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

Group-theoretical foundations of classical and quantum mechanics. II. Elementary systems

Luis Martı́nez Alonso

Published Feb 1, 1979• 10 citations
Publisher

Abstract

Classical and quantum elementary systems are investigated from the point of view of invariance under a connected Lie group. The classification and characterization of elementary systems are considered in a unified way. The representation theory of symmetric and enveloping algebras is used as a tool in order to characterize the observables physically and also to analyze the analogies between classical and quantum mechanics. The results obtained are applied to the Galilei, Poincaré, and Weyl Lie groups.

DOI
10.1063/1.524068

Related papers

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

Quasiclassical and Quantum Systems of Angular Momentum. Part II. Quantum Mechanics on Lie Groups and Methods of Group Algebras

Jan Jerzy Sławianowski, Vasyl Kovalchuk, Agnieszka Martens +2 more

In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Basing on differential geometry enables one to introduce explicitly representation of important physical quantities and formulate the general ideas of quasiclassical representation and classical analogy.

arXiv:1008.0512•Aug 3, 2010•12 citations

GROUP-THEORETICAL FORMALISM OF QUANTUM-MECHANICS AND CLASSICAL-QUANTUM CORRESPONDENCE

Yingfei Gu

This article presents a mathematical framework for group-theoretical formalism of quantum mechanics and discusses the geometric quantization of the classical systems associated with a Lie group. The classical-quantum correspondence is realized by identifying quantum observable algebra and its classical analogue with the set of distributions with compact supports on a Lie group and on the associated Lie algebra respectively, both having a convolution-type associative algebraic structure. The general mathematical constructs are illustrated by studying systems associated with the Heisenberg-Weyl group. It is shown that the exponential mapping from Heisenberg-Weyl algebra to the corresponding Lie group gives Weyl quantization.

•Jan 1, 1992•2 citations

State spaces for classical and quantal, relativistic and nonrelativistic elementary particles

N. Giovannini

We apply a recently developed mathematical formalism to the kinematical classification of elementary physical systems, more precisely to the derivation of the possible state spaces for single (massive) particles and to their corresponding group theoretical interpretation. We show in particular that in all considered cases, relativistic or not, we find in a unified way two solutions, a quantal one and a classical one.

Journal of Mathematical Physics•Nov 1, 1981•2 citations

Quantum Mechanics via Lie Algebras

Arnold Neumaier, Dennis B Westra

This monograph introduces mathematicians, physicists, and engineers to the ideas relating quantum mechanics and symmetries - both described in terms of Lie algebras and Lie groups. The exposition of quantum mechanics from this point of view reveals that classical mechanics and quantum mechanics are very much alike. Written by a mathematician and a physicist, this book is (like a math book) about precise concepts and exact results in classical mechanics and quantum mechanics, but motivated and discussed (like a physics book) in terms of their physical meaning. The reader can focus on the simplicity and beauty of theoretical physics, without getting lost in a jungle of techniques for estimating or calculating quantities of interest.

De Gruyter eBooks•Sep 10, 2024•1 citation

Invariance Groups in Classical and Quantum Mechanics

D. J. Simms

Abstract This is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

Zeitschrift für Naturforschung A•Apr 1, 1973•4 citations
Classical and quantum elementary systems are investigated from the point of view of invariance under a connected Lie group. The classification and characterization of elementary systems are considered in a unified way. The representation theory of symmetric and enveloping algebras is used as a tool in order to characterize the observables physically and also to analyze the analogies between classical and quantum mechanics. The results obtained are applied to the Galilei, Poincaré, and Weyl Lie groups.
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.
In Part I of this series we presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment was there very "ascetic" in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Basing on differential geometry enables one to introduce explicitly representation of important physical quantities and formulate the general ideas of quasiclassical representation and classical analogy.
This article presents a mathematical framework for group-theoretical formalism of quantum mechanics and discusses the geometric quantization of the classical systems associated with a Lie group. The classical-quantum correspondence is realized by identifying quantum observable algebra and its classical analogue with the set of distributions with compact supports on a Lie group and on the associated Lie algebra respectively, both having a convolution-type associative algebraic structure. The general mathematical constructs are illustrated by studying systems associated with the Heisenberg-Weyl group. It is shown that the exponential mapping from Heisenberg-Weyl algebra to the corresponding Lie group gives Weyl quantization.
This monograph introduces mathematicians, physicists, and engineers to the ideas relating quantum mechanics and symmetries - both described in terms of Lie algebras and Lie groups. The exposition of quantum mechanics from this point of view reveals that classical mechanics and quantum mechanics are very much alike. Written by a mathematician and a physicist, this book is (like a math book) about precise concepts and exact results in classical mechanics and quantum mechanics, but motivated and discussed (like a physics book) in terms of their physical meaning. The reader can focus on the simplicity and beauty of theoretical physics, without getting lost in a jungle of techniques for estimating or calculating quantities of interest.