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

Let W2 denote the Weyl algebra generated by self-adjoint elements {pjtQj}j=i,2 satisfying the canonical commutation relations. In this paper we discuss *-representations [TT] ofW2 such that x(pj) and X(QJ) (/=!, 2) are essentially self-adjoint operators but x is not exponentiable to a representation of the associated Weyl system. We first construct a class of such *-representations of W2 by considering a non-simply connected space Q = R2\{a\, , aN} and a one-dimensiona l representations of the fundamental group n\(Q). Non-exponentiability of those *-representations comes from the geometry of the universal covering space Q of Q. Then we show that our *-representations of W2 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 a\, •-, UN, 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 W2, we calculate the defect numbers which measure the distance to the exponentiability.