The Ponzano-Regge cylinder and Propagator for 3d quantum gravity

TL;DR

This paper studies the propagator in 3d quantum gravity using a discrete path integral approach, analyzing its properties and potential for quantum simulation.

Contribution

It introduces a new formulation of the 3d quantum gravity propagator as a Ponzano-Regge amplitude on a solid cylinder, with analysis of its eigen-modes and boundary effects.

Findings

The propagator distinguishes subspaces with different total spins.

It may select the zero total spin sector at late times.

Potential applications to quantum circuits and experimental simulations.

Abstract

We investigate the propagator of 3d quantum gravity, formulated as a discrete topological path integral. We define it as the Ponzano-Regge amplitude of the solid cylinder swept by a 2d disk evolving in time. Quantum states for a 2d disk live in the tensor products of N spins, where N is the number of holonomy insertions connecting to the disk boundary. We formulate the cylindric amplitude in terms of a transfer matrix and identify its eigen-modes in terms of spin recoupling. We show that the propagator distinguishes the subspaces with different total spin and may select the vanishing total spin sector at late time depending on the chosen cylinder boundary data. We discuss applications to quantum circuits and the possibility of experimental simulations of this 3d quantum gravity propagator.