The Weyl Representation on the Torus

Abstract

We construct reflection and translation operators on the Hilbert space corresponding to the torus by projecting them from the plane. These operators are shown to have the same group properties as their analogues on the plane. Decomposition of operators on the basis of reflections corresponds to the Weyl or center representation, conjugate to the chord representation, which is based on quantized translations. Thus, the symbol of any operator on the torus is derived as the projection of the symbol on the plane. The group properties allow us to derive the product law for an arbitrary number of operators in a simple form. The analogy of the center and the chord representations on the torus to those on the plane is then exploited to treat Hamiltonian systems defined on the torus and to formulate a path integral representation of the evolution operator. We derive its semiclassical approximation.