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Journal of Mathematical Physics

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

Asao Arai

Published Jun 1, 1995• 12 citations
Publisher

Abstract

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.

DOI
10.1063/1.531051

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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.
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.
A noncanonical quantum system, consisting of two nonrelativistic particles, interacting via a harmonic potential, is considered. The center-of-mass position and momentum operators obey the canonical commutation relations, whereas the internal variables are assumed to be the odd generators of the Lie superalgebra sl(1,2). This assumption implies a set of constraints in the phase space, which are explicitly written in the paper. All finite-dimensional irreducible representations of sl(1,2) are considered. Particular attention is paid to the physical representations, i.e., the representations corresponding to Hermitian position and momentum operators. The properties of the physical observables are investigated. In particular, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. The distance between the constituents is preserved in time. It can take no more than three different values. For any non-negative integer or half-integer l there exists a representation, where the orbital momentum is l (in units of 2ℏ). The position of any one of the particles cannot be localized, since the operators of the coordinates do not commute with each other. The constituents are smeared with a certain probability within a finite surface, which moves with a constant velocity together with the center of mass.