Gauging Time Reversal Symmetry in Quantum Gravity: Arrow of Time from a Confinement--Deconfinement Transition

Abstract: The question of the origin of time's arrow is a major outstanding problem in physics. Here we present a mechanism for the emergence of a cosmological arrow of time from a confinement--deconfinement transition in a $ Z_2 $ lattice gauge theory living on the spin-network states of Loop Quantum Gravity. Following Chen and Vishwanath \cite{Chen2015Gauging}, who showed that time-reversal symmetry can be gauged on tensor network states, and using the spin-network/tensor-network correspondence \cite{Qi2013Exact,Han2016Loop}, we introduce a $ Z_2 $ gauge field on spin networks encoding a local time-reversal symmetry. The effective theory of this gauge field contains a confined phase -- corresponding to a pre-geometric ``quantum gravitational foam'' with no coherent arrow of time -- and a deconfined phase -- corresponding to semiclassical spacetime with a uniform cosmological arrow. The emergence of the arrow of time is identified with the confinement--deconfinement transition, detected by the Wilson loop order parameter. The deconfined phase is further shown to correspond to a symmetry-protected topological (SPT) phase of the CZX type, whose topological order provides additional stability of the coherent time orientation against local perturbations. We conjecture that the topologically protected surface excitations of this SPT phase give rise to fermionic matter degrees of freedom.