Canonical commutation relations, the Weierstrass Zeta function, and infinite dimensional Hilbert space representations of the quantum group Uq(sl2)

Abstract

A two‐dimensional quantum system of a charged particle interacting with a vector potential determined by the Weierstrass Zeta function is considered. The position and the physical momentum operators give a representation of the canonical commutation relations with two degrees of freedom. If the charge of the particle is not an integer (the case corresponding to the Aharonov–Bohm effect), then the representation is inequivalent to the Schrödinger representation. It is shown that the inequivalent representation induces infinite‐dimensional Hilbert space representations of the quantum group Uq(sl). Some properties of these representations of Uq(sl) are investigated.