  We consider the ambiguity associated with the choice of clock in time
reparameterization invariant theories. This arbitrariness undermines the goal
of prescribing a fixed set of physical laws, since a change of time variable
can completely alter the predictions of the theory. We review the main features
of the clock ambiguity and our earlier work on its implications for the
emergence of physical laws in a statistical manner. We also present a number of
new results: We show that (contrary to suggestions in our earlier work) time
independent Hamiltonians may quite generally be assumed for laws of physics
that emerge in this picture. We also further explore the degree to which the
observed Universe can be well approximated by a random Hamiltonian. We discuss
the possibility of predicting the dimensionality of space, and also relate the
2nd derivative of the density of states to the heat capacity of the Universe.
This new work adds to the viability of our proposal that strong predictions for
physical laws may emerge based on statistical arguments despite the clock
ambiguity, but many open questions remain.
