Standard quantization using, for example, path integration of field theory models, includes paths... more Standard quantization using, for example, path integration of field theory models, includes paths of momentum and field reach infinity in the Hamiltonian density, while the Hamiltonian itself remains finite. That fact causes considerable difficulties. In this paper, we represent π(x) by k(x)/ϕ(x). To insure proper values for π(x) it is necessary to restrict 0 < |ϕ(x)| < ∞ as well as 0 ≤ |k(x)| < ∞. Indeed that leads to Hamiltonian densities in which ϕ(x) p , where p can be even integers between 4 and ∞. This leads to a completely satisfactory quantization of field theories using situations that involve scaled behavior leading to an unexpected,h 2 /φ(x) 2 which arises only in the quantum aspects. Indeed, it is fair to claim that this symbol change leads to valid field theory quantizations.
After the success of affine quantization in proving through Monte Carlo analysis that the covaria... more After the success of affine quantization in proving through Monte Carlo analysis that the covariant euclidean scalar field theory, φ r n , where r denotes the power of the interaction term and n ¼ s þ 1 with s the spatial dimension and 1 adds imaginary time, such that r ≥ 2n=ðn − 2Þ can be acceptably quantized and the resulting theory is nontrivial, unlike what happens using canonical quantization, we show here that the same has to be expected for r > 2 and any n even for the ultralocal field theory. In particular we consider the ultralocal φ 4 2 model and study its renormalized properties for both the scaled canonical quantization version and the scaled affine quantization version through path integral Monte Carlo.
After the success of affine quantization in proving through Monte Carlo analysis that the covaria... more After the success of affine quantization in proving through Monte Carlo analysis that the covariant euclidean scalar field theory, ϕ r n , where r denotes the power of the interaction term and n = s + 1 with s the spatial dimension and 1 adds imaginary time, such that r ≥ 2n/(n − 2) can be acceptably quantized and the resulting theory is nontrivial, unlike what happens using canonical quantization, we show here that the same has to be expected for r > 2 and any n even for the ultralocal field theory. In particular we consider the ultralocal ϕ 4 2 model and study its renormalized properties for both the scaled canonical quantization version and the scaled affine quantization version through path integral Monte Carlo.
This paper is aimed to show the essential role played by the theory of quasi-analytic functions i... more This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasianalytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.
The continuous-time regularization scheme for defining phase-space path integrals is briefly revi... more The continuous-time regularization scheme for defining phase-space path integrals is briefly reviewed as a method to define a quantization procedure that is completely covariant under all smooth canonical coordinate transformations. As an illustration of this method, a limited set of transformations is discussed that have an image in the set of the usual squeezed states. It is noteworthy that even this limited set of transformations offers new possibilities for stationary phase approximations to quantum mechanical propagators.
The Berezin-Lieb inequalities provide upper and lower bounds for a partition function based on ph... more The Berezin-Lieb inequalities provide upper and lower bounds for a partition function based on phase space integrals that involve the Glauber-Sudarshan and Husimi representations, respectively. Generalizations of these representations have recently been introduced by the present authors, and in this article, we extend the use of these new representations to develop numerous analogs of the Berezin-Lieb inequalities that may offer improved bounds. Several examples illustrate the use of the new inequalities. Although motivated by problems in quantum mechanics, these results may also find applications in time-frequency analysis, a valuable cross fertilization that has been profitably used at various times in the past.
From the user's point of view, what are the desirable characteristics of pushbuttons for use in 5... more From the user's point of view, what are the desirable characteristics of pushbuttons for use in 500-type telephone sets? The studies reported bear on this question and also on questions of how people process information when keying telephone numbers. Four categories of design features were studied: key arrangement, force-displacement characteristics, button-top design and central office factors. The results indicate that considerable latitude exists for key set design in terms of user performance; however, the preference judgments are more selective. The studies also showed that the manner in which the person acquired and keyed the telephone number influenced performance appreciably. Technological progress in recent years has brought pushbutton signaling from the telephone set within sight of economic feasibility. What, from the user's viewpoint, are the desirable operating characteristics of the key set which should guide development and manufacture? And how do people process information when they key a telephone number? 1. HUMAN FACTORS PROBLEM
Quantum‐mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II. [Journ... more Quantum‐mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II. [Journal of Mathematical Physics 26, 2239 (1985)]. Ingrid Daubechies, John R. Klauder. Abstract. The coherent‐state representation ...
Restricting the support of relativistic quantum fields to lightlike hyperplanes (e.g. x ~ + x 3 =... more Restricting the support of relativistic quantum fields to lightlike hyperplanes (e.g. x ~ + x 3 = const) we find examples of such fields to exist as well-defined self-adjoint operators with properties however that differ vastly from those of fields on the usual space]ike surfaces. We show that on a lightlike hyperplane: 1) the free-field algebra is irreducible (instead of Abelian, and in contrast to what one would expect of data on a characteristic surface) and 2) fields with different masses become unitarily equiw~lent (whereas they are iuequivalent on spacelike planes). Furthermore %he field algebra restricted to the space-time slab between two parallel lightlike planes is always irreducible {while there are counterexamples for spacelikc slabs). We establish this directly for generalized free fields and rederive it for Wightman fields in general.
... Fundamentals of quantum optics. Post a Comment. CONTRIBUTORS: Author: Klauder, John R. Author... more ... Fundamentals of quantum optics. Post a Comment. CONTRIBUTORS: Author: Klauder, John R. Author: Sudarshan, EC. ... VOLUME/EDITION: PAGES (INTRO/BODY): xi, 279 p. SUBJECT(S): Quantum optics. DISCIPLINE: No discipline assigned. LC NUMBER: QC476.2 .K55. HTTP ...
We investigate modifications of the discrete-time lattice action, for a quantum mechanical partic... more We investigate modifications of the discrete-time lattice action, for a quantum mechanical particle in one spatial dimension, that vanish in the naïve continuum limit but which, nevertheless, induce nontrivial effects due to quantum fluctuations. These effects are seen to modify the geometry of the paths contributing to the path-integral describing the time evolution of the particle, which we investigate through numerical simulations. In particular, we demonstrate the existence of a modified lattice action resulting in paths with any fractal dimension, d f , between one and two. We argue that d f = 2 is a critical value, and we exhibit a type of lattice modification where the fluctuations in the position of the particle becomes independent of the time step, in which case the paths are interpreted as superdiffusive Lévy flights. We also consider the jaggedness of the paths, and show that this gives an independent classification of lattice theories.
Eigenvectors of two particles&#x27; relative position and total momentum. Fan Hong-yi China C... more Eigenvectors of two particles&#x27; relative position and total momentum. Fan Hong-yi China Center of Advanced Science and Technology (World Laboratory), PO Box 8730, Beijing, 100080, People&#x27;s Republic of China Department ...
... 4, it was recast in the following form: 7-_ r d2Z Z ) (ZI =IT1 f d2z:exp(- z z12+zat+z *a-ata... more ... 4, it was recast in the following form: 7-_ r d2Z Z ) (ZI =IT1 f d2z:exp(- z z12+zat+z *a-ata)l where an integration over the complex z plane is implied. This is yet another example of the newformalism developed in Ref. 4. 7Fan Hong-Yi and Guo Guan-Can, Chinese Phys. ...
We show that the coherent state matrix elements of the quantum mechanical propagator for all quad... more We show that the coherent state matrix elements of the quantum mechanical propagator for all quadratic Hamiltonians may be represented as the limit of path integrals with respect to appropriately modified Wiener measures as the associated diffusion constant tends to infinity.
Journal of Physics A: Mathematical and Theoretical, 2007
Introducing asymmetry into the Weyl representation of operators leads to a variety of phase-space... more Introducing asymmetry into the Weyl representation of operators leads to a variety of phase-space representations and new symbols. Specific generalizations of the Husimi and the Glauber-Sudarshan symbols are explicitly derived.
Standard quantization using, for example, path integration of field theory models, includes paths... more Standard quantization using, for example, path integration of field theory models, includes paths of momentum and field reach infinity in the Hamiltonian density, while the Hamiltonian itself remains finite. That fact causes considerable difficulties. In this paper, we represent π(x) by k(x)/ϕ(x). To insure proper values for π(x) it is necessary to restrict 0 < |ϕ(x)| < ∞ as well as 0 ≤ |k(x)| < ∞. Indeed that leads to Hamiltonian densities in which ϕ(x) p , where p can be even integers between 4 and ∞. This leads to a completely satisfactory quantization of field theories using situations that involve scaled behavior leading to an unexpected,h 2 /φ(x) 2 which arises only in the quantum aspects. Indeed, it is fair to claim that this symbol change leads to valid field theory quantizations.
After the success of affine quantization in proving through Monte Carlo analysis that the covaria... more After the success of affine quantization in proving through Monte Carlo analysis that the covariant euclidean scalar field theory, φ r n , where r denotes the power of the interaction term and n ¼ s þ 1 with s the spatial dimension and 1 adds imaginary time, such that r ≥ 2n=ðn − 2Þ can be acceptably quantized and the resulting theory is nontrivial, unlike what happens using canonical quantization, we show here that the same has to be expected for r > 2 and any n even for the ultralocal field theory. In particular we consider the ultralocal φ 4 2 model and study its renormalized properties for both the scaled canonical quantization version and the scaled affine quantization version through path integral Monte Carlo.
After the success of affine quantization in proving through Monte Carlo analysis that the covaria... more After the success of affine quantization in proving through Monte Carlo analysis that the covariant euclidean scalar field theory, ϕ r n , where r denotes the power of the interaction term and n = s + 1 with s the spatial dimension and 1 adds imaginary time, such that r ≥ 2n/(n − 2) can be acceptably quantized and the resulting theory is nontrivial, unlike what happens using canonical quantization, we show here that the same has to be expected for r > 2 and any n even for the ultralocal field theory. In particular we consider the ultralocal ϕ 4 2 model and study its renormalized properties for both the scaled canonical quantization version and the scaled affine quantization version through path integral Monte Carlo.
This paper is aimed to show the essential role played by the theory of quasi-analytic functions i... more This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasianalytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.
The continuous-time regularization scheme for defining phase-space path integrals is briefly revi... more The continuous-time regularization scheme for defining phase-space path integrals is briefly reviewed as a method to define a quantization procedure that is completely covariant under all smooth canonical coordinate transformations. As an illustration of this method, a limited set of transformations is discussed that have an image in the set of the usual squeezed states. It is noteworthy that even this limited set of transformations offers new possibilities for stationary phase approximations to quantum mechanical propagators.
The Berezin-Lieb inequalities provide upper and lower bounds for a partition function based on ph... more The Berezin-Lieb inequalities provide upper and lower bounds for a partition function based on phase space integrals that involve the Glauber-Sudarshan and Husimi representations, respectively. Generalizations of these representations have recently been introduced by the present authors, and in this article, we extend the use of these new representations to develop numerous analogs of the Berezin-Lieb inequalities that may offer improved bounds. Several examples illustrate the use of the new inequalities. Although motivated by problems in quantum mechanics, these results may also find applications in time-frequency analysis, a valuable cross fertilization that has been profitably used at various times in the past.
From the user's point of view, what are the desirable characteristics of pushbuttons for use in 5... more From the user's point of view, what are the desirable characteristics of pushbuttons for use in 500-type telephone sets? The studies reported bear on this question and also on questions of how people process information when keying telephone numbers. Four categories of design features were studied: key arrangement, force-displacement characteristics, button-top design and central office factors. The results indicate that considerable latitude exists for key set design in terms of user performance; however, the preference judgments are more selective. The studies also showed that the manner in which the person acquired and keyed the telephone number influenced performance appreciably. Technological progress in recent years has brought pushbutton signaling from the telephone set within sight of economic feasibility. What, from the user's viewpoint, are the desirable operating characteristics of the key set which should guide development and manufacture? And how do people process information when they key a telephone number? 1. HUMAN FACTORS PROBLEM
Quantum‐mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II. [Journ... more Quantum‐mechanical path integrals with Wiener measure for all polynomial Hamiltonians. II. [Journal of Mathematical Physics 26, 2239 (1985)]. Ingrid Daubechies, John R. Klauder. Abstract. The coherent‐state representation ...
Restricting the support of relativistic quantum fields to lightlike hyperplanes (e.g. x ~ + x 3 =... more Restricting the support of relativistic quantum fields to lightlike hyperplanes (e.g. x ~ + x 3 = const) we find examples of such fields to exist as well-defined self-adjoint operators with properties however that differ vastly from those of fields on the usual space]ike surfaces. We show that on a lightlike hyperplane: 1) the free-field algebra is irreducible (instead of Abelian, and in contrast to what one would expect of data on a characteristic surface) and 2) fields with different masses become unitarily equiw~lent (whereas they are iuequivalent on spacelike planes). Furthermore %he field algebra restricted to the space-time slab between two parallel lightlike planes is always irreducible {while there are counterexamples for spacelikc slabs). We establish this directly for generalized free fields and rederive it for Wightman fields in general.
... Fundamentals of quantum optics. Post a Comment. CONTRIBUTORS: Author: Klauder, John R. Author... more ... Fundamentals of quantum optics. Post a Comment. CONTRIBUTORS: Author: Klauder, John R. Author: Sudarshan, EC. ... VOLUME/EDITION: PAGES (INTRO/BODY): xi, 279 p. SUBJECT(S): Quantum optics. DISCIPLINE: No discipline assigned. LC NUMBER: QC476.2 .K55. HTTP ...
We investigate modifications of the discrete-time lattice action, for a quantum mechanical partic... more We investigate modifications of the discrete-time lattice action, for a quantum mechanical particle in one spatial dimension, that vanish in the naïve continuum limit but which, nevertheless, induce nontrivial effects due to quantum fluctuations. These effects are seen to modify the geometry of the paths contributing to the path-integral describing the time evolution of the particle, which we investigate through numerical simulations. In particular, we demonstrate the existence of a modified lattice action resulting in paths with any fractal dimension, d f , between one and two. We argue that d f = 2 is a critical value, and we exhibit a type of lattice modification where the fluctuations in the position of the particle becomes independent of the time step, in which case the paths are interpreted as superdiffusive Lévy flights. We also consider the jaggedness of the paths, and show that this gives an independent classification of lattice theories.
Eigenvectors of two particles&#x27; relative position and total momentum. Fan Hong-yi China C... more Eigenvectors of two particles&#x27; relative position and total momentum. Fan Hong-yi China Center of Advanced Science and Technology (World Laboratory), PO Box 8730, Beijing, 100080, People&#x27;s Republic of China Department ...
... 4, it was recast in the following form: 7-_ r d2Z Z ) (ZI =IT1 f d2z:exp(- z z12+zat+z *a-ata... more ... 4, it was recast in the following form: 7-_ r d2Z Z ) (ZI =IT1 f d2z:exp(- z z12+zat+z *a-ata)l where an integration over the complex z plane is implied. This is yet another example of the newformalism developed in Ref. 4. 7Fan Hong-Yi and Guo Guan-Can, Chinese Phys. ...
We show that the coherent state matrix elements of the quantum mechanical propagator for all quad... more We show that the coherent state matrix elements of the quantum mechanical propagator for all quadratic Hamiltonians may be represented as the limit of path integrals with respect to appropriately modified Wiener measures as the associated diffusion constant tends to infinity.
Journal of Physics A: Mathematical and Theoretical, 2007
Introducing asymmetry into the Weyl representation of operators leads to a variety of phase-space... more Introducing asymmetry into the Weyl representation of operators leads to a variety of phase-space representations and new symbols. Specific generalizations of the Husimi and the Glauber-Sudarshan symbols are explicitly derived.
Canonical quantization has served well, but there are many problems it is designed not to handle.... more Canonical quantization has served well, but there are many problems it is designed not to handle. A new procedure, called affine quantization, opens the door to formulate and correctly solve many new problems. This paper outlines the new procedures and points to several problems for which affine quantization can provide valid results This paper is designed to introduce what this new procedure is and what it can do for difficult problems, like quantum fields and quantum gravity.
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Papers by John Klauder