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

Journal of Mathematical Physics

On some representations of the Poincaré group on phase space. II

S. Twareque Ali

Published Apr 1, 1980• 18 citations
Publisher

Abstract

Representations of the proper Poincaré group on spaces of functions of the phase space variables q and p have been studied, by starting with the rotation subgroup SO(3), and carrying out an inducing procedure. A complete decomposition theory, for the physically interesting representations, have been developed. Explicit computations of the forms of the generators and certain systems of imprimitivity have been carried out. The role of the Newton–Wigner position and momentum operators, in the setting of phase space representations, has been discussed. The work is a continuation of an earlier paper on phase space representations, inspired ultimately by the possibility of developing a consistent relativistic quantum mechanics for single particles on phase space.

DOI
10.1063/1.524461

Related papers

Is a particle an irreducible representation of the Poincaré group?

Adam Caulton

The claim that a particle is an irreducible representation of the Poincaré group -- what I call \emph{Wigner's identification} -- is now, decades on from Wigner's (1939) original paper, so much a part of particle physics folklore that it is often taken as, or claimed to be, a definition. My aims in this paper are to: (i) clarify, and partially defend, the guiding ideas behind this identification; (ii) raise objections to its being an adequate definition; and (iii) offer a rival characterisation of particles. My main objections to Wigner's identification appeal to the problem of interacting particles, and to alternative spacetimes. I argue that the link implied in Wigner's identification, between a spacetime's symmetries and the generator of a particle's space of states, is at best misleading, and that there is no good reason to link the generator of a particle's space of states to symmetries of any kind. I propose an alternative characterisation of particles, which captures both the relativistic and non-relativistic setting. I further defend this proposal by appeal to a theorem which links the decomposition of Poincaré generators into purely orbital and spin components with canonical algebraic relations between position, momentum and spin.

arXiv:2410.02354•Oct 3, 2024•0 citations

Covariant Representation of Spin and Entanglement—A Review and Reformulation

Martin Land

A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector nμ and then inducing a representation of SL(2,C), based on pμ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space {(xμ,pμ),(ζμ,πμ)}, inducing a representation on the momentum πμ, thus providing a novel dynamical interpretation of the timelike unit vector nμ=πμ/M. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments.

Symmetry•Nov 4, 2024•2 citations

The Wigner-Newton state localization and the commutativity of the configuration space

R. M. Mir‐Kasimov

The commutativity requirement of the components of the position operator is implicitly included in the set of the Wigner-Newton postulates defining the localized state of the relativistic particle. By omitting this constraint, we obtain a more general concept of localization based on the representation of the Poincare group.

Physics of Particles and Nuclei Letters•Oct 1, 2006•5 citations

De Sitter to Poincaré contraction and relativistic coherent states

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

We continue the analysis of relativistic phase space, identified with the quotient of the Poincare group in 1+1 dimensions by the time translation subgroup. Proceeding by contraction from the corresponding (Anti) de Sitter group SO 0 (2, 1), we obtain a realisation of the Poincare representation P(m) of mass m in a space of functions defined on phase space. The contraction singles out a privileged section in the Poincare group, with a unique left and right invariant measure. Using that section, we show that the representation P(m) is square integrable over the coset space, i.e. phase space. From this we build a new set of Poincare coherent states, and more generally weighted coherent states, which have all the usual properties, resolution of the identity, overcompleteness, reproducing kernel, orthogonality relations. Finally we derive the corresponding Wigner transform On poursuit l'analyse de l'espace de phase relativiste, identifie au quotient du groupe de Poincare en dimension 1+1 par le sous-groupe des translations temporelles. Par contraction a partir du groupe Anti-de Sitter correspondant SO 0 (2, 1), on obtient une realisation de la representation P(m) de masse m du groupe de Poincare dans un espace de fonctions definies sur l'espace de phase. La contraction selectionne une section privilegiee dans le groupe de Poincare, possedant une unique mesure invariante a la fois a gauche et a droite. Avec ce choix de section, on montre que la representation P(m) est de carre sommable sur l'espace quotient, c'est-a-dire l'espace de phase. A partir de la, on construit une nouvelle famille d'etats coherents de Poincare et, plus generalement, d'etats coherents avec poids, qui ont toutes les proprietes usuelles: resolution de l'identite, surcompletude, noyau reproduisant, relations d'orthogonalite. Enfin, on derive la transformation de Wigner correspondante

Annales De L Institut Henri Poincare-physique Theorique•Jan 1, 1990•7 citations

Relativistic quantum kinematics in the Moyal representation

José F. Cariñena, JoséM. Garcia-Bondia, Joseph C. Várilly

The authors obtain the phase-space quantisation for relativistic spinning particles. The main tool is what they call a 'Stratonovich-Weyl quantiser' which relates functions on phase space to operators on a suitable Hilbert space, and has the essential properties of covariance (under a group representation) and traciality. Their phase spaces are coadjoint orbits of the restricted Poincare group; they compute and explicitly coordinatise the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincare group; they show that there is a unique Stratonovich-Weyl quantiser from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly.

Journal of Physics A Mathematical and General•Mar 21, 1990•79 citations

Poincaré Group and Operators of Position and Spin

Alexander J. Silenko, Pengfei Zhang, Liping Zou

Abstract Fundamental properties of the position and spin operators in relativistic quantum mechanics are defined with the Poincaré group. Quantum-mechanical counterparts of the classical position and spin variables are the corresponding operators in the Foldy–Wouthuysen representation but not in the Dirac one. The probabilistic interpretation is valid only for Foldy–Wouthuysen wave functions. The relativistic operators of the position and spin are discussed.

Physics of Particles and Nuclei•Nov 29, 2023•0 citations
Representations of the proper Poincaré group on spaces of functions of the phase space variables q and p have been studied, by starting with the rotation subgroup SO(3), and carrying out an inducing procedure. A complete decomposition theory, for the physically interesting representations, have been developed. Explicit computations of the forms of the generators and certain systems of imprimitivity have been carried out. The role of the Newton–Wigner position and momentum operators, in the setting of phase space representations, has been discussed. The work is a continuation of an earlier paper on phase space representations, inspired ultimately by the possibility of developing a consistent relativistic quantum mechanics for single particles on phase space.
The claim that a particle is an irreducible representation of the Poincaré group -- what I call \emph{Wigner's identification} -- is now, decades on from Wigner's (1939) original paper, so much a part of particle physics folklore that it is often taken as, or claimed to be, a definition. My aims in this paper are to: (i) clarify, and partially defend, the guiding ideas behind this identification; (ii) raise objections to its being an adequate definition; and (iii) offer a rival characterisation of particles. My main objections to Wigner's identification appeal to the problem of interacting particles, and to alternative spacetimes. I argue that the link implied in Wigner's identification, between a spacetime's symmetries and the generator of a particle's space of states, is at best misleading, and that there is no good reason to link the generator of a particle's space of states to symmetries of any kind. I propose an alternative characterisation of particles, which captures both the relativistic and non-relativistic setting. I further defend this proposal by appeal to a theorem which links the decomposition of Poincaré generators into purely orbital and spin components with canonical algebraic relations between position, momentum and spin.
A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector nμ and then inducing a representation of SL(2,C), based on pμ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space {(xμ,pμ),(ζμ,πμ)}, inducing a representation on the momentum πμ, thus providing a novel dynamical interpretation of the timelike unit vector nμ=πμ/M. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments.
We continue the analysis of relativistic phase space, identified with the quotient of the Poincare group in 1+1 dimensions by the time translation subgroup. Proceeding by contraction from the corresponding (Anti) de Sitter group SO 0 (2, 1), we obtain a realisation of the Poincare representation P(m) of mass m in a space of functions defined on phase space. The contraction singles out a privileged section in the Poincare group, with a unique left and right invariant measure. Using that section, we show that the representation P(m) is square integrable over the coset space, i.e. phase space. From this we build a new set of Poincare coherent states, and more generally weighted coherent states, which have all the usual properties, resolution of the identity, overcompleteness, reproducing kernel, orthogonality relations. Finally we derive the corresponding Wigner transform On poursuit l'analyse de l'espace de phase relativiste, identifie au quotient du groupe de Poincare en dimension 1+1 par le sous-groupe des translations temporelles. Par contraction a partir du groupe Anti-de Sitter correspondant SO 0 (2, 1), on obtient une realisation de la representation P(m) de masse m du groupe de Poincare dans un espace de fonctions definies sur l'espace de phase. La contraction selectionne une section privilegiee dans le groupe de Poincare, possedant une unique mesure invariante a la fois a gauche et a droite. Avec ce choix de section, on montre que la representation P(m) est de carre sommable sur l'espace quotient, c'est-a-dire l'espace de phase. A partir de la, on construit une nouvelle famille d'etats coherents de Poincare et, plus generalement, d'etats coherents avec poids, qui ont toutes les proprietes usuelles: resolution de l'identite, surcompletude, noyau reproduisant, relations d'orthogonalite. Enfin, on derive la transformation de Wigner correspondante
The authors obtain the phase-space quantisation for relativistic spinning particles. The main tool is what they call a 'Stratonovich-Weyl quantiser' which relates functions on phase space to operators on a suitable Hilbert space, and has the essential properties of covariance (under a group representation) and traciality. Their phase spaces are coadjoint orbits of the restricted Poincare group; they compute and explicitly coordinatise the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincare group; they show that there is a unique Stratonovich-Weyl quantiser from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly.