Fundamental decoherence from relational time in discrete quantum gravity: Galilean covariance

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.