The “Problem of Time” in time-parametrized theories appears at the quantum level in various ways, depending on the quantization procedure. In (Dirac) canonical quantization, the usual Schrodinger evolution equation is replaced by a constraint on the physical states HΨ[q] = 0; such states depend on the generalized coordinates {qα}, but not on any additional time evolution parameter t. In path-integral formulation, one integrates over all paths, extending over all possible proper time lapses, which connect the initial and final configurations. The resulting Green's function G[qf,q0], like the physical state wavefunctions Φ[qα], has no dependence on an extra time parameter t. In certain theories, e.g. parametrized non-relativistic quantum mechanics, or the case of a relativistic particle moving in flat Minkowski space, it is possible to identify one of the generalized coordinates as the time variable, and to associate with that variable a conserved and positive-definite probability measure. In other theories, such as a relativistic particle moving in an arbitrary curved background spacetime, or in the case of quantum gravity, it has proven very difficult to identify an appropriate evolution parameter, and a unique, positive, and conserved probability measure. This is the “Problem of Time”, reviewed in ref. [1].

Our proposal for resolving this problem begins with the rather trivial observation that, at the classical level, there is no difference between the action S and the action S′ = const, × S; these actions obviously have the same Euler- Lagrange equations.