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

Representations of a local current algebra in nonsimply connected space and the Aharonov–Bohm effect

Gerald A. Goldin, Ralph Menikoff, D. H. Sharp

Published Aug 1, 1981• 311 citations
Publisher

Abstract

A recent paper established technical conditions for the construction of a class of induced representations of the nonrelativistic current group 𝒮Λ𝒦, where 𝒮 is Schwartz’s space of rapidly decreasing C∞ functions, and 𝒦 is a group of C∞ diffeomorphisms of Rs. Bose and Fermi N-particle systems were recovered as unitarily inequivalent induced representations of the group by lifting the action of 𝒦 on an orbit Δ⊆𝒮′ to its universal covering space δ̃. For s⩾3, δ̃ is the coordinate space for N particles, which is simply connected. In two-dimensional space, however, the coordinate space is multiply connected, implying induced representations other than those describing the usual Bose or Fermi statistics; these are explored in the present paper. Likewise the Aharonov–Bohm effect is described by means of induced representations of the local observables, defined in a nonsimply connected region of Rs. The vector potential plays no role in this description of the Aharonov–Bohm effect.

DOI
10.1063/1.525110

Related papers

Particle statistics from induced representations of a local current group

Gerald A. Goldin, Ralph Menikoff, D. H. Sharp

Representations of the nonrelativistic current group 𝒮-𝒦 are studied in the Gel’fand–Vilenkin formalism, where 𝒮 is Schwartz’ space of rapidly decreasing functions, and 𝒦 is a group of diffeomorphisms of Rs. For the case of N identical particles, information about particle statistics is contained in a representation of 𝒦F (the stability group of a point F∈𝒮′) which factors through the permutation group SN. Starting from a quasi-invariant measure μ concentrated on a 𝒦 orbit Δ in 𝒮′, together with a suitable representation of 𝒦F for F∈Δ, sufficient conditions are developed for inducing a representation of 𝒮-𝒦. The Hilbert space for the induced representation consists of square-integrable functions on a covering space of Δ, which transform in accordance with a representation of 𝒦F. The Bose and Fermi N-particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representation of 𝒮-𝒦 determines a well-defined representation of 𝒦F; (2) equivalent representations of 𝒮-𝒦 determine equivalent representations of 𝒦F; (3) a representation of 𝒦F induces a representation of 𝒮-𝒦; and (4) equivalent representations of 𝒦F determine equivalent induced representations.

Journal of Mathematical Physics•Apr 1, 1980•169 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

Path-integrals in multiply-connected spaces and the Aharonov-Bohm effect

G. Morandi, E Menossi

The motion of a nonrelativistic charged particle in a plane, multiply-connected region is studied within the framework of Feynman's path-integral approach to quantum mechanics. In particular, the authors study the simple case when the multiply-connected region is obtained by excluding a disc from the plane. If a nonzero magnetic flux is confined inside the disc, this is shown to include a change in the one-dimensional unitary representation of the fundamental group of the space which enters a proper definition of the path-integral. In this way, a simple explanation of the Aharonov-Bohm effect is shown to arise.

European Journal of Physics•Jan 1, 1984•46 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

Gauge-independent description of the Aharonov-Bohm effect

Xiang Li, T. H. Hansson, Wei Ku

The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential $\mathbf{A}$, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics $\mathbf{A}$ is in some sense more ``fundamental'' than $\mathbf{B}$ in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant $\mathbf{B}$ field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the $\mathbf{B}$ field in the region inaccessible to the particle.

arXiv:2209.11091•Physical Review A•Sep 21, 2022•9 citations

Canonical commutation relations, the Weierstrass Zetafunction, and infinite dimensional Hilbert space representations of the quantum group Uq (sl2)

Asao Arai

A two-dimensional quantum system of a charged particle interacting with a vector potential determined by the Weierstrass Zeta function is considered. The position and the physical momentum operators give a representation of the canonical commutation relations with two degrees of freedom. If the charge of the particle is not an integer (the case corresponding to the Aharonov–Bohm effect), then the representation is inequivalent to the Schrödinger representation. It is shown that the inequivalent representation induces infinite-dimensional Hilbert space representations of the quantum group Uq(sl2). Some properties of these representations of Uq(sl2) are investigated.

Preprint Series of Department of Mathematics, Hokkaido University•Nov 1, 1995•1 citation
A recent paper established technical conditions for the construction of a class of induced representations of the nonrelativistic current group 𝒮Λ𝒦, where 𝒮 is Schwartz’s space of rapidly decreasing C∞ functions, and 𝒦 is a group of C∞ diffeomorphisms of Rs. Bose and Fermi N-particle systems were recovered as unitarily inequivalent induced representations of the group by lifting the action of 𝒦 on an orbit Δ⊆𝒮′ to its universal covering space δ̃. For s⩾3, δ̃ is the coordinate space for N particles, which is simply connected. In two-dimensional space, however, the coordinate space is multiply connected, implying induced representations other than those describing the usual Bose or Fermi statistics; these are explored in the present paper. Likewise the Aharonov–Bohm effect is described by means of induced representations of the local observables, defined in a nonsimply connected region of Rs. The vector potential plays no role in this description of the Aharonov–Bohm effect.
Representations of the nonrelativistic current group 𝒮-𝒦 are studied in the Gel’fand–Vilenkin formalism, where 𝒮 is Schwartz’ space of rapidly decreasing functions, and 𝒦 is a group of diffeomorphisms of Rs. For the case of N identical particles, information about particle statistics is contained in a representation of 𝒦F (the stability group of a point F∈𝒮′) which factors through the permutation group SN. Starting from a quasi-invariant measure μ concentrated on a 𝒦 orbit Δ in 𝒮′, together with a suitable representation of 𝒦F for F∈Δ, sufficient conditions are developed for inducing a representation of 𝒮-𝒦. The Hilbert space for the induced representation consists of square-integrable functions on a covering space of Δ, which transform in accordance with a representation of 𝒦F. The Bose and Fermi N-particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representation of 𝒮-𝒦 determines a well-defined representation of 𝒦F; (2) equivalent representations of 𝒮-𝒦 determine equivalent representations of 𝒦F; (3) a representation of 𝒦F induces a representation of 𝒮-𝒦; and (4) equivalent representations of 𝒦F determine equivalent induced representations.
The motion of a nonrelativistic charged particle in a plane, multiply-connected region is studied within the framework of Feynman's path-integral approach to quantum mechanics. In particular, the authors study the simple case when the multiply-connected region is obtained by excluding a disc from the plane. If a nonzero magnetic flux is confined inside the disc, this is shown to include a change in the one-dimensional unitary representation of the fundamental group of the space which enters a proper definition of the path-integral. In this way, a simple explanation of the Aharonov-Bohm effect is shown to arise.
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.
The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region inaccessible to the particle. Classically, such a nonlocal effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle and the electromagnetic vector potential A\mathbf{A}A, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics A\mathbf{A}A is in some sense more ``fundamental'' than B\mathbf{B}B in spite of the former being gauge dependent and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB effect can be fully accounted for by considering only the gauge-invariant B\mathbf{B}B field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge-invariant formulation is that we must give up locality---the AB phase for the particle will arise from the change in the action for the B\mathbf{B}B field in the region inaccessible to the particle.
A two-dimensional quantum system of a charged particle interacting with a vector potential determined by the Weierstrass Zeta function is considered. The position and the physical momentum operators give a representation of the canonical commutation relations with two degrees of freedom. If the charge of the particle is not an integer (the case corresponding to the Aharonov–Bohm effect), then the representation is inequivalent to the Schrödinger representation. It is shown that the inequivalent representation induces infinite-dimensional Hilbert space representations of the quantum group Uq(sl2). Some properties of these representations of Uq(sl2) are investigated.