Geometric Construction of *-Representations of the Weyl Algebra with Degree 2

Abstract

Let _W_2 denote the Weyl algebra generated by self-adjoint elements {pj,qj}j=1,2 satisfying the canonical commutation relations. In this paper we discuss *-representations {π} of _W_2 such that π(pj) and π(qj) (j=1, 2) are essentially self-adjoint operators but π is not exponentiable to a representation of the associated Weyl system. We first construct a class of such *-representations of _W_2 by considering a non-simply connected space Ω= R2\{a1, ⋯, an} and a one-dimensional representations of the fundamental group π1(Ω). Non-exponentiability of those *-representations comes from the geometry of the universal covering space Ω~ of Ω. Then we show that our *-representations of _W_2 are related, by unitary equivalence, with Reeh-Arai's ones, which are based on a quantum system on the plane under a perpendicular magnetic field with singularities at a1, ⋯, an, and, by doing that, we classify the Reeh-Arai's *-representations up to unitary equivalence. We further discuss extension and irreducibility of those * -representations. Finally, for the * -representations of _W_2, we calculate the defect numbers which measure the distance to the exponentiability.