Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physica Scripta

Dirac distribution and Dirac constraint quantizations

Mario Cesar Baldiotti, R. Fresneda, Jean‐Pierre Gazeau

Published Jun 1, 2015• 4 citations
Publisher

Abstract

We explore two ways of quantizing constraints through integral quantizations based on the Weyl–Heisenberg group. The first one is based on the quantization of distributions on phase space expressing geometric constraints, like Dirac or Heaviside distributions. The second one is implemented in the spirit of Dirac. In both cases we complete the study with a semiclassical analysis through the use of lower symbols. For the sake of simplicity, we restrict our study to the complex plane viewed as the phase space of the motion on the line, or as the phase space for classical electromagnetic radiation (quadratures). Despite the fact that the two procedures do not in general yield identical results, we show interesting features in exploring aspects of their quantum and semiclassical versions.

DOI
10.1088/0031-8949/90/7/074039

Related papers

Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

Hervé Bergeron, Jean‐Pierre Gazeau

Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

Entropy•Oct 13, 2018•12 citations

Dirac versus reduced phase space quantization

Mikhail S. Plyushchay, A. V. Razumov

The relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of Hamiltonian systems having no gauge conditions. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, it is shown that there is a class of constrained systems, which can be quantized only by the Dirac method. A possible interpretation of the gauge degrees of freedom is given. Introduction There are two main methods to quantize the Hamiltonian systems with first class constraints: the Dirac quantization [1] and the reduced phase space quantization [2], whereas two other methods, the path integral method [3, 2] and the BRST quantization [4] being the most popular method for the covariant quantization of gauge-invariant systems, are based on and proceed from them [2, 5]. The basic idea of the Dirac method consists in imposing quantum mechanically the first class constraints as operator conditions on the states for singling out the physical ones [1]. The reduced phase space quantization first identifies the physical degrees of freedom at the classical level by the factorization of the constraint surface with respect to the action of the gauge group, generated by the constraints. Then the resulting Hamiltonian system is quantized as a usual unconstrained system [2]. Naturally, the problem of the relationship of these two methods arises. It was discussed in different contexts in literature [6], and there is an opinion that the differences between the two quantization methods can be traced out to a choice of factor ordering in the construction of various physical operators.

arXiv:hep-th/9412137•Cambridge University Press eBooks•Jan 5, 1995•11 citations

Universality of the Weyl–Heisenberg symmetry and its covariant quantizations

Jean‐Pierre Gazeau, Célestin Habonimana, Romain Murenzi +1 more

The Weyl–Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalizations. In this paper, we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time–frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.

International Journal of Geometric Methods in Modern Physics•Feb 11, 2025•0 citations

Weyl-Heisenberg integral quantization(s): a compendium

Hervé Bergeron, Evaldo M. F. Curado, Jean‐Pierre Gazeau +1 more

We present a list of formulae useful for Weyl-Heisenberg integral quantizations, with arbitrary weight, of functions or distributions on the plane. Most of these formulae are known, others are original. The list encompasses particular cases like Weyl-Wigner quantization (constant weight) and coherent states (CS) or Berezin quantization (Gaussian weight). The formulae are given with implicit assumptions on their validity on appropriate space(s) of functions (or distributions). One of the aims of the document is to accompany a work in progress on Weyl-Heisenberg integral quantization of dynamics for the motion of a point particle on the line.

arXiv:1703.08443•Mar 24, 2017•10 citations

Integral quantizations with POVM and some applications

R. Fresneda, Jean‐Pierre Gazeau

We present quantizations of functions (or distributions) on a measure space viewed as a classical set. They are based on positive operator-valued measures. To illustrate the large range of potentialities of the method, we develop examples where the classical sets are the group SU(2), the (projective) Weyl-Heisenberg group, and the affine group. Applications to quantum cosmology and to the quantization of constraints are outlined.

Journal of Physics Conference Series•Apr 13, 2015•3 citations

Path Integrals, and Classical and Quantum Constraints

John R. Klauder

Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.

arXiv:quant-ph/0507222•Jul 22, 2005•2 citations
We explore two ways of quantizing constraints through integral quantizations based on the Weyl–Heisenberg group. The first one is based on the quantization of distributions on phase space expressing geometric constraints, like Dirac or Heaviside distributions. The second one is implemented in the spirit of Dirac. In both cases we complete the study with a semiclassical analysis through the use of lower symbols. For the sake of simplicity, we restrict our study to the complex plane viewed as the phase space of the motion on the line, or as the phase space for classical electromagnetic radiation (quadratures). Despite the fact that the two procedures do not in general yield identical results, we show interesting features in exploring aspects of their quantum and semiclassical versions.
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.
The relationship between the Dirac and reduced phase space quantizations is investigated for spin models belonging to the class of Hamiltonian systems having no gauge conditions. It is traced out that the two quantization methods may give similar, or essentially different physical results, and, moreover, it is shown that there is a class of constrained systems, which can be quantized only by the Dirac method. A possible interpretation of the gauge degrees of freedom is given. Introduction There are two main methods to quantize the Hamiltonian systems with first class constraints: the Dirac quantization [1] and the reduced phase space quantization [2], whereas two other methods, the path integral method [3, 2] and the BRST quantization [4] being the most popular method for the covariant quantization of gauge-invariant systems, are based on and proceed from them [2, 5]. The basic idea of the Dirac method consists in imposing quantum mechanically the first class constraints as operator conditions on the states for singling out the physical ones [1]. The reduced phase space quantization first identifies the physical degrees of freedom at the classical level by the factorization of the constraint surface with respect to the action of the gauge group, generated by the constraints. Then the resulting Hamiltonian system is quantized as a usual unconstrained system [2]. Naturally, the problem of the relationship of these two methods arises. It was discussed in different contexts in literature [6], and there is an opinion that the differences between the two quantization methods can be traced out to a choice of factor ordering in the construction of various physical operators.
The Weyl–Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalizations. In this paper, we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time–frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.
We present a list of formulae useful for Weyl-Heisenberg integral quantizations, with arbitrary weight, of functions or distributions on the plane. Most of these formulae are known, others are original. The list encompasses particular cases like Weyl-Wigner quantization (constant weight) and coherent states (CS) or Berezin quantization (Gaussian weight). The formulae are given with implicit assumptions on their validity on appropriate space(s) of functions (or distributions). One of the aims of the document is to accompany a work in progress on Weyl-Heisenberg integral quantization of dynamics for the motion of a point particle on the line.
Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.