Relational time in anyonic systems

TL;DR

This paper introduces a novel approach to relational time in quantum systems by encoding logical clocks into anyons within Chern--Simons theories, enabling a Hamiltonian-independent description of time evolution.

Contribution

It reformulates the Page and Wootters model using anyonic qubits, applicable to theories without Hamiltonian dynamics, and explores how braiding properties determine timing resolution.

Findings

Relational time can be encoded using anyonic states in topological quantum field theories.

The timing resolution depends on the universality of the anyonic braid group.

Discreteness of time emerges naturally in non-universal anyonic models.

Abstract

In a seminal paper (Page and Wootters 1983) Page and Wootters suggest time evolution could be described solely in terms of correlations between systems and clocks, as a means of dealing with the "problem of time" stemming from vanishing Hamiltonian dynamics in many theories of quantum gravity. Their approach to relational time centres around the existence of a Hamiltonian and the subsequent constraint on physical states. In this paper we present a "state-centric" reformulation of the Page and Wootters model better suited to theories which intrinsically lack Hamiltonian dynamics, such as Chern--Simons theories. We describe relational time by encoding logical "clock" qubits into anyons---the topologically protected degrees of freedom in Chern--Simons theories. The timing resolution of such anyonic clocks is determined by the universality of the anyonic braid group, with non-universal models naturally exhibiting discrete time. We exemplify this approach using SU(2)$_2$ anyons and discuss generalizations to other states and models.

Figures