open access

https://hdl.handle.net/2115/13682

Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials: Canonical commutation relations, quantum algebras, and reduction to lattice quantum systems

英語

quantum theory

1998-05

American Institute of Physics

Journal of Mathematical Physics

39(5)

2476-2498

Some representation-theoretic aspects of a two-dimensional quantum system of a charged particle in a vector potential A, which may be singular on an infinite discrete subset D of R^2 are investigated. For each vector v in a set V(D)⊂ R^2\{0}, the projection Pv of the physical momentum operator P := p-αA to the direction of v is defined by Pv := v・P as an operator acting in L^2(R^2), where p = (-iDx,-iDy)[(x,y)∈ R^2] with Dx (resp., Dy) being the generalized partial differential operator in the variable x (resp., y) and α∈ R is a parameter denoting the charge of the particle. It is proven that Pv is essentially self-adjoint and an explicit formula is derived for the strongly continuous one-parameter unitary group {e^it[P-bar]v}t∈ R generated by the self-adjoint operator [P-bar]v (the closure of Pv), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in R^2 \ D are also considered. Conjugately to Pv and Pw [w∈V(D)], a self-adjoint multiplication operator Qv,w is introduced, which is a linear combination of the position operators x and y, such that, if A is flat on R^2 \ D, then pi_v,w^A := {Qv,w,Qw,v,Pv,Pw} gives a representation of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation pi_v,w^A are analyzed. In particular, a necessary and sufficient condition for pi_v,w^A to be unitarily equivalent (or inequivalent) to the Schrödinger representation of CCR is established. The case where pi_v,w^A is inequivalent to the Schrödinger representation corresponds to the Aharonov-Bohm effect. Quantum algebraic structures [quantum plane and the quantum group Uq(sl2)] associated with the pair {[P-bar]v,[P-bar]w} are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice L(ω 1,ω 2) := {mω 1 + nω2|m,n∈ Z} [the case D = L(ω 1,ω 2)], where ω1∈R^2 and ω 2∈R^2 are linearly independent, it is shown that the magnetic translations e^{i[P-bar]ω j}, j = 1,2, with A replaced by a modified vector potential are reduced by the Hilbert space l^2(L(ω1,ω 2)) identified with a closed subspace of L^2(R^2). This result, which may be regarded as one of the most important novel results of the present paper, establishes a connection of continuous quantum systems in vector potentials to lattice ones.

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Copyright © 1998 American Institute of Physics

0022-2488

1089-7658