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

### Hideki Kurose

Fukuoka University, Japan### Hiroshi Nakazato

Hirosaki University, Japan

## Abstract

Let $W_{2}$ 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 $∗$-representations ${π}$ of $W_{2}$ such that $π(p_{j})$ and $π(q_{j})$ $(j=1,2)$ 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 $Ω=R_{2}∖{a_{1},⋯,a_{N}}$ 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 $W_{2}$ 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 $a_{1},⋯,a_{N}$, 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.

## Cite this article

Hideki Kurose, Hiroshi Nakazato, Geometric Construction of $∗$-Representations of the Weyl Algebra with Degree 2. Publ. Res. Inst. Math. Sci. 32 (1996), no. 4, pp. 555–579

DOI 10.2977/PRIMS/1195162712