Quantum Gauge Equivalence in QED

TL;DR

This paper investigates gauge transformations in QED, demonstrating how different gauges can be transformed into a common form with identical particle excitation properties, unifying their time evolution and operator structures.

Contribution

It introduces a method to transform QED in various gauges into a common form, aligning their operator structures and time evolution, thus unifying gauge formulations.

Findings

QED in different gauges can be transformed into a common form.

All gauges share a unified time-evolution operator.

Gauge transformations combined with representation changes unify the theory.

Abstract

We discuss gauge transformations in QED coupled to a charged spinor field, and examine whether we can gauge-transform the entire formulation of the theory from one gauge to another, so that not only the gauge and spinor fields, but also the forms of the operator-valued Hamiltonians are transformed. The discussion includes the covariant gauge, in which the gauge condition and Gauss's law are not primary constraints on operator-valued quantities; it also includes the Coulomb gauge, and the spatial axial gauge, in which the constraints are imposed on operator-valued fields by applying the Dirac-Bergmann procedure. We show how to transform the covariant, Coulomb and spatial axial gauges to what we call ``common form,'' in which all particle excitation modes have identical properties. We also show that, once that common form has been reached, QED in different gauges has a common time-evolution operator that defines time-translation for states that represent systems of electrons and photons. By combining gauge transformations with changes of representation from standard to common form, the entire apparatus of a gauge theory can be transformed from one gauge to another.