WIGNER-WEYL FORMALISMS FOR TOROIDAL GEOMETRIES

The phase-space formulation of quantum mechanics due to Weyl and Wigner is based on the assumption that the space in which the particles move is Euclidean. In the standard formulation of quantum mechanics, to which this formalism is equivalent, the wave functions are therefore assumed to satisfy natural boundary conditions (sufficiently fast decay at infinity). However. these conditions are not satisfied when, for physical reasons or due to the approximations employed, the underlying geometry is that of a torus (periodic boundary conditions). We discus s the corresponding modifications of the Wigner-Weyl formalism by comparing the following three cases: (A) motion on the real line, (B) motion on the circle or on an infinite one-dimensional lattice, and (C) motion on a finite set of points. For each of these three different situations we first list the basic equations of an appropriate phase-space formalism. We then relate our approach to previous derivations of this formalism, if such exist in literature, and compare our formalism to equivalent but different schemes that were proposed for this type of geometry in the past. Finally, we discuss to what extent one scheme may be considered as an approximation of another one. The mathematical tools used in this paper are operator algebras and their bases. (C) 1994 Academic Press, Inc.