A new proposal for group field theory I: the 3d case

TL;DR

This paper introduces a new 3D group field theory model aligned with standard quantum field theory principles, enabling causal dynamics and coupling to matter, with implications for quantum gravity research.

Contribution

It proposes a novel 3D GFT model that aligns with conventional QFT, introduces causality, and extends to matter coupling and categorification for 4D.

Findings

The new GFT model exhibits causal dynamics.

Coupling to point matter yields a non-commutative matter field theory.

Categorification insights pave the way for 4D generalizations.

Abstract

In this series of papers, we propose a new rendition of 3d and 4d state sum models based upon the group field theory (GFT) approach to non-perturbative quantum gravity. We will see that the group field theories investigated in the literature to date are, when judged from the position of quantum field theory, an unusual manifestation of quantum dynamics. They are one in which the Hadamard function for the field theory propagates a-causally the physical degrees of freedom of quantum gravity. This is fine if we wish to define a scalar product on the physical state space, but it is not what we generally think of as originating directly from a field theory. We propose a model in 3d more in line with standard quantum field theory, and therefore the field theory precipitates causal dynamics. Thereafter, we couple the model to point matter, and extract from the GFT the effective non-commutative field theory describing the matter dynamics on a quantum gravity background. We identify the symmetries of our new model and clarify their meaning in the GFT setting. We are aided in this process by identifying the category theory foundations of this GFT which, moreover, propel us towards a categorified version for the 4d case.