Emergent non-commutative matter fields from Group Field Theory models of quantum spacetime

TL;DR

The paper addresses how continuum spacetime and matter fields can emerge from a fundamental quantum spacetime described by Group Field Theories (GFTs). By treating GFT as a many-quanta system and applying a condensed-matter perspective, the authors derive effective non-commutative field theories for matter on Lie-algebra type spacetimes, using a non-commutative Fourier transform to relate group momentum to algebra coordinates. They present explicit constructions in 3d and 4d: in 3d, a NC scalar theory lives on $\mathfrak{su}(2)$ with a Majid Fourier transform, while in 4d, the momentum space on $AN_3$ yields a κ-Minkowski/DSR framework, culminating in a κ-Poincaré-invariant scalar field theory with interactions after restricting to the appropriate sector. This work bridges spin foam/LQG formalisms with non-commutative geometry and quantum gravity phenomenology, offering a concrete mechanism to connect microscopic, discrete quantum spacetime to effective continuum gravity and matter dynamics, and outlining a program for further renormalization and phenomenological analysis.

Abstract

We offer a perspective on some recent results obtained in the context of the group field theory approach to quantum gravity, on top of reviewing them briefly. These concern a natural mechanism for the emergence of non-commutative field theories for matter directly from the GFT action, in both 3 and 4 dimensions and in both Riemannian and Lorentzian signatures. As such they represent an important step, we argue, in bridging the gap between a quantum, discrete picture of a pre-geometric spacetime and the effective continuum geometric physics of gravity and matter, using ideas and tools from field theory and condensed matter analog gravity models, applied directly at the GFT level.