Let \mathscr{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 \mathscr{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 \mathscr{W}
2 by considering a non-simply connected space Ω=
R2{\backslash}{
a1, …,
aN} and a one-dimensional representations of the fundamental group π
1(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 \mathscr{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 \mathscr{W}
2, we calculate the defect numbers which measure the distance to the exponentiability.
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