Finiteness and Dual Variables for Lorentzian Spin Foam Models

TL;DR

This paper proves the finiteness of the Lorentzian Barrett-Crane spin foam model's partition function for various triangulations and introduces a dual variables formulation that could facilitate numerical computations in quantum gravity.

Contribution

It extends finiteness proofs for the Lorentzian Barrett-Crane model and introduces a dual variables approach using hyperboloid variables instead of face representations.

Findings

Finiteness of the partition function for all non-degenerate 4-manifold triangulations.

A new dual variables formulation using hyperboloid variables.

Potential for improved numerical computation methods.

Abstract

We describe here some new results concerning the Lorentzian Barrett-Crane model, a well-known spin foam formulation of quantum gravity. Generalizing an existing finiteness result, we provide a concise proof of finiteness of the partition function associated to all non-degenerate triangulations of 4-manifolds and for a class of degenerate triangulations not previously shown. This is accomplished by a suitable re-factoring and re-ordering of integration, through which a large set of variables can be eliminated. The resulting formulation can be interpreted as a ``dual variables'' model that uses hyperboloid variables associated to spin foam edges in place of representation variables associated to faces. We outline how this method may also be useful for numerical computations, which have so far proven to be very challenging for Lorentzian spin foam models.