Papers by Daniel Sternheimer
arXiv (Cornell University), Apr 16, 2023
arXiv (Cornell University), Dec 29, 2022
Symmetries and their applications always played an important role in I.E. Segal's work. I shall e... more Symmetries and their applications always played an important role in I.E. Segal's work. I shall exemplify this, starting with his correct proof (at the Lie group level) of what physicists call the "O'Raifeartaigh theorem", continuing with his incidental introduction in 1951 of the (1953) Inönü-Wigner contractions, of which the passage from AdS (SO(2,3)) to Poincaré is an important example, and with his interest in conformal groups in the latter part of last century. Since the 60s Flato and I had many fruitful interactions with him around these topics. In a last section I succintly relate these interests in symmetries with several of ours, especially elementary particles symmetries and deformation quantization, and with an ongoing program combining both.
International audienceWe start with some general remarks related to Julian Schwinger’s works in, ... more International audienceWe start with some general remarks related to Julian Schwinger’s works in, and attitude to, physics, in particular in relation with Moshe Flato and me. Then we recall some notable events with Moshe (70s to 90s) and (since the 80s) with Noriko Sakurai, and also with Clarice Schwinger and me this millennium. A few photos illustrated the talk, most of which are reproduced at the end of this contribution
Modern Group Theoretical Methods in Physics, 1995
... Inst. Henri Poincare A 32, 1-31 (1980). Page 286. 264 DANIEL STERNHEIMER 62. F. Hansen. The M... more ... Inst. Henri Poincare A 32, 1-31 (1980). Page 286. 264 DANIEL STERNHEIMER 62. F. Hansen. The Moyal product and spectral theory for a class of infinite-dimensional matrices, Publ. ... 92. A. fc J. Unterberger. Quantification et analyse pseudodifferentielle, Ann. Sc. Ec. Norm. Sup. ...
Basic definitions and some results in the recently developed theory of nonlinear Lie group repres... more Basic definitions and some results in the recently developed theory of nonlinear Lie group representations in Banach and Fréchet spaces are presented. Using cohomological methods, this framework permits a study of the linearizability of covariant nonlinear evolution equations. Formal linearizability is proved under some conditions on the linear part of the representation, for massive and for massless Poincaré covariant equations. In particular, pure Yang-Mills equations supplemented with a relativistic gauge condition are formally linearizable.
Letters in Mathematical Physics, 2005
After a short presentation of the difference in motivation between the Berezin and deformation qu... more After a short presentation of the difference in motivation between the Berezin and deformation quantization approaches, we start with a reminder of Berezin’s view of quantization as a functor followed by a brief overview of deformation quantization in contrast with the latter. We end by a short survey of two main avatars of deformation quantization, quantum groups and quantum spaces
Czechoslovak Journal of Physics, 1982
The theory of nonlinear evolution equations developed by M. Flato, J. Simon and a few others is r... more The theory of nonlinear evolution equations developed by M. Flato, J. Simon and a few others is reviewed. The method of construction of global solutions is described and the cohomological and analytic properties of linearizability of these equations are described.
Poisson Geometry, Deformation Quantisation and Group Representations
Table of Contents Introduction 1. Projections on convex cones 2. Classification by lowest K-types... more Table of Contents Introduction 1. Projections on convex cones 2. Classification by lowest K-types 3. Relation to the unitary classification 4. Relation to ordinary parabolic induction 5. The reduction step in the classification 6. The set B z u (G) of unitarily small K-types 7. Unitarily small K-types and the Dirac operator 8. Topology of the unitary dual References
Quantum Theories and Geometry, 1988
After presenting the kinematical basis for interpreting massless particles as composed of two Dir... more After presenting the kinematical basis for interpreting massless particles as composed of two Dirac singletons in de Sitter space, we review the gauge structure (triple extension of representations) of the Rac and photon fields, and their relationship to one another. Then we show how it is possible to quantize the Rac field in a way consistent with both the usual quantization of the photon field and conformal invariance, and apply it to QED.
Harmonic Analysis and Representations of Semisimple Lie Groups, 1980
When do we utilize the structure of a deformed Lie algebra in Physics? As a matter of fact and a ... more When do we utilize the structure of a deformed Lie algebra in Physics? As a matter of fact and a posteriori one can say that almost always when we passed from one level of description of phenomena to another one the deformation structure was present. This is the case when one passes from non-relativistic Galilean mechanics to relativistic (Lorentz invariant) mechanics, as the Poincare Lie algebra is a deformation of the Galilean Lie algebra (when the light velocity c becomes finite). This is also the case, as will be shown in the next chapter, when one passes from classical mechanics, which is described by the infinite-dimensional Lie algebra of C ∞ functions on phase-space endowed with the Poisson bracket structure, to a phase-space formulation of quantum mechanics (in this case it is ℏ - the Planck constant - which becomes a finite positive number).
Harmonic Analysis and Representations of Semisimple Lie Groups, 1980
Quantum mechanics is usually formulated in terms of linear operators on a Hilbert space of physic... more Quantum mechanics is usually formulated in terms of linear operators on a Hilbert space of physical states, while classical mechanics relies on the algebra N of functions over phase-space. In one of the most naive presentations, it is said that the passage from classical to quantum mechanics is realized through the correspondence principle of Bohr, according to which one replaces the position variable q by the operator of multiplication by q, momentum p by iℏ∂/∂q where ℏ is the Planck constant, ‘and the Poisson bracket by the commutator’; the classical limit of a quantum theory being obtained by letting ℏ→0. This smells of deformation theory - but how can one ‘deform’ a function into an operator? Due to the fundamental difference in the nature of the observables in classical and quantum theories, it may seem at first sight hopeless to interpret quantum mechanics as a deformation. We shall nevertheless show in this chapter that one can give an autonomous phase-space formulation of quantum mechanics, in the framework of which computations can be made, and for which quantum mechanics will appear naturally as a differentiable deformation of classical mechanics. Quantization will manifest itself in a deformation of the algebra of observables rather than in a radical change of their nature. The link between this formulation of quantum mechanics and the usual one is provided by the Weyl application, that we shall explain in Section 1.
We review the developments in the past twenty years (which are based on our deformation philosoph... more We review the developments in the past twenty years (which are based on our deformation philosophy of physical theories) dealing with elementary particles composed of singletons in anti De Sitter space-time. The study starts with the kinematical aspects (especially for massless particles) and extends to the beginning of a field theory of composite elementary particles and its relations with conformal field theory (including very recent developments).
Collection of Papers on Geometry, Analysis and Mathematical Physics, 1997
... Let G be a group, U a linear representation of G in E, g, g'6 G. We define 1-cocycles as... more ... Let G be a group, U a linear representation of G in E, g, g'6 G. We define 1-cocycles as maps R: G> E such that Rgg «= Rg+ UgRg, and denote their space by Z1 (G, E). 1-coboundaries are 1-cocycles for which there exists a (0-cochain) if 6 E such that Rg= Ugif-if and we ...
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Papers by Daniel Sternheimer