Remarks on Gauge Invariance and First-Class Constraints

TL;DR

This paper discusses the relationship between gauge invariance and first-class constraints, emphasizing that non-trivial gauge symmetries act non-trivially on dynamical variables, with a focus on modifications of gravity involving time reparametrization invariance.

Contribution

It clarifies the conditions under which gauge symmetries lead to first-class constraints, especially in the context of modified gravity theories with deformed Hamiltonian constraints.

Findings

Gauge symmetries act non-trivially on-shell for non-trivial cases.

Subtleties arise in modified gravity models with deformed Hamiltonian constraints.

The relationship between gauge invariance and constraints is nuanced in these models.

Abstract

Gauge symmetries lead to first-class constraints. This assertion is of course true only for non trivial gauge symmetries, i.e., gauge symmetries that act non trivially on-shell on the dynamical variables. We illustrate this well-appreciated fact for time reparametrization invariance in the context of modifications of gravity -suggested in a recent proposal by Horava-in which the Hamiltonian constraint is deformed by arbitrary spatial diffeomorphism invariant terms, where some subtleties are found to arise.