Principles of Discrete Time Mechanics: II. Classical field Theory

TL;DR

This paper develops a formalism for discrete time classical and quantum field theories, deriving equations of motion, conserved quantities, and exploring fundamental limits on energy and momentum, with applications to Schrödinger, Klein-Gordon, and electrodynamics.

Contribution

It introduces a novel discrete time framework for classical and quantum field theories, including derivations of equations of motion and conservation laws, and explores fundamental physical bounds.

Findings

Discrete time Schrödinger solutions match standard energy eigenvalues with a fundamental energy limit.

Derived equations of motion and conserved quantities for Klein-Gordon and electromagnetic fields.

Identified an upper bound on linear momentum for particle-like solutions.

Abstract

We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrodinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrodinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell's electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantisation of discrete time mechanics and discrete time Quantum Electrodynamics.