Limitations of gauge invariance

TL;DR

This paper discusses how gauge transformations can alter physical interpretations despite preserving observable values, highlighting the importance of gauge choice in classical and quantum contexts.

Contribution

It demonstrates that gauge transformations are not generally unitary and shows the significance of gauge choice for consistent physical interpretations.

Findings

Gauge transformations can change physical interpretations.

Potential energy is gauge-dependent, unlike kinetic energy.

Coulomb gauge maintains consistency with laboratory field generation.

Abstract

Although gauge invariance preserves the values of physical observables, a gauge transformation can introduce important alterations of physical interpretations. To understand this, it is first shown that a gauge transformation is not, in general, a unitary transformation. Also, physical interpretations are based on both kinetic energy and potential energy expressions. While the kinetic energy is a measurable quantity, and hence gauge-invariant, the potential energy is gauge-dependent. Two basic examples are examined; one classical and the other quantum-mechanical. The aim is to show that the use of the Coulomb (or radiation) gauge is always consistent with the way that fields are generated in the laboratory. Upon transformation out of the Coulomb gauge, this connection is lost, and physical interpretations can give rise to misleading inferences.