Relational time in anyonic systems

Abstract

In a seminal paper [Phys. Rev. D 27, 2885 (1983)], Page and Wootters suggest that 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 seeks to identify relational dynamics given a Hamiltonian constraint on the physical states. Here we present a ``state-centric'' reformulation of the Page and Wootters model better suited to cases where the Hamiltonian constraint is satisfied, such as anyons emerging in Chern--Simons theories. We describe relational time by encoding logical ``clock'' qubits into topologically protected anyonic degrees of freedom. The minimum temporal increment of such anyonic clocks is determined by the universality of the anyonic braid group, with nonuniversal models naturally exhibiting discrete time. We exemplify this approach by using $\mathrm{SU}{(2)}_{2}$ anyons and discuss generalizations to other states and models.