open access

https://doi.org/10.14943/83531

https://hdl.handle.net/2115/69135

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

1997-06-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

385

1-32

Investigated are 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 . For each vector v in a set V(D) C R2 \ {O}, the projection Pv of the physical momentum operator P := p - aA to the di­rection of vis defined by Pv := v · P as an operator acting in L2 (R2 ), where p = (-iDx, -iDy ) [(x, y) E R2] with Dx (resp. Dy ) being the generalized par­tial differential operator in the variable x (resp. y) and a E 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 { eitPv heR generated by the self-adjoint operator E'v ( 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 E 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 1r¢".w := {Qv,w,Qw,v,Pv,Pw} gives a representa­tion of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation 1r¢".w are analyzed. In particular, established is a necessary and sufficient condition for 1r¢".w to be unitarily equivalent ( or inequivalent) to the Schrodinger representation of CCR. The case where rr􀀊 w is inequivalent to the Schrodinger representation corresponds to the Aharon􀀍v­Bohm effect. Quantum algebraic structures [quantum plane and the quantum group Uq(sl2)] associated with the pair {Pv,.Pw} are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice L(w1,w2) := {mw1 + nw2 lm, n E Z} [the case D = L(w1,w2)], where w E R2 and w2 E R2 are linearly independent, it is shown that the magnetic translations eiPwi, j = 1, 2, with A replaced by a modified vector potential are reduced by the Hilbert space £2(L(w1,w2)) 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.

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