The ``evolving constants'' method of defining the quantum dynamics of
time-reparametrization-invariant theories is investigated for a particular
implementation of parametrized non-relativistic quantum mechanics (PNRQM). The
wide range of time functions that are available to define evolving constants
raises issues of interpretation, consistency, and the degree to which the
resulting quantum theory coincides with, or generalizes, the usual
non-relativistic theory. The allowed time functions must be restricted for the
predictions of PNRQM to coincide with those of usual quantum theory. They must
be restricted to have a notion of quantum evolution in a time-parameter
connected to spacetime geometry. They must be restricted to prevent the theory
from making inconsistent predictions for the probabilities of histories.
Suitable restrictions can be introduced in PNRQM but these seem unlikely to
apply to a reparametrization invariant theory like general relativity.Comment: 18pages, 1postscript figure in separate file, uses REVTEX 3.