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

Abstract

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 R2 are investigated. For each vector v in a set V(D)⊂R2\{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 L2(R2), where p=(−iDx,−iDy)[(x,y)∈R2] 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 {eitP̄v}t∈R generated by the self-adjoint operator P̄v (the closure of Pv), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in R2\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 R2\D, then πv,wA≔{Qv,w,Qw,v,Pv,Pw} gives a representation of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation πv,wA are analyzed. In particular, a necessary and sufficient condition for πv,wA to be unitarily equivalent (or inequivalent) to the Schrödinger representation of CCR is established. The case where πv,wA 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 {P̄v,P̄w} are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice L(ω1,ω2)≔{mω1+nω2|m,n∈Z} [the case D=L(ω1,ω2)], where ω1∈R2 and ω2∈R2 are linearly independent, it is shown that the magnetic translations eiP̄ωj, j=1,2, with A replaced by a modified vector potential are reduced by the Hilbert space l2(L(ω1,ω2)) identified with a closed subspace of L2(R2). 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.