In Which Sense Can We Say That First-Class Constraints Generate Gauge Transformations?

TL;DR

This paper explores the nuanced relationship between first-class constraints and gauge transformations in electromagnetism, proposing an extended notion of gauge transformation that may apply to broader theories and aligns with quantum formulations.

Contribution

It introduces an extended notion of gauge transformation that clarifies the role of first-class constraints and supports Dirac's conjecture in certain theories.

Findings

Extended gauge transformation differs from the original but is non-trivial.

Dirac's conjecture may hold for some physically reasonable theories.

The extended notion aligns with quantum gauge theory construction.

Abstract

In this paper, I consider a recent controversy about whether first-class constraints generate gauge transformations in the case of electromagnetism. I argue that there is a notion of gauge transformation, the extended notion, which is different from the original gauge transformation of electromagnetism, but at the same time not trivial, which allows the making of that claim. I further argue that one can expect that this claim can be extended to more general theories, and that Dirac's conjecture may be true for some physically reasonable theories and only in this sense of gauge transformation. Finally, I argue that the extended notion of gauge transformation seems unnatural from the point of view of classical theories, but that it nicely fits with the way quantum versions of gauge theories are constructed.