Clock ambiguity and the emergence of physical laws

Abstract

The process of identifying a time variable in time reparameterization invariant theories results in great ambiguities about the actual laws of physics described by a given theory. A theory set up to describe one set of physical laws can equally well be interpreted as describing any other laws of physics by making a different choice of time variable or "clock". In this article we demonstrate how this "clock ambiguity" arises and then discuss how one might still hope to extract specific predictions about the laws of physics even when the clock ambiguity is present. We argue that a requirement of quasi-separability should play a critical role in such an analysis. As a step in this direction, we compare the Hamiltonian of a local quantum field theory with a completely random Hamiltonian. We find that any random Hamiltonian (constructed in a sufficiently large space) can yield a "good enough" approximation to a local field theory. Based on this result we argue that theories that suffer from the clock ambiguity may in the end provide a viable fundamental framework for physics in which locality can be seen as a strongly favored (or predicted) emergent behavior. We also speculate on how other key aspects of known physics such as gauge symmetries and Poincare invariance might be predicted to emerge in this framework.

Lorentz symmetric dispersion relation from a random Hamiltonian

“…We posed this question in earlier work [1,2] from the point of view of a comparison of the energy eigenvalue spectrum, and found an affirmative answer. In that work we considered only field theories with a dispersion relation given by Lorentz symmetry.…”

“…In [1] we addressed that question at the level of the spectrum of energy eigenvalues. We found that the spectrum of a sufficiently large random Hamiltonian could approximate that of a free field theory over a sufficiently large range of energies to allow the former to be approximated by the latter on scales relevant to our data about the physical world.…”

“…In [1] we considered specific values for the parameters in Eqn. 7 based on properties of the observed physical world and found that ∆ 2 is small for realistic values.…”

“…In this work we extend our previous analysis to include a one parameter family of possible dispersion relations. We also include the dimension of space d (as was already analyzed in [2]). Our analysis shows that choosing a Lorentz invariant dispersion relation makes it easier to approximate the field theory density of states by that of a random Hamiltonian.…”

Note on the existence theorem in ‘emergent quantum mechanics and emergent symmetries’

“…The view that a random Hamiltonian might serve as a starting point to explain emergent physical laws has recently been expressed in Ref. [16] (see also further references therein), studying consequences of the clock ambiguity in time reparametrization invariant theories.…”