Discrete-time quantum mechanics. II. Systems with several degrees of freedom

In a previous paper we used the method of finite elements to formulate consistent, unitary, discrete-time quantum-mechanical systems having one degree of freedom. In this paper we extend the treatment to systems having two degrees of freedom. The proof of consistency is more delicate: It is nontrivial to show that there are no operator-ordering problems and that independent degrees of freedom remain independent at subsequent lattice sites. The construction of purely bosonic lattice systems is a straightforward application of the finite-element prescription. However, it is surprising that quantum-mechanical systems having interacting fermions and bosons are only unitary if the interaction is modified on the lattice by a term which vanishes as the lattice spacing approaches zero. The modified interaction is determined by an interesting nonlinear condition. In all cases we give an explicit formula for the lattice transfer operator.