Point Form Electrodynamics and the Construction of Conserved Current Operators

TL;DR

This paper presents a general method for constructing conserved electromagnetic current operators in quantum systems, applicable to both finite and infinite degrees of freedom, without relying on equations of motion.

Contribution

It introduces a novel, non-local construction of current operators using four-momentum commutators, expanding the theoretical framework of electrodynamics.

Findings

Constructed conserved current operators without equations of motion.

Applicable to systems with finite and infinite degrees of freedom.

Provided explicit examples demonstrating the construction process.

Abstract

A general procedure for constructing conserved electromagnetic current operators, for both finite and infinite degree of freedom systems, is given. A four-momentum operator consisting of matter, photon, and electromagnetic interactions is assumed to be polynomial in photon creation and annihilation operators. Commutators of the four-momentum operator with the four-vector potential operator at the space-time point zero give the electromagnetic field tensor and current operator at the space-time point zero. In this construction there are no equations of motion to be satisfied and field operators at an arbitrary space-time point-defined as the four-momentum translates of the corresponding operators at the space-time point zero-are not local operators. Several examples are given to show how the construction is carried out.