Fundamental theorem on gauge fixing at the action level

TL;DR

This paper establishes a rigorous condition under which gauge fixing at the action level is valid, proving that complete gauge fixing ensures equivalence of equations, and clarifies misconceptions in modified gravity theories.

Contribution

It provides a fundamental theorem linking gauge fixing completeness to the validity of deriving equations from the gauge-fixed action.

Findings

Complete gauge fixing guarantees equivalence of equations.

Incomplete gauge fixing can lead to inconsistencies.

Application to scalar-tensor theories clarifies gauge fixing issues.

Abstract

Regardless of the long history of gauge theories, it is not well recognized under which condition gauge fixing at the action level is legitimate. We address this issue from the Lagrangian point of view, and prove the following theorem on the relation between gauge fixing and Euler-Lagrange equations: In any gauge theory, if a gauge fixing is complete, i.e., the gauge functions are determined uniquely by the gauge conditions, the Euler-Lagrange equations derived from the gauge-fixed action are equivalent to those derived from the original action supplemented with the gauge conditions. Otherwise, it is not appropriate to impose the gauge conditions before deriving Euler-Lagrange equations as it may in general lead to inconsistent results. The criterion to check whether a gauge fixing is complete or not is further investigated. We also provide applications of the theorem to scalar-tensor theories and make comments on recent relevant papers on theories of modified gravity, in which there are confusions on gauge fixing and counting physical degrees of freedom.