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

Quantum Studies Mathematics and Foundations

On the discrete Heisenberg group and commutative modular variables in quantum mechanics: I. The Abelian symplectic shadow and integrality of area

Elias Zafiris, Albrecht von Müller

Published Jun 30, 2021• 3 citations
Publisher
DOI
10.1007/s40509-021-00251-z

Related papers

The Equiareal Archimedean Synchronization Method of the Quantum Symplectic Phase Space: II. Circle-Valued Moment Map, Integrality, and Symplectic Abelian Shadows

Elias Zafiris

Foundations of Physics•Apr 1, 2022•2 citations

Quantum holonomies and the Heisenberg group

J. E. Nelson, R. F. Picken

Quantum holonomies of closed paths on the torus [Formula: see text] are interpreted as elements of the Heisenberg group [Formula: see text]. Group composition in [Formula: see text] corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group [Formula: see text] of [Formula: see text], making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of [Formula: see text] adjust these signed areas, and the discrete symplectic transformations of [Formula: see text] generate the modular group of [Formula: see text].

arXiv:1808.08812•Modern Physics Letters A•Jul 22, 2019•0 citations

Quantum mechanics as a matrix symplectic geometry

A. E. F. Djemai

International Journal of Theoretical Physics•Feb 1, 1997•3 citations

Projective representations of the inhomogeneous symplectic group: Quantum symmetry origins of the Heisenberg commutation relations

Stephen G. Low

Quantum symmetries that leave invariant physical transition probabilities are described by projective representations of Lie groups. The mathematical theory of projected representations for topologically connected Lie groups is reviewed and applied to the inhomogeneous symplectic group ISp(2n). The projective representations are given in terms of the ordinary unitary representations of the central extension of the group with respect to its first homotopy group direct product with its second cohomology group. The second cohomology group of ISp(2n) is the one dimensional abelian group and the central extension turns the 2n dimension abelian normal subgroup of translations of ISp(2n) into the Weyl-Heisenberg group. This is the quantum symmetry origin of the Heisenberg commutation relations that are the Hermitian representation of the Lie algebra of the Weyl-Heisenberg group.

arXiv:1909.11126•Sep 24, 2019•0 citations

Quantum mechanics as a matrix symplectic geometry

A. E. F. Djemai

International Journal of Theoretical Physics•Mar 1, 1996•7 citations

Towards a Group-Theoretical Interpretation of Mechanics

Gabriel Catren

Abstract. We argue that the classical description of a symplectic manifold endowed with a Hamil-tonian action of an abelian Lie group G and the corresponding quantum theory can be understood as different aspects of the unitary representation theory of G. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry (notably, Souriau’s moment map and the Mardsen-Weinstein symplectic reduction formalism) and group representation theory (notably Kirillov’s orbit method). The proposed argumentative line strongly relies on the conjecture pro-posed by Guillemin and Sternberg according to which “quantization commutes with (symplectic) reduction”. By using the generalization of this conjecture to non-zero coadjoint orbits, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric un-derpinning, namely the symplectic reduction formalism. This fact points towards a gauge-theoretic interpretation of Heisenberg indeterminacy principle. Key words: quantum mechanics; gauge theories; moment map; symplectic reduction. 1.

•Nov 30, 2013•1 citation
Quantum holonomies of closed paths on the torus [Formula: see text] are interpreted as elements of the Heisenberg group [Formula: see text]. Group composition in [Formula: see text] corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group [Formula: see text] of [Formula: see text], making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of [Formula: see text] adjust these signed areas, and the discrete symplectic transformations of [Formula: see text] generate the modular group of [Formula: see text].
Quantum symmetries that leave invariant physical transition probabilities are described by projective representations of Lie groups. The mathematical theory of projected representations for topologically connected Lie groups is reviewed and applied to the inhomogeneous symplectic group ISp(2n). The projective representations are given in terms of the ordinary unitary representations of the central extension of the group with respect to its first homotopy group direct product with its second cohomology group. The second cohomology group of ISp(2n) is the one dimensional abelian group and the central extension turns the 2n dimension abelian normal subgroup of translations of ISp(2n) into the Weyl-Heisenberg group. This is the quantum symmetry origin of the Heisenberg commutation relations that are the Hermitian representation of the Lie algebra of the Weyl-Heisenberg group.
Abstract. We argue that the classical description of a symplectic manifold endowed with a Hamil-tonian action of an abelian Lie group G and the corresponding quantum theory can be understood as different aspects of the unitary representation theory of G. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry (notably, Souriau’s moment map and the Mardsen-Weinstein symplectic reduction formalism) and group representation theory (notably Kirillov’s orbit method). The proposed argumentative line strongly relies on the conjecture pro-posed by Guillemin and Sternberg according to which “quantization commutes with (symplectic) reduction”. By using the generalization of this conjecture to non-zero coadjoint orbits, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric un-derpinning, namely the symplectic reduction formalism. This fact points towards a gauge-theoretic interpretation of Heisenberg indeterminacy principle. Key words: quantum mechanics; gauge theories; moment map; symplectic reduction. 1.