PRINCIPLES OF DISCRETE-TIME MECHANICS - III - QUANTUM-FIELD THEORY

We apply the principles discussed in earlier papers to the constructio n of discrete time quantum field theories. We discuss some of the issu es concerned with the loss of Lorentz covariance and its recovery in t he appropriate limit. We use the Schwinger action principle to find th e discrete time free field commutators for scalar fields, which allows us to set up the reduction formalism for discrete time scattering pro cesses. Then we derive the discrete time analogue of the Feynman rules for a scalar field with a cubic self-interaction and give examples of discrete time scattering amplitude calculations. We find overall cons ervation of total linear momentum and overall conservation of total th eta parameters, which is the discrete time analogue of energy conserva tion and corresponds to the existence of a Logan invariant for the sys tem. We find that temporal discretization leads to softened vertex fac tors, modifies propagators and gives a natural cut-off for physical pa rticle momenta.