Quantum Mechanics and Discrete Time from "Timeless" Classical Dynamics

TL;DR

This paper explores how classical Hamiltonian systems with a discrete physical time, derived from timeless models, can be reformulated as quantum models with emergent Hamiltonians related to the Liouville operator.

Contribution

It introduces a method to derive quantum models from classical systems with discrete time using path-integral formulation, highlighting the emergence of quantum behavior from classical dynamics.

Findings

Classical systems can be described as unitary quantum models.

Emergent quantum Hamiltonians are related to the Liouville operator.

Regularization is necessary for bounded spectra and stability.

Abstract

We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. - This is motivated by the ``timeless'' reparametrization invariant model of a relativistic particle with two compactified extradimensions. In this example, discrete physical time is constructed based on quasi-local observables. - Generally, employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. The emergent quantum Hamiltonian is derived from the underlying classical one. It is closely related to the Liouville operator. We demonstrate in several examples the necessity of regularization, in order to arrive at quantum models with bounded spectrum and stable groundstate.