On state sums, internalisation and unification

TL;DR

This paper reviews higher category theory concepts crucial for developing advanced four-dimensional quantum geometric models, aiming to improve existing state sum models for Quantum Gravity by addressing degeneracies and exploring unification principles.

Contribution

It introduces the notion of internalisation in higher categories to enhance quantum geometric models and suggests a pathway toward a unified theory incorporating spacetime and matter duality.

Findings

Improved quantum gravity models with fewer degeneracies.

Categorical internalisation offers new insights into unification.

Spacetime-matter duality principle guides model development.

Abstract

In this mostly expository article, elements of higher category theory essential to the construction of a class of four dimensional quantum geometric models are reviewed. These models improve current state sum models for Quantum Gravity, such as the Barrett-Crane model, in that they appear, for instance, to remove degeneracies which swamp the partition function. Much work remains to be done before a complete construction is reached, but the crucial categorical notion of internalisation already illuminates the idea that a full unified model may result from few, albeit as yet poorly understood, additional principles. In particular, a spacetime and matter duality principle is employed through an understanding of the role of pseudomonoidal objects in categorified cohomology.