Remarks on gauge invariance and first-class constraints

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 Hořava – in which the Hamiltonian constraint is deformed by arbitrary spatial diffeomorphism invariant terms, where some subtleties are found to arise.