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

arXiv:2107.03623 · Journal of Physics A Mathematical and Theoretical

Projective representation of the Galilei group for classical and quantum–classical systems*

A D Bermúdez Manjarres

Published Sep 21, 2021• 9 citations
arXivHTMLPDF

Abstract

Abstract A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in the Koopman–von Neumann formulation of classical mechanics. This classical representation is characterized by the vanishing of the central charge of the Galilei algebra. This is in contrast to the quantum case where the mass plays the role of the central charge. Here we show, by direct construction, that classical mechanics also allows for a projective representation of the Galilei group where the mass is the central charge of the algebra. We extend the result to certain kind of quantum–classical hybrid systems.

DOI
10.1088/1751-8121/ac28cc

Related papers

Classical and mixed classical-quantum systems from van Hove’s unitary representation of contact transformations

Marcel Reginatto, A. D. Bermúdez Manjarres, Sebastian Ulbricht

Abstract Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used to develop a novel formulation of classical mechanics in Hilbert space. This formulation differs from the Koopman-von Neumann theory in many ways. Classical observables are represented by van Hove operators, which satisfy a commutation algebra isomorphic to the Poisson algebra of functions in phase space. Moreover, these operators are both observables and generators of transformations, which makes it unnecessary to introduce unobservable auxiliary operators as in the Koopman-von Neumann theory. In addition, for consistency with classical mechanics, a constraint must be imposed that fixes the phase of the wavefunction. The approach can be extended to hybrid mixed classical-quantum systems in Hilbert space. The formalism is applied to the measurement of a quantum two-level system (qubit) by a classical apparatus.

arXiv:2503.02480•Journal of Physics Conference Series•Jun 1, 2025•0 citations

Galilei Group and Nonrelativistic Quantum Mechanics

Jean-Marc Lévy-Leblond

This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.

Journal of Mathematical Physics•Jun 1, 1963•241 citations

From a ’’laboratory’’ Galilei−Hilbert bundle to an algebra of observables

Maria Cristina Abbati, Renzo Cirelli, Franco Gallone

After a schematic examination of the ’’physical’’ representations of the Galilei group, a quantum mechanical description is drawn from a ’’laboratory’’ one in the case of a free elementary system with spin, assuming that the kinematical group is the Galilei group. In fact, the ’’laboratory states’’ of the system naturally fit in a Galilei−Hilbert bundle. Then, according to the general theory of the unitary representations of groups in the framework of Hilbert bundles which is outlined in this paper, a unitary representation of the Galilei group is constructed which is shown to contain all the physical Galilei representations. The quantum mechanical description which has been obtained in this way gives rise to an algebra of observables by means of a general procedure which connects a unitary representation of a Lie group with a Hilbert representation of the corresponding Lie algebra. The inner energy is shown to be a superobservable for the system under consideration. Moreover, the kinematical properties of such a system are independent of the values of the inner energy.

Journal of Mathematical Physics•Apr 1, 1975•1 citation

Central extensions and realizations of one-dimensional Galilean systems and quantization

Miguel A. Martín, M. A. del Olmo

The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei group, (1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, (1 + 1). As an application we construct the Stratonovich - Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.

Journal of Physics A Mathematical and General•Feb 7, 1996•6 citations

Phase space ensembles for classical and quantum-classical systems

A. D. Bermúdez Manjarres

We develop a so-called theory of ensembles in phase space and use it to investigate the construction of a quantum-classical hybrid theory. We use Galilei covariance and the Lie algebra of the Galilei group as a guide to constructing the hybrid model presented here. In particular, we chose the interaction term between the classical and the quantum sector so that the equations are Galilei covariant. Our approach points out a possible connection between two previously unrelated hybrid systems.

arXiv:2305.01880•May 3, 2023•0 citations

Galilean covariance of quantum-classical hybrid systems of the Sudarshan type

A. D. Bermúdez Manjarres, Nathaly Marín–Medina

We revisit quantum-classical hybrid systems of the Sudarshan type under the light of Galilean covariance. We show that these kinds of hybrids cannot be given as a unitary representation of the Galilei group and at the same time conserve the total linear momentum unless the interaction term only depends on the relative canonical velocities.

arXiv:2012.02324•Physical Review A•Oct 22, 2020•5 citations
Abstract A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in the Koopman–von Neumann formulation of classical mechanics. This classical representation is characterized by the vanishing of the central charge of the Galilei algebra. This is in contrast to the quantum case where the mass plays the role of the central charge. Here we show, by direct construction, that classical mechanics also allows for a projective representation of the Galilei group where the mass is the central charge of the algebra. We extend the result to certain kind of quantum–classical hybrid systems.
Abstract Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used to develop a novel formulation of classical mechanics in Hilbert space. This formulation differs from the Koopman-von Neumann theory in many ways. Classical observables are represented by van Hove operators, which satisfy a commutation algebra isomorphic to the Poisson algebra of functions in phase space. Moreover, these operators are both observables and generators of transformations, which makes it unnecessary to introduce unobservable auxiliary operators as in the Koopman-von Neumann theory. In addition, for consistency with classical mechanics, a constraint must be imposed that fixes the phase of the wavefunction. The approach can be extended to hybrid mixed classical-quantum systems in Hilbert space. The formalism is applied to the measurement of a quantum two-level system (qubit) by a classical apparatus.
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.
After a schematic examination of the ’’physical’’ representations of the Galilei group, a quantum mechanical description is drawn from a ’’laboratory’’ one in the case of a free elementary system with spin, assuming that the kinematical group is the Galilei group. In fact, the ’’laboratory states’’ of the system naturally fit in a Galilei−Hilbert bundle. Then, according to the general theory of the unitary representations of groups in the framework of Hilbert bundles which is outlined in this paper, a unitary representation of the Galilei group is constructed which is shown to contain all the physical Galilei representations. The quantum mechanical description which has been obtained in this way gives rise to an algebra of observables by means of a general procedure which connects a unitary representation of a Lie group with a Hilbert representation of the corresponding Lie algebra. The inner energy is shown to be a superobservable for the system under consideration. Moreover, the kinematical properties of such a system are independent of the values of the inner energy.
The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei group, (1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, (1 + 1). As an application we construct the Stratonovich - Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.