  We have recently argued that if one introduces a relational time in quantum
mechanics and quantum gravity, the resulting quantum theory is such that pure
states evolve into mixed states. The rate at which states decohere depends on
the energy of the states. There is therefore the question of how this can be
reconciled with Galilean invariance. More generally, since the relational
description is based on objects that are not Dirac observables, the issue of
covariance is of importance in the formalism as a whole. In this note we work
out an explicit example of a totally constrained, generally covariant system of
non-relativistic particles that shows that the formula for the relational
conditional probability is a Galilean scalar and therefore the decoherence rate
is invariant.
