The clock ambiguity: Implications and new developments
  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. 
Andreas Albrecht and Alberto Iglesias