Quantum boundary conditions for torus maps

The quantum states of a dynamical system whose phase space is the two-torus are periodic up to phase factors under translations by the fundamental per iods of the torus in the position and momentum representations. These phase s, theta(1) and theta(2), are conserved quantities of the quantum evolution . We show that for a large and important class of quantum maps, theta(1) an d theta(2) are restricted to bring the coordinates of the fixed points of t he automorphism induced on the fundamental group of the torus by the underl ying classical dynamics. As a consequence, if the classical map commutes wi th lattice translations in R-2 it can be quantized for any choice of the ph ases, but otherwise it can be quantized for only a finite set. This result is a special case of a more general condition on the phases, which is also derived. The cat maps, perturbed cat maps, and the kicked Harper map are di scussed as specific examples.