Extended symmetry of the Maxwell theory with a gauge coupling constant as a conserved charge

TL;DR

The paper investigates promoting the Maxwell coupling constant to a conserved charge by introducing a dynamical scalar $\sigma$ and an auxiliary vector $\chi^\mu$, encoded in $S[A_\mu,\sigma,\chi_\mu] = -\tfrac14\int d^4x\, \sigma(F_{\mu\nu}F^{\mu\nu}-\partial_\mu\chi^\mu)$. A Dirac constraint analysis reveals second-class constraints that break some local gauge invariances, motivating a BFT embedding to restore all local symmetries. By introducing an infinite tower of auxiliary fields, the authors construct an extended first-class theory with a fully gauge-invariant action $\tilde{S}$, whose local symmetries yield no new conserved charges beyond the original ones; the physical Maxwell theory is recovered in a suitable gauge. The work clarifies how coupling constants can be treated as conserved charges without altering physical content and illustrates the capabilities and limits of the BFT method in simple gauge theories.

Abstract

It has been proposed that any coupling constant in a covariant action can be treated as a conserved charge by promoting the coupling constant to auxiliary fields, typically realized by a scalar field paired with a higher-form gauge field. However, the procedure may break local symmetries, which can be explicitly shown in a simpler setting such as Maxwell theory. The Hamiltonian analysis of Maxwell theory with the auxiliary fields reveals that some of the constraints are second-class. Applying the BFT formalism, we restore the broken local symmetries and obtain a fully symmetric action defined on an extended configuration space. Despite the restoration of the local symmetries, no additional conserved charges are associated with the recovered symmetries. Consequently, the original theory turns out to be the gauge-fixed version of the extended theory.