A Mermin--Wagner theorem on Lorentzian triangulations with quantum spins

TL;DR

This paper proves a Mermin--Wagner theorem for quantum spins on infinite Lorentzian triangulations, showing the absence of spontaneous symmetry breaking in this quantum gravity-inspired model.

Contribution

It establishes a Mermin--Wagner type result for quantum particles on random Lorentzian triangulations, extending classical symmetry results to a quantum gravity context.

Findings

Proves a Mermin--Wagner theorem for the model

Shows absence of spontaneous symmetry breaking

Extends quantum statistical mechanics results to random geometries

Abstract

We consider infinite random casual Lorentzian triangulations emerging in quantum gravity for critical values of parameters. With each vertex of the triangulation we associate a Hilbert space representing a bosonic particle moving in accordance with standard laws of Quantum Mechanics. The particles interact via two-body potentials decaying with the graph distance. A Mermin--Wagner type theorem is proven for infinite-volume reduced density matrices related to solutions to DLR equations in the Feynman--Kac (FK) representation.