Revisiting the relaxation of constraints in gauge theories

TL;DR

This paper clarifies misconceptions about gauge theory quantization, showing that relaxing constraints in the path integral is a natural consequence of gauge fixing at the Lagrangian level, with parallels in Hamiltonian methods.

Contribution

It demonstrates that constraint relaxation in gauge theories arises naturally during gauge fixing in the Lagrangian formalism, correcting prior misconceptions and drawing analogies with Hamiltonian approaches.

Findings

Constraint relaxation occurs during gauge fixing at the Lagrangian level.

Path integral quantization does not require relaxing constraints.

Analogies between Lagrangian gauge fixing and Hamiltonian second class systems.

Abstract

Recently, there were works claiming that path integral quantisation of gauge theories necessarily requires relaxation of Lagrangian constraints. As has also been noted in the literature, it is of course wrong since there perfectly exist gauge field quantisations respecting the constraints, and at the same time the very idea of changing the classical theory in this way has many times appeared in other works. On the other hand, what was done in the path integral approach is fixing a gauge in terms of zero-momentum variables. We would like to show that this relaxation is what normally happens when one fixes such a gauge at the level of action principle in a Lagrangian theory. Moreover, there is an interesting analogy to be drawn. Namely, one of the ways to quantise a gauge theory is to build an extended Hamiltonian and then add new conditions by hand such as to make it a second class system. The constraints' relaxation occurs when one does the same at the level of the total Hamiltonian, i.e. a second class system with the primary constraints only.