DISCRETE-TIME QUANTUM-MECHANICS

This paper summarizes a research program that has been underway for a decade. The objective is to find a fast and accurate scheme for solvin g quantum problems which does not involve a Monte Carlo algorithm. We use an alternative strategy based on the method of finite elements. We are able to formulate fully consistent quantum-mechanical systems dir ectly on a lattice in terms of operator difference equations. One adva ntage of this discretized formulation of quantum mechanics is that the ambiguities associated with operator ordering are eliminated. Further more, the scheme provides an easy way in which to obtain the energy le vels of the theory numerically. A generalized version of this discreti zation scheme can be applied to quantum field theory problems. The dif ficulties normally associated with fermion doubling are eliminated. Al so, one can incorporate local gauge invariance in the finite-element f ormulation. Results for some field theory models are summarized. In pa rticular, we review the calculation of the anomaly in two-dimensional quantum electrodynamics (the Schwinger model). Finally, we discuss non abelian gauge theories.