Time without time: a stochastic clock model

TL;DR

This paper develops a model where time emerges from a classical, reparametrization-invariant system with a nonrelativistic particle in higher dimensions, leading to quantum evolution at low energies.

Contribution

It introduces a method to define an emergent time from internal degrees of freedom using statistical and information-theoretic tools in a classical system.

Findings

Emergent time is related to proper time on average.

Low-energy external motion follows Schrödinger evolution.

The model demonstrates how classical systems can exhibit quantum behavior.

Abstract

We study a classical reparametrization-invariant system, in which ``time'' is not a priori defined. It consists of a nonrelativistic particle moving in five dimensions, two of which are compactified to form a torus. There, assuming a suitable potential, the internal motion is ergodic or more strongly irregular. We consider quasi-local observables which measure the system's ``change'' in a coarse-grained way. Based on this, we construct a statistical timelike parameter, particularly with the help of maximum entropy method and Fisher-Rao information metric. The emergent reparametrization-invariant ``time'' does not run smoothly but is simply related to the proper time on the average. For sufficiently low energy, the external motion is then described by a unitary quantum mechanical evolution in accordance with the Schr\"odinger equation.