# 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.

## Key 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.