  It is often conjectured that a choice of time function merely sets up a frame
for the quantum evolution of gravitational field, meaning that all choices
should be in some sense compatible. In order to explore this conjecture (and
the meaning of compatibility), we develop suitable tools for determining the
relation between quantum theories based on different time functions. First, we
discuss how a time function fixes a canonical structure on the constraint
surface. The presentation includes both the kinematical and the reduced
perspective, and the relation between them. Second, we formulate twin theorems
about the existence of two inequivalent maps between any two
deparameterizations, a {\it formal canonical} and a {\it coordinate} one. They
are used to separate the effect of choice of clock from other effects. We show,
in an example, how the spectra of quantum observables are transformed under the
change of clock and prove, via a general argument, the existence of
choice-of-time-induced semiclassical effects. Finally, we study an example, in
which we find that the semiclassical discrepancies can in fact be arbitrarily
large for dynamical observables. We conclude that the values of critical energy
density or critical volume in the bouncing scenarios of quantum cosmology
cannot in general be at the Planck scale and always need to be given with
reference to a specific time function.
