A Mermin–Wagner theorem on Lorentzian triangulations with quantum spins

Abstract

We consider infinite random causal 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 the 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.