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

Abstract

This is the first in a series of papers dealing with a discrete phase space formulation for classical and quantum fields. This formulation leads to a representation of quantum field theory that is covariant and possesses singularity free S-matrix. In this paper, 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.