Spin Foam Models of n-dimensional Quantum Gravity and Non-Archimedean and Non-Commutative Formulations

TL;DR

This paper unifies n-dimensional quantum gravity spin foam models, extends evaluations to Lorentzian and Newtonian limits, and introduces non-archimedean and non-commutative formulations based on discrete spacetime structures.

Contribution

It proposes a comprehensive framework for evaluating spin networks in various quantum gravity models, including Lorentzian, Newtonian, and p-adic formulations, with potential regularization benefits.

Findings

Unified evaluation method for spin networks in n-dimensional quantum gravity.

Extension of spin foam models to Lorentzian and Newtonian limits.

Introduction of non-archimedean and non-commutative formulations based on discrete spacetime.

Abstract

This paper is twofold. First of all a complete unified picture of $n$-dimensional quantum gravity is proposed in the following sense: In spin foam models of quantum gravity the evaluation of spin networks play a very important role. These evaluations correspond to amplitudes which contribute in a state sum model of quantum gravity. In \cite{fk}, the evaluation of spin networks as integrals over internal spaces was described. This evaluation was restricted to evaluations of spin networks in $n$-dimensional Euclidean quantum gravity. Here we propose that a similar method can be considered to include Lorentzian quantum gravity. We therefore describe the the evaluation of spin networks in the Lorentzian framework of spin foam models. We also include a limit of the Euclidean and Lorentzian spin foam models which we call Newtonian. This Newtonian limit was also considered in \cite{jm}. Secondly, we propose an alternative formulation of spin foam models of quantum gravity with its corresponding evaluation of spin networks. This alternative formulation is a non-archimedean or $p$-adic spin foam model. The interest on this description is that it is based on a discrete space-time, which is the expected situation we might have at the Planck length; this description might lead us to an alternative regularisation of quantum gravity. Moreover a non-commutative formulation follows from the non-archimedean one.