Principles of Discrete Time Mechanics: III. Quantum Field Theory

TL;DR

This paper develops a framework for discrete time quantum field theories, deriving modified Feynman rules, scattering amplitudes, and conservation laws, with implications for natural momentum cutoffs and system invariants.

Contribution

It introduces a discrete time formulation of quantum field theory, including new Feynman rules and conservation principles, extending previous principles to quantum fields.

Findings

Discrete time Feynman rules derived for scalar fields

Conservation of total linear momentum and theta parameters established

Discretisation introduces natural momentum cutoffs and modifies propagators

Abstract

We apply the principles discussed in earlier papers to the construction of discrete time quantum field theories. We use the Schwinger action principle to find the discrete time free field commutators for scalar fields, which allows us to set up the reduction formalism for discrete time scattering processes. 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 conservation of total linear momentum and overall conservation of total theta parameters, which is the discrete time analogue of energy conservation and corresponds to the existence of a Logan invariant for the system. We find that temporal discretisation leads to softened vertex factors, modifies propagators and gives a natural cutoff for physical particle momenta.