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Journal of Mathematical Physics

Momentum operators with gauge potentials, local quantization of magnetic flux, and representation of canonical commutation relations

Asao Arai

Published Oct 1, 1992• 18 citations
Publisher

Abstract

Commutation properties of two-dimensional momentum operators with gauge potentials are investigated. A notion of local quantization of magnetic flux is introduced to characterize physically the strong commutativity of the momentum operators. In terms of the notion, a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the Schrödinger representation of the canonical commutation relations.

DOI
10.1063/1.529938

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Commutation properties of two-dimensional momentum operators with gauge potentials are investigated. A notion of local quantization of magnetic flux is introduced to characterize physically the strong commutativity of the momentum operators. In terms of the notion, a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the Schrödinger representation of the canonical commutation relations.
Considered is a quantum system of N(≥2)N(\ge2)N(≥2) charged particles moving in the plane R2{\mathbb R}^2R2 under the influence of a perpendicular magnetic field concenrated on the positions where the particle exsists. The gauge potential which gives this magnetic field is called a winding gauge potential. Properties of the momentum operators with a winding gauge potential are investigated. The momentum operators with a winding gauge potential are represented by the fibre direct integral of Arai's momentum operators [1]. Using this fibre direct integral decomposition, commutation properties of the momentum operators are investigated. A notion of local quantization of the magnetic flux is introduced to characterize the strong commutativity of the momentum operators. Aspects of the representation of the canonical commutation relations (CCR) are discussed. There is an interesting relation between the representation of the CCR with respect to this system and Arai's representation. Some applications of those results are also 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 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.
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 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.
We study quantum—classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than ħ is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.
A general proof of several formulas used in three-dimensional quantum problems is presented. By considering the two vector operators A and B having Cartesian components that satisfy the canonical commutation relations—as the position and momentum operators do—we obtain commutation rules and relations for the projection onto each another. All of these rules and relations have symmetry under A↔B exchange. This symmetry suggests, in the particular case of position and momentum operators, a new operator, the projection of r onto p, which seems to be useful in dissipative quantum problems.