The Weyl representation on the torus

Details The Weyl representation on the torus The Weyl representation on the torus

1999-04-09

arxiv.org/abs/quant-ph/9904041v1

Annals Phys. 276 (1999) 223-256

Physics

Quantum Physics

26 pages, latex, 4 figures Submitted to Annals of Physics

Scientific paper

10.1006/aphy.1999.5942

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 analogue on the plane. The decomposition of operators in 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 between 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.

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