Schrödinger Evolution for the Universe: Reparametrization

We develop a new framework for constructing the quantum observables of theories invariant under global time reparametrizations. Our starting point is a generalised Hamilton-Jacobi formalism for totally constrained systems. Our end result is a quantum formalism which simultaneously preserves: i) genuine evolution; and ii) the constraint structure of the classical formalism in the semi-classical limit. We thus offer a novel resolution of the 'problem of time' for theories invariant under global time reparametrization. There is the potential for the application of our proposal to the quantization of gravity when understood in terms of the Shape Dynamics formalism.