point is in one of the pure states Pv or P-v with weight Al and A2 respectively. Although this mixed state and the pure state Pw are ontologically very different states, they cannot experimentally be distinguished by means of the elastic experiments. The reason that they cannot is connected to the linearity of the whole scheme. This means that if we would be able to realise a nonlinear evolution of a quantum entity, the distinction between the density operator representing a pure non atomic state (as the interior point of the sphere in the case of the quantum machine) and the density operator representing a mixture (as the weighted average over two diametrically oposed surface points in the case of the quantum machine) could be detected experimentally. Hence, if one quantum entity of a pair of two entangled quantum entities could be experimentally subjected to a nonlinear evolution, it could be tested experimentally whether this quantum entity 'is' in a pure ontologie state, described by the density operator. Indeed, the non-linear evolution will have an effect on the atomic states (the points of the surface of the sphere in the quantum machine example) and an effect on the non-atomic pure states (the interior points of the sphere in the quantum machine example), and both effects will no cancel out in general, like it is the case for a linear evolution, such that it can be 200 DIEDERIK AERTS made out whether the entangled quantum entity - being entangled - is in an ontologically pure state. We invite the experimentalists to engage in this challenge. 10. FRO M E U C LID TOR I E MAN N: THE QUA N TUM MECHANICAL EQUIVALENT We have called section 9 'Paradoxes and failing axioms'. Indeed a possible conclusion for the result that we have exposed in this section is that the covering law (and some of the other axioms) are just no good and that we should look for a more general formalism than standard quantum mechanics. We are more and more convinced that this must be the case, also because it is again the covering law that makes it impossible to describe a continuous change from quantum to classical passing through the intermediate situations that we have mentioned in section 6.4 (see [5], [25] and [27] and the paper of Thomas Durt in this book). In this sense we want to come back now to the suggestive idea that was proposed in the introduction. The idea of relativity theory, to take the points of space-time as representing the events of reality, goes back to a long tradition. It was Euclid who for the first time synthesised the descriptions that the Greek knew about the properties of space by using as basic concepts points, lines and planes. Let us remember that Euclid constructed an axiomatic system consisting of five axioms-now called Euclid's axioms-for Euclidean space and its geometric structure. All classical physical theories have later, without hesitation, taken the Euclidean space as theatre for reality. From a purely axiomatic point of view there has been a long and historic discussion about the independence of the fifth Euclidean axiom. Some scientists have pretended to be able to derive it from the four other axioms. The problem was resolved in favour of the independence by Gauss, Bolyai and Lobachevski by constructing explicit models of non-euclidean geometries. Felix Klein proposed a classification along three fundamental types: an elliptic geometry, the one originally proposed by Gauss, the geometry of the surface of a sphere for example, a hyperbolic geometry, proposed by Lobachevski, the geometry of the surface of a saddle for example, and a parabolic geometry, that lies in between both. It was Riemann who proposed a complete theory of non-euclidean geometries, the geometry of curved space, that was later used by Einstein to formulate general relativity theory. It is interesting to remark that the programme of general relativity-to introduce the force fields of physics as properties of the metric of space-was already put forward explicitly by Riemann. He could however not have found the solution of general relativity because he was looking for a solution in STRUCTURES AND PARADOXES 201 three dimensional space, and general relativity has to be constructed in four dimensional space-time. In the formulation of Einstein, which is the one of Riemann applied to the four dimensional space-time continuum, the events are represented by the points and the metric tensor describes the nature of the geometry. Let us now see whether we can find an analogy with our analysis of quantum mechanics by means of our axiomatic approach. Here the basic concepts are states, properties and probability. In geometry a set of points forms a space. In quantum mechanics the set of states and properties form a state property space. Just like the Euclidean space is not just any space, the state property space of quantum mechanics is not just any state property space. We have outlined 8 axioms that make an arbitrary state property space into a quantum mechanical state property space. We have remarked already that the purpose of quantum axiomatics was not to change standard quantum mechanics, exactly as the purpose of Euclid was certainly not to formulate an alternative geometry: he wanted to construct an operational theory about the structure of space. The operational axiomatization of quantum theory has taken a long time, the axiom of 'plane transitivity' relies on a result of Maria Pia Soler of 1995 [48]. Since however the general problems related to axiomatization are nowadays better known, there has been little discussion about the independence of the axioms. But the physical meaning of certain axioms (e.g., the failing covering law) remained obscure. These were merely axioms introduced to recover the complete Hilbert space structure which includes the linearity of the state space. Within this development of the axiomatic structure of quantum theory it has been shown now that the compound entity of two separated quantum entities cannot be described. The axiom that makes such a description impossible is the covering law, equivalent to the linearity of the state space. Do we have to work out a general quantum theory, not necessarily satisfying the covering law, as Riemann has given us a general theory of space? We believe so, but we know that a lot of work has to be done. In [6] a humble general scheme is put forward that may be a start for the elaboration of such a general quantum theory. ACKNOWLEDGEMENTS I want to thank Jan Broekaert and Bart Van Steirteghem to read and discuss with me parts of the text of this article. Their remarks and suggestions have been of great value. 202 DIEDERIK AERTS AFFILIATION Diederik Aerts, CLEA-FUND, Brussels Free University, Brussels, Belgium 11. REFERENCES [1] von Neumann, J., Mathematische Grundlagen der Quanten-Mechanik, Springer-Verlag, Berlin, 1932. [2] Birkhoff, G. and von Neumann, J., "The logic of quantum mechanics", Annals of Mathematics, 37, 1936, p. 823. [3] Foulis, D., "A Half-Century of Quantum Logic-What have we learned?" in Quantum Structures and the Nature of Reality, the Indigo Book of Einstein meets Magritte, eds.,Aerts, D. and Pykacz, J., Kluwer Academic, Dordrecht, 1998. [4] Jammer, M., The Philosophy of quantum mechanics, Wiley and Sons, New York, Sydney, Toronto, 1974. [5] Aerts, D. and Durt, T., "Quantum. Classical and Intermediate, an illustrative example", Found. Phys. 24, 1994, p. 1353. [6] Aerts, D., "Foundations of Physics: a general realistic and operational approach", to be published in International Journal of Theoretical Physics. [7] Aerts, D., "A possible explanation for the probabilities of quantum mechanics and example of a macroscopic system that violates Bell inequalities", in Recent developments in quantum logic, (eds.), Mittelstaedt, P. and Stachow, E.W., Grundlagen der Exakten Naturwissensehaften, band 6, Wissenschaftverlag, Bibliografisches Institut, Mannheim, 1985. [8] Aerts, D., "A Possible Explanation for the Probabilities of quantum mechanics", J. Math. Phys., 27, 1986, p. 202. [9] Aerts, D., "Quantum structures: an attempt to explain their appearance in nature", Int. J. Theor. Phys., 34, 1995, p. 1165. [10] Gerlach, F. and Stern, 0., "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld", Zeitschrift fUr Physik 9, 1922, p.349. [11] Bell, J.S., Rev. Mod. Phys., 38, 1966, p. 447. [12] Jauch, J.M. and Piron, C., Helv. Phys. Acta, 36, 1963, p. 827. [13] Gleason, A.M., J. Math. Meeh., 6, 1957, p. 885. STRUCTURES AND PARADOXES 203 [14] Kochen, S. and Specker, E.P., J. Math. Mech., 17, 1967, p. 59. [15] Gudder, S.P., Rev. Mod. Phys., 40, 1968, p. 229. [16] Accardi, L., Rend. Sem. Mat. Univ. Politech. Torino, 1982, p. 24l. [17] Accardi, L. and Fedullo, A., Lett. Nuovo Cimento, 34, 1982, p. 161. . [18] Aerts, D., "Example of a macroscopical situation that violates Bell inequalities", Lett. Nuovo Cimento, 34, 1982, p. 107. [19] Aerts, D., "The physical origin of the EPR paradox", in Open questions in quantum physics, (eds.), Tarozzi, G. and van der Merwe, A., Reidel, Dordrecht, 1985. [20] Aerts, D., "The physical origin of the Einstein-Podolsky-Rosen paradox and how to violate the Bell inequalities by macroscopic systerns", in Proceedings of the Symposium on the Foundations of Modern Physics, (eds.), Lahti, P. and Mittelstaedt, P., World Scientific, Singapore, 1985. [21] Aerts, D., "Quantum Structures, Separated Physical Entities and Probability", Found. Phys. 24, 1994, p. 1227. [22] Coecke, B., "Hidden Measurement Representation for Quantum Entities Described by Finite Dimensional Complex Hilbert Spaces" , Found. Phys., 25, 1995, p. 203. [23] Coecke, B., "Generalization of the Proof on the Existence of Hidden Measurements to Experiments with an Infinite Set of Outcomes", Found. Phys. Lett., 8, 1995, p. 437. [24] Coecke, B., "New Examples of Hidden Measurement Systems and Outline of a General Scheme", Tatra Mountains Mathematical Publications, 10, 1996, p. 203. [25] Aerts, D. and Durt, T., "Quantum, classical and intermediate: a measurement model" , in Montonen C. (ed.), Editions Frontieres, Gives Sur Yvettes, France, 1994. [26] Aerts, D., Durt, T. and Van Bogaert, B., "A physical example of quantum fuzzy sets, and the classical limit" , in the proceedings of the International Conference on Fuzzy Sets, Liptovsky, Tatra mountains, 1993, p. 5. [27] Aerts, D., Durt, T. and Van Bogaert, B., "Quantum Probability, the Classical Limit and Non-Locality", in the proceedings of the International Symposium on the Foundations of Modern Physics 1992, Helsinki, Finland, ed. T. Hyvonen, World Scientific, Singapore, 1993, p.35. 204 DIEDERIK AERTS [28] Randall, C. and Foulis, D., "Properties and operational propositions in quantum mechanics", Found. Phys., 13, 1983, p. 835. [29] Foulis, D., Piron, C. and Randall, C., "Realism, operationalism, and quantum mechanics", Found. Phys., 13, 1983, p. 813. [30] Aerts, D., Colebunders, E., Van der Voorde, A. and Van Steirteghem, B., "State property systems and closure spaces: a study of categorical equivalence", Int. 1. Theor. Phys., to appear 1998. [31] Aerts, D., Colebunders, E., Van der Voorde, A. and Van Steirteghem, B., "Categorical study of the state property systems and closure spaces", preprint, FUND - TOPO, Brussels Free University. [32] Aerts, D. and Van Steirteghem, B., "Quantum Axiomatics and a Theorem of M.P. Soler", preprint, FUND, Brussels Free University. [33] Van Steirteghem, B., "Quantum Axiomatics: Investigation of the structure of the category of physical entities and Soler's theorem", graduation thesis, FUND, Brussels Free University. [34] Piron, C., "Axiomatique Quantique" , Helv. Phys. Acta, 37, 1964, p.439. [35] Amemiya, I. and Araki, H., "A remark of Piron's paper", Publ. Res. Inst. Math. Sci., A2, 1966, p. 423. [36] Zierler, N., "Axioms for non-relativistic quantum mechanics", Pac. J. Math., 11, 1961, p. 1151. [37] Varadarajan, V., Geometry of Quantum Theory, Van Nostrand, Princeton, New Jersey, 1968. [38] Piron, C., Foundations of Quantum Physics, Benjamin, Reading, Massachusetts, 1976. [39] Wilbur, W., "On characterizing the standard quantum logics", . Trans. Am. Math. Soc., 233, 1977, p. 265. [40] Keller, H.A., "Ein nicht-klassicher Hilbertsher Raum", Math. Z., 172, 1980, p. 41. [40] Aerts, D., "The one and the many", Doctoral Thesis, Brussels Free University, Brussels, 1981. [41] Aerts, D., "Description of many physical entities without the paradoxes encountered in quantum mechanics", Found. Phys., 12, 1982, p. 1131. [42] Aerts, D., "Classical theories and Non Classical Theories as a Special Case of a More General Theory", J. Math. Phys., 24, 1983, p. 2441. [43] Valckenborgh, F., "Closure Structures and the Theorem of Decomposition in Classical Components", Tatra Mountains Mathematical Publications, 10, 1997, p. 75. STRUCTURES AND PARADOXES 205 [44] Aerts, D., "The description of one and many physical systems", in Foundations of quantum mechanics, eds. C. Gruber, A.V.C.P., Lausanne, 1983, p. 63. [45] Aerts, D., "Construction of a structure which makes it possible to describe the joint system of a classical and a quantum system", Rep. Math. Phys., 20, 1984, p. 421. [46] Piron, C., Mecanique Quantique: Bases et applications, Presse Polytechnique et Universitaire Romandes, Lausanne, 1990. [47] Pulmannova, S., "Axiomatization of Quantum Logics", Int. J. Theor. Phys., 35, 1995, p. 2309. [48] Soler, M.P., "Characterization of Hilbert spaces with Orthomodular spaces", Comm. Algebra, 23, 1995, p. 219. [49] Holland Jr, S.S., "Orthomodularity in Infinite Dimensions: a theorem of M. Soler", Bull. Aner. Math. Soc., 32, 1995, p. 205. [50J Aerts, D., "Construction of the tensor product for lattices of properties of physical entities", J. Math. Phys., 25, 1984, p. 1434. [51J Aerts, D. and Daubechies, I., "Physical justification for using the tensor product to describe two quantum systems as one joint system", Helv. Phys. Acta 51, 1978, p. 661. [52J Jauch, J., Foundations of quantum mechanics, Addison-Wesley, Reading, Mass, 1968. [53J Cohen-Tannoudji, C., Diu, B. and Laloe, F., Mecanique Quantique, Tome I, Hermann, Paris, 1973. [54J Aerts, D., "A mechanistic classical laboratory situation violating the Bell inequalities with yI2, exactly 'in the same way' as its violations by the EPR experiments", Helv. Phys. Acta, 64, 1991, p. 1. THOMAS DURT ORTHOGONALITY RELATIONS: FROM CLASSICAL TO QUANTUM INTRODUCTION We developed in Brussels a heuristic model, the E-model, in which it is possible to simulate a continuous transition between a classical (deterministic) and a quantum (probabilistic) regime. The lattice of properties related to this model was intensively studied [3, 5 to 10] and it appears that it exhibits a continuous transition between the classical and the quantum lattices. These lattices are representative of classical situations at one side and of quantum situations at the other side. It is not presently known if a continuous transition between the classical, deterministic, and the quantum, fuzzy, regimes does occur in nature. Even if this would occur, no one exactly knows the border-line between the two regions. This is an aspect of the so-called problem of measurement which deals with the comprehension of the way in which the macroscopic, deterministic and sharp world to which observers and measuring apparata belong coexists with the microscopic, unsharp, quantum world. In our model, we assume that "hidden variables" are present at the level of the apparatus, which is a non-standard hypothesis. The reader may thus consider this paper as a good example of how speculative ideas can be implemented in the framework of quantum logics. In particular, we shall study two ways of defining an orthogonality relation inside the set of states of the system described in the E-model. In lattice theory, the orthogonality relation is important at least for two reasons: - It is possible to associate to an orthogonality relation what is called an orthocomplementation, as we shall show it. The existence of an orthocomplementation is postulated in many attempts made in order to generalise the classical (Boolean) paradigm. For instance it is an unavoidable element of the representation theorem of Piron [18 to 20]. - The orthocomplementation can be interpreted as a generalised logical negation. We study here three orthogonality relations in the framework of the Emodel, show that they evolve continuously during the classical-quantum 207 © 1999 Kluwer Academic Publishers. 208 THOMAS DURT transition, and discuss whether the lattice of properties is orthocomplemented by them. 1. THE E-MODEL From now on, we shall consider that the state of the system is represented by a point on the surface of the unit 3-dimensional sphere (the so-called Poincare-sphere). We invite the interested reader to consult the appendix where we present some details about the connections which exist between a spin 1/2, which is described in a 2-dimensional Hilbert space, and its representation on the surface of the Poincare-sphere. For information, the properties of a 2-dimensional Hilbert space as well as the lattice of properties associated to it are discussed in great detail in the paper of D. Aerts (same volume). All what is necessary for our present purposes is that this direction on the sphere somehow represents the direction along which the system that we consider here, which could be a particle, "turns around itself" . For instance, the spin of the Earth, if our system was the Earth, would be represented by the North Pole. If the Earth was spinning in the opposite direction (with the Sun rising in the West), its spin would be represented by the South Pole. 1.1. Guiding Principles of the f.-Model The f.-model aims at simulating the interaction which occurs during a measurement process. It is based on the following guiding principles: - The measuring apparatus is essentially classical and interacts with the observed system in a deterministic way. - The state of this apparatus undergoes statistical fluctuations, responsible for a dispersion in the results of measurement [2J. - The amplitude of these fluctuations is quantified by a real parameter f. comprised between 0 and 1. - When f. equals 0, the fluctuations vanish, when f. equals 1, they are maximal. - The classical limit is reached when we can neglect these fluctuations (f. equals 0); intuitively, this means that the scale of the disturbance caused by the measuring apparatus is smaller than the scale of the observed system. - When the scale of the disturbance of the measuring apparatus is larger than the scale of the observed system, f. is not negligible anymore, the dispersion of the experimental results increases, and when f. equals 1, we recover a distribution of results equal to the quantum one. ORTHOGONALITY RELATIONS 209 1.2. Simple Version of the f.-Model Let us briefly describe how the model works (this is an ultrasimplified version, see [3, 6, 16] for more sophisticated presentations). We advice the readers who are not familiar with this kind of models to consult the article of D. Aerts (same volume), where the measurement process is illustrated by the "metaphor of the elastic" that we summarize now. The measurement process is assumed to occur in two steps: the particle "falls" on an elastic rod that joins the poles p and -p and sticks there for a while; thereafter, the elastic, which is assumed to be fragile in a certain zone 1, breaks at random in this zone, and the part of the elastic on which the particle is located projects the particle on the pole to which it is attached. A picture and detailed explanations are given in the article of D. Aerts. Let us now give, in an abstract style, a presentation which contains the essential features of the model. - The f.-model describes a generalised spin-measurement. - The state of the generalised spinning particle is represented by a point q of the Poincare-sphere (a direction). The set of states of the system is thus the surface of the Poincare sphere. - The generalised Stern-Gerlach device is represented by another point p of the Poincare-sphere and a hidden variable cpo - We assume that the hidden variable cP is submitted to uncontrollable fluctuations, so that it is a random variable homogeneously spread over the real interval [CPmin, CPmaxl (with 0 ~ CPmin ~ CPmax ~ 1). - The measurement process forces the state to collapse towards two possible directions: the direction of the device p, or its opposite -po We shall then say that the spin of the system is "up" or "down" along p. - Let us denote () the angle between p and q. We suppose now that the result of the measurement is determined as follows: - If cP appears to be strictly less than cos2~, the result of the measurement is "spin-up" and the state q collapses on p. - If cP appears to be greater than or equal to cos 2 ~, the result of the measurement is "spin-down" and the state q collapses on -po We obtain after averaging on the variable cP, for each value of the pair (CPmin, CPmax) , a transition probability which depends on the angle between the initial and the final states (before and after the measurement process). The probability of getting spin-up P(plq) can be shown to depend on the angle () between p and q as follows: 1 In an interval of width 2</>max - 2</>min centered around a point located at a distance </>min + </>max from the south pole in our case. 210 THOMAS DURT - It is equal to 1 when 0 ::; B < Bup , 0 when 7r 2 B 2 7r - Bdown' where cos2~ = cPmax and sin2~ = cPmin. The subscripts up and down are justified by the fact that these angles are the extents of the classical zones (probability 0 or 1) around the poles in which the spin measurements are predetermined with a probability equal to hundred percent. - In between it is superposition of the two possible results, in a zone of angular opening Bsup. The subscript sup is justified by the fact that Bsup is the extent of the superposition zone (probability neither 0, nor 1) between the classical zones. We have, in the superposition zone (see pictures at the end of the paper), that: P(plq ) = cosB + COSBdown + COSBdown· cosBup (1) The three angles Bsup , Bup , Bdown fulfill the following relation: ()sup + + Bdown = 7r. We define the classical (deterministic) and quantum limits of the model by imposing that Bsup is respectively equal to zero and 7r. Bup p -p Figure 1. The up and down zones around p. The classical case corresponds to a purely deterministic situation (the probability of getting spin up takes only values 0 or 1), while in the quantum case we recover the quantum probability associated to the measurement of the spinning direction of a spin 1/2 particle 2. Historically 2 The idea to associate hidden variables to the apparatus in the quantum context was originally presented in [2]. ORTHOGONALITY RELATIONS 211 [3], the parameter E which is at the origin of the name of the model (see section 1) was introduced firstly. The relation with the angles introduced here is given by the following relation: sin( (}s2 P ) = E. 2. THE IMPORT ANCE OF ABSOL UTE TRUTH An important aspect of the lattice theory (at least in the Geneva formulation [18 to 20, 1]) is that in order to deduce from experimental data the "propositions", which are the objects studied in the Geneva approach to quantum logics, all the relevant information is contained in the socalled eigenstate sets, the sets of states for which the results of a given measurement can be deterministically foreseen (with probability one or zero). For instance, in the E-model, the eigenstate sets of the property "the spin is up (down) when we measure it along the direction p" are spherical sectors of opening ()up (()down) along p (-p). We will not explain here the procedure which allows us to build the lattice of properties once we know the eigenstate sets of all the experiments that can be performed on the system, but it is useful to draw the attention of the reader on this fact: when the lattice is built, on the basis of experiments, all the states which are not eigenstates of one of the outcomes are simply discarded and do not contribute in any manner to the properties of the structure. This remark is trivial in classical logics, because in classical physics the results of arbitrary measurements are assumed to be "sharply" predetermined, so to say, determined with absolute certainty (probability one or zero) whatever the state of the system is. This is no longer true in quantum mechanics where new kinds of states appear, the so-called superposition states, for which all what we can predict is the probability (neither zero nor one) assigned to the different possible outcomes of the measurement. This major difference between the classical and quantum worlds is reflected in the E-model by the existence of a superposition zone between the eigenstate sets, which are spherical sectors of angular opening ()up, (()down) around the direction of the generalised spin measurement (its antipode). Quantum logics, in the Geneva approach, is thus an attempt made in order to generalise classical logics by considering only the deterministic part of probabilistic theories (only the states for which propositions are absolutely true or false, so to say the states which realise an outcome of a given experiment with probability 1 or 0, are considered during the building of the lattice). Note that for experimental physicists, these superposition states are important because the probabilities of occurrence assigned to the different outcomes of which they are superposition can be measured through repeated measurements and contain relevant information about the phys- 212 THOMAS DURT ical properties of the system (dynamics of the system, the influences that it underwent before the measurement and so on). The fact that in quantum logics one can discard all the information contained in intermediate probabilities is related to a property of Hilbert space quantum mechanics that we will sketch here without entering into technicalities [22, 23]: once we know the eigenstate set of a particular outcome of a given measurement, the projection postulate makes it possible to compute all the probabilities assigned to this possible outcome of the measurement, for arbitrary initial states of the system. Considered so, intermediate (strictly between 0 and 1) probabilities are thus second hand properties which can be recovered once we know the states which realise a given outcome with probabilities 0 or 1. This is a peculiarity of quantum mechanics (and also of the E-model): tell me what is certain (actual) and I can quantify the uncertain (potential). This is why the lattice of properties of these systems contains potentially all the information that comes from experiments. Note that these considerations are trivial in the classical regime where the uncertain simply vanishes. We will now show how it is possible to generalise the classical concept of negation on the basis of the knowledge of the eigenstate sets of the experimental outcomes that can be obtained by submitting a system to measurements. This generalised negation will thus be deduced from the knowledge of the states which realise experimental outcomes with probability 0 or 1 only, a survival of the special status given to "absolutely true and certain" propositions in classical logics. 3. THE ORTHOGON ALITY RELATION 3.1. The Negation Let us consider the following example: the system that we study is a car, that we select among a given population of cars. For one or another reason, we study the colour of this car. We classify the colours into a set which contains eight elements: black, white, yellow, orange, red, purple, blue, and green. This would correspond to what we call experimental outcomes in the Geneva approach. The part of the population that has a given colour would correspond to what we call state in the Geneva approach. Normally, it is easy to assign unambiguously one of these colours to a car arbitrarily chosen in the population. In this case, the result of a measurement is deterministic, and the negation of a property is well defined: if a car is not red, then it is black or white or yellow or orange or purple or blue or green, and we made use here of an "exclusive or". We shall from now on refer to this situation as the sharp one. ORTHOGONALITY RELATIONS 213 In reality, things could be more complicated and some subjectivity could appear: if a car is grey, is it white or black?; if it is dark orange, is it (pale) red or orange? The answer to such questions could for instance depend on the person who realises the test, or even on the meteorological conditions (grey or sunny sky) during the test. If we do not control the choice of these parameters, we are in a situation similar to the one described in the €-model, with superposition states (here intermediate colours). We shall refer to this situation as the unsharp one. What are then, in the unsharp situation, the subpopulations (states) that we can classify according to the test formulated by the property: "This car is not red?" In a first approach, one could say that if we submit a given car to the test of colours and that we get a positive answer to the question "the car is red" , even with a very low probability of occurrence, we may not say that this car is not red. Now, what can we do if the car is grey? According to our first approach, such a car is not red, but if the car is grey with a touch of red, so that some confusion is still possible, we may not say that our car is not red. As we see, the situation becomes complex because we have in this approach infinite number of different states (colours). Now, it could happen (this depends on the experimental protocol) that an embarassed tester simply does not answer to the test when he doubts the answer, in which case we would have nine possible colours (the eight forementioned colours plus the "indefinite" colour). In this second approach, a car is not red except when it is either red or of indefinite colour. There would be nine different states in this case, and the "negation of the red state" would be, according to the previous suggestion, true when the state is black or white or yellow or orange or purple or blue or green. These two ways of defining the negation in a fuzzy context suggest that we can define the negation as a relation between states. The example furnished by the classification of cars in function of their colour suggests that, in the sharp situation, this relation would be the relation of difference: if a car is not red, then it is black or white or yellow or orange or purple or blue or green, as we noticed it already, which means in this case that the "negation of the state red" is true in the set of all the states which differ from the red state. Furthermore, the state (colour of a given car) can be checked at once, in a unique experiment, because no probabilistic element is present here. Note that the difference is a symmetric and antirefiexive relation: if p differs from q, then q differs from p and p does never differ from itself. This corresponds to the classical logical properties: "if a proposition A is not true whenever the proposition B is true, then the proposition B is 214 THOMAS DURT not true whenever A is true", and "the proposition cannot possibly be true and not true simultaneously". We will now generalise this relation to unsharp situations and represent the generalised negation thanks to a symmetric and antireflexive relation: the orthogonality relation. The question to know if such generalisations are formal speculations or really mean that the logics itself must be generalised [12] is let to the appreciation of the reader. For reasons that we will clarify later, when this orthogonality relation is the relation of difference (two states are orthogonal iff they are different), we will say that the orthogonality is Boolean. The previous discussion suggests that in the unsharp situation at least two definitions can be adopted for deducing an orthogonality relation from the knowledge of experimental data. Let us firstly study one of them, the Aerts orthogonality. 3.2. The Aerts Orthogonality D. Aerts [1] proposed to deduce an antireflexive and symmetrical function from the experiments in the following way. Definition 1: Two states are A-orthogonal iff they are eigenstates of two different outcomes of the same experiment. This relation is clearly symmetrical, and antireflexive because in general an experiment performed on a given system prepared in a given state cannot have two different outcomes at the same time. 3.3. The E-Orthogonality Let us now present a second definition of the orthogonality relation originally introduced in [16]' that we shall call the E-orthogonality, because it can be shown [16] that this is a definition which is perfectly adapted to the E-model. Definition 2: Two states are E-orthogonal iff there exists at least one experiment that does not induce a transition from one of the states to the other. In other words, the probability of transition from one state to the other is zero for at least one experiment. This definition was inspired by the work of Grib and Zapatrin who proposed the following orthogonality relation [4, 16, 17]: two states p, q are G-Z-orthogonal iff no experiment exists which induces a transition from one of the states to the other. It can be shown that this definition is not convenient in the deterministic limit, so that we shall not discuss it in details (see [16J for more information). The definition of the G-Zorthogonality as well as the definition of the E-orthogonality implicitly ORTHOGONALITY RELATIONS 215 presuppose some kind of collapse ("jump" between different states). The hypothesis of the existence of a collapse is of quantum inspiration and is not always realised, so that the G-Z and €-orthogonality relations are less general than Aerts orthogonality. Nevertheless, in the €-model, we recover the quantum property that after a spin measurement along a direction, the state of the system aligns itself (collapses) along this direction or its antipode. We can thus study the €-orthogonality in this framework. Note that the E-relation is given in a form symmetrical in p, q, because the probability of transition is symmetrical in general (in quantum mechanics, this is an aspect of microreversibility). Furthermore, it is antirefiexive, because in general the probability of transition of a state towards itself is equal to 1. Remark also that, in accordance with the discussion of the second section, the two definitions of the negation presented here require the knowledge of the "non-superposition" states (which realise an outcome with probability 1 or 0) only: the A-orthogonality is based on the eigenstate sets of outcomes (probability 1), the €-orthogonality is based on the impossible transitions (probability 0). This illustrates that quantum logics in the Geneva approach can be considered as an attempt to build a mathematical structure which generalises the classical (Boolean) structure but is still built on the basis of the "actual, absolutely true and certain" propositions which survive in probabilistic theories 3 . 4. THE A AND EORTHOGONALITY RELATIONS AND THE E-MODEL Before discussing the orthogonality, it is useful to recall some properties of the €-model: - The measurement process forces the state to collapse along two possible directions: the direction of the device p ("up"), or its opposite -p ("down"). - The probability of getting spin-up P(plq) depends on the angle () between p and q only and yields 1 when 0 ::; () < ()up, 0 when 7l' ~ () ~ 7l' - ()down' - In between it is superposition of the two possible results, in a zone of angular opening ()sup (with sin(oS2 P ) = E). We can now apply the definition of the A-orthogonality. We obtain the following: 3 This is also true for what concerns the Birkhoff-von Neumann approach [12). 216 THOMAS DURT Theorem 1: Two states p and q are A-orthogonal iff they make an angle superior or equal to ()sup when € # 0, iff they are different when ()sup = € = o. The proof, of technical nature, is given in [16]. As a consequence, the set of states A-orthogonal to a given state is a spherical sector around the antipode of this state, of opening 1[" - ()sup, closed if € # 0, and open in the deterministic case (€ = 0; ()sup = 0). It is then the Boolean (set-theoretical) complement of the state and we recover the classical relation (of difference) in this case. In the quantum case, the fuzziness is maximal and the orthogonal of a state is reduced to a unique point: its antipode. p ......... .......... ............. .............. ................. .................. ..................... ...................... ......................... .......................... ............................. .............................. ................................. ................................ ............................... ............................... ............................... .............................. ............................. ............................ ........................... .......................... ......................... ........................ ....................... ...................... ..................... .................. ................. .............. ·······;.;.;.·····i p A Figure 2. The A-orthogonal of p. If we apply the definition of the €-orthogonality, we obtain the following: Theorem 2: - When € # 0, the €-orthogonals of a state are closed spherical sectors of opening max(()down , ()up) around the antipode of this state. - When € = 0 and ()down ~ ()up, the €-orthogonals of a state are closed spherical sectors of opening ()down around the antipode of the state. - When € = 0 and ()down < ()up, the €-orthogonals of a state are open spherical sectors of opening ()down around the antipode of the state. A first difference appears relatively to the A-orthogonality: the deterministic limit does not necessarily imply that the €-orthogonality relation is ORTHOGONALITY RELATIONS 217 the difference (the Boolean relation). This is true only if (}down = O. Note that the E-orthogonality relation can be tested by a unique experimental configuration: if one places the measuring apparatus along the direction of the state when (}down 2: (}up (of its opposite otherwise), the states which collapse along the opposite with probability one belong to a sector of opening (}down (Bup) , so to say, they are orthogonal to this state. This property, which presents much interest [16] at the level of the lattice of properties is not fulfilled in the case of the A-orthogonality relation. Nevertheless, both definitions coincide in the quantum case. p 1t - max (S down,S up) Figure 3. The E-orthogonal of p. 4.1. The Orthocomplementation In the Geneva approach to quantum logics, the propositions are in general in one to one correspondence with the sets of states for which these propositions are actual (true with probability one). It is thus necessary, if one wants to generalise the classical negation, to define it as a relation between sets of states, rather than a relation between states. This can be realised as follows. Let us assume that we know a relation * between the states (the set of states is denoted ~), such that this relation is symmetric and antireflexive: Vp, q E ~, : p * q => q * p and II p E ~ : p * p. (2) Then, we can associate to any subset K of ~ its orthogonal defined by: K* = {p E ~ : p * q , V q E K} (3) 218 THOMAS DURT It is possible [13] to build, on the basis of the orthogonality relation *, what is called an orthocomplementation, defined as follows. Let us consider a set of subsets of E, denoted X (X c P(E)), partially ordered by set-inclusion in E, then X is said to be orthocomplemented if there exists a bijection', from X to X, such that VA, B EX: i) A c B "* B' c A' ii) (A')' = A iii) A n A' = 0 (4) Thanks to the fact that in our approach, the propositions are in one to one correspondence with subsets of the set of states (the sets of states which "actualise" the proposition), we will say that a lattice is orthocomplemented if an orthocomplementation relation exists between the sets of states which represent the proposition (this representation is called the Cartan map of the lattice). For a Boolean lattice, by definition (section 3.1), two states are orthogonal iff they are different. This implies that the complement of a set is its Boolean complement or set-theoretical complement. In Boolean logics, this complementation is associated to the classical logical negation. Note that it is possible to build a set of subsets of the set of states [13] which is orthocomplemented under the set orthogonality *. In order to build this set, one must consider the map (denoted here clorth in accordance with the conventions made in the refs. [6, 16]) from the set of states E onto itself defined by: clorth(K) = (K*)*. If one defines Forth as the subset of P(E) which is self-closed under clorth: Forth = {F E P(E) : clorth(F) = F}, it is possible to show [13] that Forth is orthocomplemented under the set-orthogonality *. This orthocomplementation generalises the properties of the Boolean complementation and presents a formal analogy with some laws of classical logics. For instance, the three properties (i, ii, iii) which enter the definition of an orthocomplementation generalise the following classical properties: i) if A implies B, the negation of B implies the negation of A ii) the negation of the negation of A is A itself iii) A is never true and false simultaneously It can be shown that the Cartan map of the lattice forms what is called a closure-structure (see [11, 14, 15] for a definition), and that this is also true for Forth. When these structures are identical, then, the lattice will be said to be orthocomplemented. ORTHOGONALITY RELATIONS 219 We also mention without proof the following theorems relative to the orthocomplementation of the lattice of properties of the E-model under the A and E-orthogonality relations [16]. Theorem 3: The lattice of properties deduced from an E-probability distribution characterised by the given of the angles (Oup , Odown) is A-orthocomplemented (so to say, the Cartan map of the lattice of properties is equal to Forth) iff Odown = o. Theorem 4: The lattice of properties deduced from an E-probability distribution characterised by the given of the angles (Oup , Odown) is E-orthocomplemented for arbitrary values of Oup and Odown. This shows that the E complementation is very well adapted to the E-model, which justifies its name. 5. APP ENDIX: PROP ER TIES OF THE S PIN ONE HALVE. Let us recall some properties of the spin 1/2. - It is defined in a two-dimensional Hilbert space and can always be expressed as a normalised superposition of a spin-up and a spindown state along an a priori direction z: 1'!fJ) = al +) + (31 - ), lal 2 + 1(31 2 = 1. (5) - Then, the spin-measurement with a Stern-Gerlach device oriented where along z is described by the observable ~O'z, = (I + )( + I) - -I)· (6) - The Pauli mapping maps bijectively the set of physical states onto the unit 3-sphere, according to the following transformation law: O'z (1-)( 1'!fJ) - t n = (2Re a* (3, 21m a* (3, lal 2 - 1(31 2) (7) (in Cartesian coordinates) and conversely: o _!::e. 2 (8) 2 where 0, cp are the polar angles of n. This allows us to visualize all states on the sphere ; the up-state for instance is sent on the north pole, the down-state on the south pole. a = cos -e ACKNOWLEDGEMENTS This work was realised in the framework of a project of the Flemish Funds of Scientific Research (F.W.O.). Some parts of it were already present in the reference [16]. It was finished during an academical visit in Gdansk, realised in the framework of the bilateral Flemish-Polish 220 THOMAS DURT project 007. Thanks to J. Pykacz for his constructive and destructive comments. AFFILIATION Thomas Dun, FUND, Brussels Free University, Brussels, Belgium REFERENCES [1] Aerts, D.: "The One and the Many", Doctoral Dissertation, Brussels Free University, Brussels, 1981. [2] Aerts, D.: "A possible explanation for the probabilities of quantum mechanics", J. Math. Phys., 27, 1986, p. 203. [3] Aerts, D., Durt, T., and Van Bogaert, B.: "A physical example of quantum fuzzy sets and the classical limit", Tatra Mountains Math. Publ., 1, 1992, p. 5. [4] Aerts, D., Durt, T., Grib, A.A., Van Bogaert, B., and Zapatrin, R.R.: "Quantum structures in macroscopic reality" , Int. J. Theor. Phys., 32, nO 3, 1993, p. 489. [5] Aerts, D., Durt, T., and Van Bogaert, B.: "Indeterminism, nonlocality and the classical limit", in uProceedings of the Symposium on the Foundations of Modern Physics, Helsinki, August 1992, World Scientific Publishing Company, Singapore, 1993, p. 154. [6] Aerts, D. and Durt, T.: "Quantum, classical and intermediate. A measurement model", in uProceedings of the Symposium on the Foundations of Modern Physics", Helsinki, August 1993, World Scientific, Singapore, 1994, p. WI. [7] Aerts, D.and Durt, T.: "Quantum, classical and intermediate. An illustrative example", Found. of Phys., 24, 1994, p. 1407. [8] Aerts, D., Aerts, S., Coecke, B., D'Hooghe, B., Durt, T., and Valckenborgh, F.: "A model with varying fluctuations in the measurement context", in New Developments on Fundamental Problems in Quantum Physics, Ferrero et al., eds, Kluwer, Dordrecht, 1997, p. 7. [9] Aerts, D., Coecke, B., Durt, T., and Valckenborgh, F.: "Quantum, classical and intermediate I: A model on the Poincare sphere", in uProceedings of the 4th Winter School on Measure Theory", Eds. A. ORTHOGONALITY RELATIONS 221 Dvurecenskij and S. Pulmannova, Tatra Mountains Math. Publ., 10, Bratislava, 1997, p. 225. [10] Aerts, D., Coecke, B., Durt, T., and Valckenborgh, F.: "Quantum, classical and intermediate II: The vanishing vector space structure", in "Proceedings of the 4th Winter School on Measure Theory", Eds. A. Dvurecenskij en S. Pulmannova, Tatra Mountains Math. Publ., 10, Bratislava, 1997, p. 24l. [11] Beltrametti, E. and Cassinelli, G.: "The Logic of Quantum Mechanics", Addison-Wesley Publishing Company, 1981. [12] Birkhoff, G. and von Neumann, J.: "The logic of quantum mechanics", Annals of Mathematics, 37, 1936, p. 823. [13] Birkhoff, G.: "Lattice Theory", third edition, Amer. Math. Soc., Colloq. Publ. Vol. XXV, Providence. [14] Crapo, H.H. and Rota, G.C.: "Geometric lattices" in "Trends in lattice Theory", ed. Abbott J.C. Van Nostrand-Reinhold, New York, 1970. [15] Crapo, H.H. and Rota, G.C.: "On the foundations of combinatorial theory (II)" in "Studies in Appl. Math. ", 49, 1970, p. 109. [16] Durt, T.: From quantum to classical, a toy model, Doctoral Dissertation, Brussels Free University, 1996. [17] Grib A.A. and Zapatrin R.R.: Int. Journal of Theor. Phys., 29, (2), 1990, p. 113. [18] Piron, C.: "Axiomatique quantique", Helv. Phys. Acta, 37, 1964, p.439. [19] Piron, C.: "Foundations of Quantum Physics", W.A. Benjamin, Inc., 1976. [20] Piron, C.: "Mecanique Quantique, Bases et Applications", Presses Polytechnique et Universitaire Romandes, 1990. [22] von Neumann, J.: "Grundlehren, Math. Wiss. XXXVIII", 1932. [23] von Neumann, J.: "Mathematische Grundlagen der Quanten-mechanik", Springer-Verlag, Berlin, 1932. 222 THOMAS DURT APPENDIX: SOME PICTURES: THE TRANSITION PROBABILITY E- We present here the graphs of the probabilities of transition for some values of Oup and 0down' The first graph corresponds to the quantum situation, and the second and third graphs correspond to the deterministic situation. 0.8 0.6 0.4 0.2 0.5 1.5 2.5 Figure 4. Probability distribution when ()up 3 = 0, the quantum case. 0.8 0.6 0.4 0.2 I.S 0.5 Figure 5. ()up 2 2.5 = ()down = 7r /2. 0.8 0.6 0.4 0.2 0.5 Figure 6. I.S ()v.p 2.S = 7r. 3 ORTHOGONALITY RELATIONS O.S 0.6 0.4 0.2 0.5 2.5 1.5 Figure 7. ()up = ()down = 1r /8. 0.8 0.6 0.4 0.2 0.5 Figure 8. 1.5 (}up /2, 2 2.5 = 7r = 1r /4, (}down (}down 3 = 7r /4. 0.8 0.6 0.4 0.2 0.5 Figure 9. 1.5 (}up 2 2.5 3 = 7r /2. 0.8 0.6 0.4 0.2 0.5 Figure 10. 1.5 (}up = 7r/6, 2 2.5 (}down 3 = 7r /3. 223 M.L. DALLA CHIARA & R. GIUNTINI QUANTUM LOGICAL SEMANTICS, HISTORICAL TRUTHS AND INTERPRETATIONS IN ART The basic features of classical logical semantics can be summarized as follows: 1) Truth behaves like the knowledge of an omniscient mind, which is at the same time non contradictory and complete. 2) Meanings behave in an atomistic and compositional way: the meaning of a whole is determined by the meaning of its parts. 3) Meanings are non ambiguous and sharp. All this renders classical semantics hardly applicable to an adequate analysis of natural languages and of artistic contexts, where holistic and ambiguous features seem to playa relevant role. However, formal semantics is not necessarily bound to classical logic. In contemporary logical research, the classical notion of truth has been transformed according to different views. For instance, truth can be identified with what is known by non omniscient minds in a universe that may be either deterministic or indeterministic. The second choice represent the starting point of a semantic characterization of quantum logic. As a consequence one obtains examples of formal semantics that permit us to model ambiguous and holistic situations. In such a framework, even the classical non contradiction principle can be violated. Have these semantic considerations any bearing for a possible formal analysis of interpretations in artistic contexts? We will refer to the case of music, and will try and answer the following question: is it possible to recognize a kind of abstract nucleus that might be common to scientific theories and musical compositions? Birkhoff and von Neumann began their famous article "The Logic of quantum mechanics" [1 J with the following claim: One of the aspects of quantum theory that has attracted the most general attention, is the novelty of the logical notions which it presupposes. After sixty years such a novelty seems to have largely crossed the boundaries of microphysics. From an intuitive point of view the basic features of classical semantics (developed by Leibniz, Frege, Tarski ... ) can be summarized as follows: 1) Truth behaves like the knowledge of an omniscient mind, which is at the same time non contradictory and complete. 225 © 1999 Kluwer Academic Publishers. 226 M . L. D ALL A CHI A R A & R. G I U N TIN I As a consequence: a) any problem is semantically decided: for any sentence A, either A or not A is true (Tertium non datur); b) A sentence A and its negation not A are not at the same time true (Non contradiction principle). 2) Meanings behave in an atomistic and compositional way: the meaning of a whole is determined by the meanings of its parts. 3) Meanings are non ambiguous and sharp. All this renders classical semantics hardly applicable to an adequate analysis of natural languages and of artistic contexts, where holistic and ambiguous features seem to playa relevant role. In this connection, one might refer to a number of different examples. For instance, let us think of the final verse of a famous poem, L'Infinito (Infinity), by Giacomo Leopardi: E'l naufragar m 'e dolce in questa mare (And drowning in this sea is sweet to me). The poetic result seems to be essentially connected with the following semantic relation: the meanings of the component expressions "naufragar" (drowning), "dolce" (sweet), "mare" (sea) do not correspond here to the most common meanings. By the way, there is no sea in Recanati, Leopardi's native village which the poem refers to. However the usual meanings of our expressions are somehow present and ambiguously correlated with the metaphorical meanings that are evoked by the whole poem. Needless to say, this represents a quite typical semantic situation in poetry. In contemporary logical research, the classical notion of truth has been transformed according to different views. For instance, truth can be identified with what is known by non omniscient minds in a universe that may be either deterministic or indeterministic. The first choice is compatible with the intuitionistic approaches to logic and mathematics, whereas the second choice represents the basic assumption of the quantum logical investigations. In both cases, the classical notion of truth a sentence A is true under a given interpretation has been replaced by the following relation: a given information forces us to assert the truth of a sentence A. QUANTUM LOGICAL SEMANTICS 227 All this can be conveniently described by using the metaphor of possi- ble worlds. What is a possible world? In the framework of an epistemic conception, a possible world can be regarded as a piece of information about a possible state of affairs: for instance, what is known by an observer about a physical system. The basic semantic relation is usually represented as follows: i pA And is read: "information i forces sentence A" . Should i represent a non contradictory and complete system of information, our forcing relation would naturally collapse into the classical notion of truth. However, knowledge is in general not complete and not necessarily consistent. Further, an information may essentially refer to other correlated information systems. As a consequence, it is expedient to suppose a whole set of pieces of information: 1= {i,j, k, ... } that admit of different correlations. In the technical jargon of possible world semantics, a correlation between pieces of information (or possible worlds) is usually called an accessibility relation. How can Birkhoff and von Neumann's quantum logic be described in the framework of a possible world semantics? The basic intuitive idea is the following: any information i represents the observer's knowledge about the quantum system under investigation (for instance the electron of a given hydrogen atom). As a limit case, i might correspond to a non contradictory maximal knowledge: in other words, i cannot be consistently extended to a more precise information. Pieces of information of this kind are usually called, both in classical and in quantum physics, pure states. In the formalism of quantum theory, pure states are mathematically represented as particular vectors in an appropriate abstract space (which is called a Hilbert space). Accessibility here means logical compatibility: i and j are accessible (or non orthogonal) if and only if they do not have contradictory consequences. On this basis, the accessibility relation turns out to be reflexive (any information i is accessible to itself) and symmetric (if i is accessible to j then j is accessible to i). What about the forcing conditions? As to the atomic sentences (that cannot be decomposed into simpler sentential parts) the following condition is assumed: i forces a sentence A if and only if any information j accessible to i has an accessible information k that forces A. 228 M . L. D ALL A CHI A R A & R. G I U N TIN I This condition represents a kind of stability requirement. Let us now consider the case of compound (molecular) sentences that are constructed by composing atomic sentences by means of logical connectives (negation, conjunction, disjunction). The behaviour of negation is governed by the following condition: an information i forces a sentence not A if and only if i has no accessible information j that forces A. In other words: i forces not A when it is impossible to transform i into a compatible information j that forces A. As a consequence, an information i will not necessarily decide a sentence A that may remain strongly indetermined for i: i forces neither A nor not A. The semantic tertium non datur fails, in accordance with the probabilistic character of quantum theory. What about conjunction and disjunction? Conjunction has a classical behaviour, governed by the usual truth-table. In other words: an information i forces A and B if and only if i forces both members A,B. At the same time, like in classical logic disjunction is supposed to be defined in terms of negation and conjunction by means of the so called de Morgan law: i forces A or B if and only if i forces not (not A and not B). Since negation does not have here the usual classical meaning, the characteristic truth-table for the connective or turns out to be violated: forcing the truth of a disjunction does not mean forcing the truth of at least one member As a consequence, the following situation is possible: i forces A or B; however i forces neither A nor B! QU ANTUM LOGICAL SEMANTICS 229 This peculiar behaviour of disjunction corresponds to a fairly typical quantum situation. In quantum theory on can mention a number of examples where an alternative is determined and true, even if both members are strongly undetermined (and hence not true). For instance, any electron has in any direction either spin up or spin down. However, owing to the uncertainty relations, an electron that has a well determined spin value in the x direction (say value up) cannot have a well determined spin value in the y direction (neither up, nor down). Birkhoff and von Neumann's quantum logic represents, in a sense, a "semiaristotelian logic". As we have seen, the semantic tertium non datur fails: a sentence is not necessarily either true or false. However, the non contradiction principle still holds. New forms of quantum logic that are even "more non aristotelian" arise in the framework of the so called unsharp approaches to quantum theory. Why moving to the unsharp approaches? From an intuitive point of view, on can say that such approaches represent, in a sense, an important step towards a kind of second degree of "fuzziness" or "ambiguity". We can try and illustrate the difference between the standard and the unsharp approaches by referring to a non scientific example. Let us consider the two following sentences, which apparently have no definite truth-value: I) II) Hamlet is 1. 70 meters tall. Brutus is an honourable man. The semantic uncertainty involved in our first example seems to depend on the incompleteness of the individual concept associated to the name "Hamlet". In other words: the property "being 1.70 meters tall" is certainly a sharp property. However, our concept of Hamlet is not able to decide whether such property is satisfied or not. Differently from real persons, literary characters have a number of undetermined properties. On the contrary, the semantic uncertainty involved in our second example, is mainly caused by the ambiguity of the concept "honourable". What does it mean "being honourable"? Needless to recall how the ambiguity of the adjective "honourable" plays an important role in the famous Mark Antony's monologue in Shakespeare's "Julius Caesar". Orthodox quantum theory and quantum logic take into consideration only examples of the first kind: properties are sharp; whereas all semantic uncertainties are due to the incompleteness of the individual concepts, which correspond to pure states of quantum objects. A characteristic of unsharp quantum theory, instead, is to investigate also examples of the second kind. 230 M . L. D ALL A CHI A R A & R. G I U N TIN I In Hilbert space quantum theory, physical properties that may be ambiguous and unsharp are mathematically represented by particular operators, called effects. At the same time, sharp properties correspond to projection operators, which are limit-cases of effects. An extreme case of an unsharp property is represented by the semitransparent effect, to which any physical state assigns probability ~. This represents a kind of paradigmatic example of a totally undetermined physical property. In the possible world semantics for unsharp quantum logic, an information i may correspond to an ambiguous knowledge. As a consequence, pieces of information turn out to be not necessarily compatible with themselves: reflexivity breaks down. On this basis, it turns out that an information may force at the same time a sentence A and its negation not A: iF A, notA. The non contradiction principle is violated! The result is a paraconsistent (or fuzzy or unsharp) form of quantum logic. Similar semantic behaviours can be recognized also in other domains of experience that, prima facie, appear to be quite far from microphysics. In fact, the notion of possible world can be successfully applied not only by logicians and by physicists. For instance, also historians naturally interact with possible worlds, at least implicitly. The role of possible worlds in historical research has been investigated in [2], [3]. We will use here a slightly different semantic approach. Similarly to physical states, even historical sources determine pieces of information about possible state of affairs. From an abstract point of view, in the simplest case, a source can be idealized as a set of sentences: a source i will assert a sentence A when A is contained in i. For instance the source The Lives of the Twelve Caesars asserts the sentence: Evident miracles had announced to Julius Caesar his violent death. More generally a source may be represented also by some non linguistic objects (monuments, tools, tombs, ... ). As a consequence, the relation "i asserts A" might have a weaker meaning, like "A is confirmed, testified by i". Similarly to a quantum logical situation, two sources may be either compatible or incompatible. We will say that two sources i and j are compatible when it is not the case that i asserts a sentence A whereas j asserts the negation not A. Since a source is not necessarily consistent, QUANTUM LOGICAL SEMANTICS 231 we will have that some sources might be incompatible with themselves. Apparently, the presence of a local contradiction is not a sufficient reason that renders a given source completely unreliable. Consequently, like in the case of unsharp quantum logic, the accessibility relation between pieces of information turns out to be symmetric, but generally not reflexive. Let I represent a set of sources that are available to a given historian. When will our historian accept the historical truth of a given statement? At first sight, one is tempted to take into consideration the two following possibilities: I) II) All sources assert A. At least one source asserts A. However, our first choice seems too strong: many events that are sometimes considered as "historical facts" are not asserted by all sources. At the same time, our second choice appears too weak: some sources might be not completely reliable. For instance, why should we trust Suetonius when he asserts that "Evident miracles had announced to Julius Caesar his violent death"? An intermediate choice between I) and II), that seems quite reasonable from the intuitive point of view, naturally leads to a quantum-logic-like truth condition: a source i forces the truth of an atomic sentence A if and only if for any source j compatible with i, there exists a compatible source k that asserts A. In other words, A represents a stable assertion. As to the case of compound sentences, again the quantum logical truth conditions appear to be quite natural. In particular, one might mention a number of examples in historical contexts, where an alternative is true, whereas both members are strongly undetermined. Finally let us ask: when will a historian accept the historical truth of a given sentence? The natural answer in our abstract semantics is the following: a historian will accept the truth of A, when A is forced by a set of sources that are judged reliable by our historian. On this basis one can conclude that truths discovered by historians and by quantum physicists turn out to have some strong formal similarities! In both cases, one is dealing with a typical holistic behaviour, against the analytic tradition of classical semantics. Holistic features can be easily recognized also in the quantum theoretical treatment of the notion of physical object. Suppose we want 232 M. L. DALLA CHIARA & R. GIUNTINI to describe a compound physical system. For instance, a two particle system 8 = 81 + 82 (say, the electron and the nucleus of a given hydrogen atom). In the happiest situation, the observer's information about the system will correspond to a maximal information: a pure state. Otherwise, it will be a mixed state or more generally a physical property corresponding to a partial description of our system. What about the component systems? Information about the parts can be derived from the information about the whole system in accordance with the mathematical formalism of quantum theory. Now, a typical quantum theoretical situation is the following: a state i provides a maximal information about the whole system. However, the information determined by i about the component systems cannot be maximal: the individual concepts of the two parts do not correspond to pure states. This is exactly what happens in the case of a compound system consisting of two indistinguishable particles of the same kind (for instance, two bosons, which obey the Bose-Einstein statistics). Since we are dealing with two indistinguishable bosons, any pure state of the whole system may have a typical form that has been called entangled: as a consequence the states of the two subsystems will not be pure and will correspond to two equal mixed states. The result is a characteristic holistic behaviour: the whole permits to determine partial individual concepts for the parts; at the same time the whole is not determined by its parts. Have these semantic considerations any bearing for a possible formal analysis of interpretations in artistic contexts? Let us refer to the case of music and let us try and ask the following question: is it possible to recognize a kind of abstract nucleus that might be common to scientific theories and to musical compositions? As is well known, according to a standard logical approach, any scientific theory can be, in principle, analyzed as a pair 1'IHlE([J)~Y = ( Formal system, Class of interpretations) where the formal system corresponds to an axiomatic version of the theory, while the interpretations (or models) represent experience domains where the sentences asserted by the formal system (the theorems of the theory) are verified. In some limit-cases a formal system is lacking, and the theory is simply identified with a language and with a class of models. Paradigmatic examples of theories in this sense are Peano arithmetic or von Neumann's axiomatic quantum theory. QUANTUM LOGICAL SEMANTICS 233 One can guess that even a musical composition can be similarly analyzed as a pair: COMJlD«Jl§TIlTION = ( Score, Class of interpretations) The score plays the role of a formal system (a particular syntactic object); at the same time all historical interpretations of the score play the role of the models. Needless to observe, musical compositions cannot be generally reduced to their mere scores. Scores and formal systems present some clear similarities and some obvious differences. For instance, like in the case of formal languages, also in a score one can naturally distinguish between atomic and compound musical expressions: atomic expressions will represent musical phrases or themes that are perceived as a whole, and that cannot be further decomposed. As an example, the first measures of the first violins and of the violas in Beethoven's 9th symphony can be reasonably regarded as an atomic expression: All the proper parts in such expression, are perceived as incomplete themes. Compound musical expressions are build up by means of complex concatenation operations in a bidimensional frame: as is well known, bidimensionality is a characteristic of the standard formal language of musical works. These concatenation operations can be regarded as the natural counterparts of logical connectives in formal systems. For a theoretician, a formal language represents a possible linguistic context: asserting a given formal system essentially means selecting a set of theorems within this language. In the same way, one could say that a composer refers to a linguistic context, where he selects a system of musical expressions. In a sense, the musical expressions written in a score play the role of the theorems in a formal system. A main difference between formal systems and scores is the following: in most formal languages well formed expressions give rise to a decidable set; in other words, one can always decide in a finite number of steps whether a given sequence of signs is a well formed expression of our language. At the same time, after Codel we know that the set of the theorems is generally undecidable. On the contrary, musical languages (as well as all artistic languages, in general) are intrinsically ambiguous and undecidable, in spite of all rigid harmony treatises. 234 M . L. D ALL A CHI A R A & R. G I U N TIN I What about the second component of a musical work (the class of all interpretations)? As is well known, interpreting a formal system means realizing the well formed expressions of the language in a given universe of discourse. Similarly a musical interpretation (a performance) realizes musical events that corresponds to musical expressions written in the score. In both cases, one is dealing with an operation that associates meanings to systems of signs. As we have seen, in classical model theory meanings have a compositional and analytic behaviour: the meaning of a whole is determined by the meanings of the parts. Of course, such a behaviour cannot be expected in the case of musical performances: generally a whole will determine the meanings of the parts, but not the other way around. Whenever one interprets (or one simply listens to) a given symphony, one cannot generally perceive all the different lines written in the score as distinguished elements. At the same time, the result of a performance does not represent, for a musically educated person, a kind of indistinct sound beam (which represents a typical unidimensional phenomenon, from the physical point of view). Generally, an interpretation decomposes a whole consisting of many expressions into a limited number of voices that are perceived as unitary elements, where each voice is active during a certain time interval. Of course voices may be either human or instrumental. Is it really useful to try and develop technically an abstract holistic semantics for musical compositions, based on these general ideas? We do not have a certain answer to this question. Anyway, the mere existence of some common abstract patterns that can be discovered both in scientific and artistic works, seems to be interesting from the theoretical point of view. AFFILIA TION Maria Luisa Dalla Chiara Roberto Giuntini Dipartimento di Filosofia Universita di Firenze, Italy REFERENCES [1] Birkhoff, G. and von Neumann, J., "The Logic of quantum mechanics", Annals of Mathematics, 37, 1936, pp. 823-843. [2) Toraldo di Francia, G., Un Universo Troppo Semplice, Feltrinelli, Milano, 1990. [3) Toraldo di Francia, G., "Historical Truth", Foundations of Science, 1, 1995-1996, pp. 417-425. 235 INDEX Lukasiewicz, 68, 91 conjunction, 70, 82 disjunction, 70, 82 equivalence, 70 implication, 70, 72 negation, 70, 82 operations on fuzzy sets, 82 accessibility relation, 227 antisymmetry, 44 Aristotle, 68 associativity, 42, 47 atomicity, 188 attribute, 6, 14, 29 actual, 14, 16, 22 irreducible, 18 logic, 15, 17, 27, 28 potential, 14, 22 principal, 16 principle of actuality, 18 tested, 16 axiomatics, 169, 176, 182, 190, 201 Bell -Kochen-Specker theorem, 104, 133 inequality, 133 theorem, 132 binary operation partiaL 42 total, 42 Birkhoff and von Neumann's quantum logic, 227 Birkhoff-von Neumann, 88, 89 Bohr, 31, 106, 142 bosons,232 certainly true domain, 122 characteristic function, 39 classical, 3, 4, 13, 14, 18, 27, 28, 30, 54-57, 67, 68, 75, 78, 81, 82, 8693,103,105,107-109,112-116,118, 119, 122, 125-127, 129-131, 136, 143-145, 153, 163-169, 185, 198, 200, 207, 208, 210-213, 215-218, 225-228, 231, 234 closure, 122 commutativity, 42, 47 compatibility pragmatic, 124 semantic, 123 complement, 121 complementarity, 31, 86, 88 completion procedure, 126 compound entities, 193 compound physical system, 125, 232 conditional hypothesis, 28 confidence algebra, 43 function, 38, 43 confidence function, 41 conjunction, 13, 18, 27, 94 Lukasiewicz, 70, 82 Zawirski, 74, 82 contexts accessible physical, 130 contradictory, 130 non-accessible physical, 130 contextuality, 104, 133 pragmatic, 131 semantic, 131 Copenhagen paradigm, 86 covering law, 188, 193, 194, 200, 201 crisp set, 76 D-poset, 7 de Broglie, 85, 142 de Morgan law, 228 decidable, 233 deduction rule, 107, 109 denial, 28 detector isotropic, 42 particle, 38 dichotomic registering device, 116 dichotomy, 44 difference poset, 48 236 INDEX Dirac, 2, 17, 18, 87, 142 disjunction, 28, 94 Lukasiewicz, 70, 82 Zawirski, 74, 82 distributivity, 49 domain, 118 effect, 38, 47, 117 algebra, 7 operator, 7 algebra, 47, 120 Hilbert space, 50 crisp, 48 operator, 7 sharp, 38, 46 unsharp,38 effects, 230 Einstein, 2, 75,84, 106, 142, 194, 196, 200, 201, 232 equivalence Lukasiewicz, 70 estimator, 24 maximum likelihood, 24 Euclidean, 58, 68,143,145,147,149, 200,201 event, 9, 91 logic, 10, 11, 28 nonoccurred, 22 nonoccurrence, 9 not tested, 22 occurred, 22 occurrence, 9 test of, 19 events compatible, 10 locally complementary, 10 orthogonal, 10 perspective, 10 experiment, 1, 7, 14, 22, 23, 27, 3739, 55, 62, 67, 83-87, 89-95, 116, 117, 121, 126, 131, 136, 149-154, 157-160, 162, 163, 167, 176-178, 191, 192, 196, 198, 199,211-214 double-slit, 83, 85 interference, 84, 85 experimental procedure, 8 proposition, 11, 27, 29, 89-9l confirmed, 12, 22 not tested, 22 refuted, 12, 22 test of, 12 propositions compatible, 13 logic of, 11, 12 orthogonal, 13 simultaneously testable, 27 extension, 118 extreme point, 20 face lattice, 21, 28 Feynman, 83 forcing conditions, 227 formal system, 232 frequency, 121 fuzzy proposition, 7 set, 76 complement, 78, 82 theory, 76 sets Lukasiewicz operations, 82 bold operations, 82 equality, 78 Giles operations, 82 inclusion, 78 intersection, 80, 82 union, 79, 82 Zadeh operations, 78, 82 Giles, 82 greatest lower bound, 13, 46, 93 Heisenberg, 26, 54, 55, 87, Ill, 141, 142 hidden measurements, 166 hidden variables, 2, 30, 31, 90, 114, 137, 163, 164, 207, 210 local, 137 theory, 114 Hilbert space, 4-7, 48,50,54,57,62, 89-92, 94, 96, 142, 143, 160, 180, INDEX 183-186, 188-190, 195-199, 201, 208, 219, 227, 230 historical sources, 230 historical truth, 231 holistic behaviour, 232 implication, 12 Lukasiewicz, 70, 72 connective, 28 relation, 13, 15, 28, 172 indistinguishable particles, 232 interference, 83 interval estimation, 23 involution, 44 join, 13 laboratory, 116 lack of knowledge, 90, 163, 165-168, 196-199 language of properties, 127 lattice, 3-7,13,21,22,27,28,49,50, 57, 62, 90, 92, 94, 126, 127, 183, 185-190, 194, 197, 207, 208, 211, 212,217-219 of face,;, 28 of facs, 21 orthomodular, 5, 6, 13, 90 law 0- 1, 47 associative, 42 De Morgan, 13, 46, 74 distributive, 49 distributivity, 54 empirical, 129 of bivalence, 69 of C'ontmniction, 74 of distributivity, 4, 90 of excluded middle, 69, 74 theoretical, 129 least upper bound, 13, 45, 93 Leopardi Giacomo, 226 locality, 132 logic 237 n-valued, 69, 71 Lukasiewicz, 68 of experimental propositions, 12 classical, 68, 75, 86, 107, 115 inductive, 23, 28 infinite-valued; 71, 81 intuitionistic, 109 many-valued, 68, 71, 86 non-classical, 68 of attributes, 15, 17, 27, 28 of conditional hypotheses, 28 of events, 10, 11, 28 of experimental propositions, 11, 28 quantum, 90, 110, 127 three-valued, 69, 87 logical connectives, 27 macroscopic, 60, 61, 83, 85, 116, 136, 144, 145, 165, 198, 207 maximal knowledge, 227 measurement, 7, 8, 25, 26, 29, 30, 37, 38, 41, 46-49, 54, 55, 59, 60, 62, 103, 104, 110, 111, 113, 114, 128, 132, 134, 137, 159, 165, 167, 207212, 215, 219 measurement yes-no, 37 meet, 13 membership degree of, 53, 75, 82 function, 76 Metatheoretical Principle classical, 130 Generalized, 130 mixed state, 232 models, 232 musical interpretation, 234 negation, 13, 92 Lukasiewicz, 70, 82 Post, 73 Non contradiction principle, 226 nonlocality, 104, 132 objectification problem, 113 238 INDEX observative language, 116 ontologie, 90, 112-115, 132, 163, 166, 167, 169, 172, 191, 199 operational, 62, 91, 126, 128, 129, 131, 136, 142, 143, 169, 173, 174, 176, 182, 190, 195, 201 operator, 2,4, 5, 7, 26, 29, 30, 54, 57, 62, 89-92, 94, 126, 134, 143, 157159, 178-181, 183-187, 189, 198, 199, 230 order inverting operation, 44 logical, 109 relation, 120 partial, 44 total, 44 orthoalgebra, 6, 48 orthocomplementation, 5, 13, 27, 28, 188-190, 193,207,217-219 orthogonality, 47, 156, 188, 190, 207, 212, 214, 215, 218 orthomodular identity, 5 lattice, 5 identity, 13 lattice, 6, 13, 90 poset, 6 orthosupplement, 40 orthosupplementation, 47 outcome, 9-12, 14-16, 19, 20, 24-29, 62, 92, 93, 111, 113, 117, 120, 121, 134, 153, 159, 160, 166, 176, 178, 211, 212, 214, 215 paraconsistent, 230 paradox, 2, 104, 106, 110, 115, 116, 119, 123, 125, 127, 135, 137, 141, 143, 149, 190, 194, 197 photon, 84 physical object, 117 point estimation, 23 possible world semantics, 227 Post, 69, 72 negation, 73 pre-order, 109, 171, 172, 183, 185,198 preclusivity relation, 122 predicate non-sharp, 76 sharp, 75, 76 preparation, 116 probability, 19, 121 measures convex combination of, 20 full set of, 19 order determining set of, 19 model, 19 projection, 4, 5, 7, 29, 30, 54, 62, 89, 91, 92, 94, 125, 137, 143, 158-160, 162, 168, 178, 183, 184, 186, 187, 189, 190, 212, 230 operators, 230 property, 38, 42, 44, 48, 54, 62, 87, 103, 104, 108, 111, 123, 125-128, 133, 137, 146, 164, 166, 167, 169173, 176-178, 181, 183-186, 188192, 196, 197, 200, 201, 211-213, 215,217,229,230,232 correlation, 126 first order, 123, 126 higher order, 123 testable, 123 theoretical, 126 proposition, 29 propositional function, 92 always-false, 92, 95 always-true, 92, 95 pure states, 227 quantum logic, 230 quantum machine, 145, 149-152, 154, 155, 162, 165-167, 169, 179, 185, 191, 198, 199 quantum mechanics, 1-6, 18, 25, 2931, 38, 48, 50, 53-55, 58, 59, 61, 62, 83, 86-91, 103-107, 109-116, 118-120, 122, 124-127, 129-137, 141-143, 145, 149, 153, 155, 158160, 162-165, 167, 169, 172, 182, 184, 186, 188-197, 199-201, 211, 212, 215, 225 quantum structures, 89, 92, 141, 142 INDEX reality, 132 received viewpoint, 104 reflexivity, 44 Reichenbach, 88 relativity theory, 142, 143, 200 Schrodinger, 2, 87, 141, 142 score, 233 239 testability, 110 transition probability, 151, 154, 155, 162, 179-181, 198, 209, 222 transitivity, 44 truth, 107, 128 access to, 107 empirical access to, 128 Tarski theory of, 107 Semantic Realism, 115, 119, 127 spin, 25, 30 experiment, 152, 157, 160 state, 2,16-18,20,22,25-27,29-31, 44, 48, 53-57, 59-62, 69, 76, 84, 87, 91-93, 95, 96, 103, 104, 108110, 114, 117-127, 129, 131-134, 137, 149, 150, 153-157, 159, 164, 166, 168-173, 175-182, 185-188, 190, 191, 195-199, 201, 207-209, 211-219, 227, 229, 230, 232 entangled, 125 first type, 125 mixed, 122 property space, 171-176, 182-187, 189-191, 195, 201 property system, 184, 185, 188, 189, 192-194 pure, 122 second type, 125 statement nontestable, 128 testable, 128 statistical hypothesis, 23 testing, 23 inference, 23 parameter, 23 parametric hypothesis, 23 Stern-Gerlach, 29,152-155,157,160, 209, 219 superposition, 18, 28 support, 20 testable, 124, 127 theoretical, 127 tensor product, 6 Tertium non datur, 226 uncertainty principle, 93 uncertainty relations, 229 undecidable, 233 unsharp approaches to quantum theory, 229 vector space, 2, 4, 89, 142, 145, 147, 148, 155-157, 162, 179, 188, 194 verificationism empirical, 110 logical, 109 von Neumann, 2, 4, 5, 29, 31, 5456,58,67,88-90,92-96,141,143, 144, 164, 215, 225, 227, 229, 232 weak modularity, 189 Zadeh, 76 operations, 78, 82 Zawirski, 74,88, 91 conjunction, 74, 82 disjunction, 74, 82