A Lorentzian Signature Model for Quantum General Relativity

TL;DR

This paper introduces a relativistic spin network model for quantum gravity using the Lorentz group and its q-deformation, extending previous models and providing new evaluation techniques for relativistic spin networks.

Contribution

It develops a Lorentzian spin network model for quantum gravity based on the Quantum Lorentz Algebra, generalizing earlier rotation group models and providing evaluation methods.

Findings

Evaluation of relativistic spin networks is finite in interesting cases

Proposed a combinatorial path integral model for quantum gravity

Conjecture that the '10J' symbol has a finite value

Abstract

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value.