Discrete Time from Quantum Physics

TL;DR

This paper explores models where classical physics with smooth spacetime structures leads to inherently discrete quantum time evolution due to multiple-valued Hamiltonians, illustrating how quantization can alter classical topology.

Contribution

It introduces models with single-valued classical equations but multiple-valued Hamiltonians, demonstrating discrete quantum time steps and the impact of quantization on topology.

Findings

Quantum time evolution can be inherently discrete in models with multiple-valued Hamiltonians.

Quantization can change classical topological properties of spacetime.

Classical models with smooth manifolds can exhibit discrete quantum displacements.

Abstract

't Hooft has recently developed a discretisation of (2+1) gravity which has a multiple-valued Hamiltonian and which therefore admits quantum time evolution only in discrete steps. In this paper, we describe several models in the continuum with single-valued equations of motion in classical physics, but with multiple-valued Hamiltonians. Their time displacements in quantum theory are therefore obliged to be discrete. Classical models on smooth spatial manifolds are also constructed with the property that spatial displacements can be implemented only in discrete steps in quantum theory. All these models show that quantization can profoundly affect classical topology.