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

CRC Press eBooks

Aspects of Gauge Quantum Mechanics on Non-Simply Connected Spaces

Asao Arai

Published Nov 27, 2020• 0 citations
Publisher

Abstract

A class of representations of the canonical commutation relations is constructed in a quantum theory of a particle moving in an external (non-Abelian) gauge field on a non-simply connected space in three space dimensions. A coupling to a Bose field of the particle is also discussed.
DOI
10.1201/9781003078296-25

Related papers

Gauge theory on a non-simply connected domain and representations of canonical commutation relations

Asao Arai

A quantum system of a particle interacting with a (non-Abelian) gauge field on the non-simply connected domain M=R2\{an}Nn=1 is considered, where an, n=1,...,N, are fixed isolated points in R2. The gauge potential A of the gauge field is a p×p anti-Hermitian matrix-valued 1-form on M, and may be strongly singular at the points an, n=1,...,N. If A is flat, then the physical momentum and the position operators {Pj,qj}2j=1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2;Cp). A necessary and sufficient condition for this representation to be the Schrödinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n=1,...,N. This also gives a characterization for the representation to be non-Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, Pj is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail.

Journal of Mathematical Physics•Jun 1, 1995•12 citations

Representation of Canonical Commutation Relations in a Gauge Theory, the Aharonov-Bohm Effect, and the Dirac-Weyl Operator

Asao Arai

We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space.A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR is given in terms of Wilson loops.A representation inequivalent to the Schrödinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect.Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.

Journal of Nonlinear Mathematical Physics•Jan 1, 1995•5 citations

Representation of canonical commutation relations in a gauge theory, the Aharonov-Bohm effect, and Dirac Weyl operator

Arai, A.

We consider a representation of canonical commutation relations (CCR) appear­ing in a (non-Abelian) gauge theory on a non-simply connected region in the two­dimensional Euclidean space. A necessary and sufficient condition for the representa­tion to be equivalent to the Schrodinger representation of CCR is given in terms of Wilson loops. A representation inequivalent to the Schrodinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect. Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.

•Sep 1, 1995•0 citations

The canonical connection in quantum mechanics

Péter Lévay, David McMullan, Izumi Tsutsui

In this paper we investigate the form of induced gauge fields that arises in two types of quantum systems. In the first we consider quantum mechanics on coset spaces G/H, and argue that G invariance is central to the emergence of the H connection as induced gauge fields in the different quantum sectors. We then demonstrate why the same connection, now giving rise to the non-Abelian generalization of Berry’s phase, can also be found in systems that have slow variables taking values in such a coset space.

arXiv:hep-th/9504116•Journal of Mathematical Physics•Feb 1, 1996•15 citations

Nonequivalent representations of canonical commutation relations in quantum mechanics: The case of the Aharonov-Bohm effect

L. Bracci, Luigi E. Picasso

We stress the role of nonequivalent representations of the canonical commutation relations in quantum mechanics. Such representations occur when the space accessible to the system is not simply connected. An example is the magnetic Aharonov-Bohm effect, which can be interpreted in terms of nonequivalent representations, without explicitly invoking the vector potential in the region accessible to the electrons.

American Journal of Physics•Feb 12, 2007•1 citation

Classical and Quantum Theory with a New Symmetry

Partha Ghose

A formal symmetry between generalized coordinates and momenta is postulated to formulate classical and quantum theories of a particle coupled to an Abelian gauge field. It is shown that the symmetry (a) requires the field to have dynamic degrees of freedom and to be a connection in a non-flat space-time manifold, and (b) leads to a quantum theory free of the measurement problem. It is speculated that gravitomagnetism could be a possible source of the gauge field.

arXiv:0905.2320•May 14, 2009•0 citations
A quantum system of a particle interacting with a (non-Abelian) gauge field on the non-simply connected domain M=R2\{an}Nn=1 is considered, where an, n=1,...,N, are fixed isolated points in R2. The gauge potential A of the gauge field is a p×p anti-Hermitian matrix-valued 1-form on M, and may be strongly singular at the points an, n=1,...,N. If A is flat, then the physical momentum and the position operators {Pj,qj}2j=1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2;Cp). A necessary and sufficient condition for this representation to be the Schrödinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n=1,...,N. This also gives a characterization for the representation to be non-Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, Pj is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail.
We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space.A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR is given in terms of Wilson loops.A representation inequivalent to the Schrödinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect.Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.
We consider a representation of canonical commutation relations (CCR) appear­ing in a (non-Abelian) gauge theory on a non-simply connected region in the two­dimensional Euclidean space. A necessary and sufficient condition for the representa­tion to be equivalent to the Schrodinger representation of CCR is given in terms of Wilson loops. A representation inequivalent to the Schrodinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect. Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.
In this paper we investigate the form of induced gauge fields that arises in two types of quantum systems. In the first we consider quantum mechanics on coset spaces G/H, and argue that G invariance is central to the emergence of the H connection as induced gauge fields in the different quantum sectors. We then demonstrate why the same connection, now giving rise to the non-Abelian generalization of Berry’s phase, can also be found in systems that have slow variables taking values in such a coset space.