Discrete phase space - I: Variational formalism for classical relativistic wave fields

TL;DR

This paper develops a variational formalism for classical relativistic wave fields in discrete phase space, establishing their invariance, covariance, and conservation laws, and deriving key physical quantities.

Contribution

It introduces a novel variational approach to formulating relativistic wave equations in discrete phase space, ensuring covariance and deriving conservation laws.

Findings

Relativistic invariance and covariance are established for the equations.

Conservation laws for momentum, energy, and charge are derived.

Difference and difference-differential equations are formulated from a variational principle.

Abstract

The classical relativistic wave equations are presented as partial difference equations in the arena of covariant discrete phase space. These equations are also expressed as difference-differential equations in discrete phase space and continuous time. The relativistic invariance and covariance of the equations in both versions are established. The partial difference and difference-differential equations are derived as the Euler-Lagrange equations from the variational principle. The difference and difference-differential conservation equations are derived. Finally, the total momentum, energy, and charge of the relativistic classical fields satisfying difference-differential equations are computed.