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

  • Hideki Kurose

    Fukuoka University, Japan
  • Hiroshi Nakazato

    Hirosaki University, Japan


Let denote the Weyl algebra generated by self-adjoint elements satisfying the canonical commutation relations. In this paper we discuss -representations of such that and are essentially self-adjoint operators but is not exponentiable to a representation of the associated Weyl system. We first construct a class of such -representations of by considering a non-simply connected space and a one-dimensional representations of the fundamental group . Non-exponentiability of those -representations comes from the geometry of the universal covering space of . Then we show that our -representations of 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 , 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 , 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