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

Abstract

Let ${\mathcal{W}}_2$ denote the Weyl algebra generated by self-adjoint elements $\{p_j,q_j{\}}_{j=1,2}$ satisfying the canonical commutation relations. In this paper we discuss $\ast$-representations $\{\pi \}$ of ${\mathcal{W}}_2$ such that $\pi (p_j)$ and $\pi (q_j)$ $(j=1,2)$ are essentially self-adjoint operators but $\pi$ is not exponentiable to a representation of the associated Weyl system. We first construct a class of such $\ast$-representations of ${\mathcal{W}}_2$ by considering a non-simply connected space $\Omega ={\mathbf{R}}^2\setminus \{{\mathbf{a}}_1,\cdots ,{\mathbf{a}}_N\}$ and a one-dimensional representations of the fundamental group ${\pi }_1(\Omega )$. Non-exponentiability of those $\ast$-representations comes from the geometry of the universal covering space $\tilde{\Omega }$ of $\Omega$. Then we show that our $\ast$-representations of ${\mathcal{W}}_2$ 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 ${\mathbf{a}}_1,\cdots ,{\mathbf{a}}_N$, and, by doing that, we classify the Reeh–Arai's $\ast$-representations up to unitary equivalence. We further discuss extension and irreducibility of those $\ast$-representations. Finally, for the $\ast$-representations of ${\mathcal{W}}_2$, we calculate the defect numbers which measure the distance to the exponentiability.