WIGNER-WEYL FORMALISMS FOR TOROIDAL GEOMETRIES

The phase-space formulation of quantum mechanics due to Weyl and Wigne r is based on the assumption that the space in which the particles mov e is Euclidean. In the standard formulation of quantum mechanics, to w hich this formalism is equivalent, the wave functions are therefore as sumed to satisfy natural boundary conditions (sufficiently fast decay at infinity). However. these conditions are not satisfied when, for ph ysical reasons or due to the approximations employed, the underlying g eometry is that of a torus (periodic boundary conditions). We discus s the corresponding modifications of the Wigner-Weyl formalism by compa ring the following three cases: (A) motion on the real line, (B) motio n on the circle or on an infinite one-dimensional lattice, and (C) mot ion on a finite set of points. For each of these three different situa tions 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 g eometry in the past. Finally, we discuss to what extent one scheme may be considered as an approximation of another one. The mathematical to ols used in this paper are operator algebras and their bases. (C) 1994 Academic Press, Inc.