Phase transitions in group field theory: The Landau perspective

TL;DR

This paper investigates phase transitions in group field theory models related to quantum gravity, finding that the Gaussian approximation breaks down for compact groups but remains valid for noncompact groups, suggesting noncompact domains may be essential for phase transitions.

Contribution

It provides a critical analysis of the Gaussian approximation in different GFT models, highlighting the importance of noncompact domains for phase transition phenomena in quantum gravity.

Findings

Gaussian approximation breaks down for 3D Euclidean GFT on SU(2)

Gaussian approximation remains valid for Lorentzian GFT on SL(2,R)

Noncompact domains may be necessary for phase transitions in GFT

Abstract

In various approaches to quantum gravity continuum spacetime is expected to emerge from discrete geometries through a phase transition. In group field theory, various indications for such a transition have recently been found but a complete understanding of such a phenomenon remains an open issue. In this work, we investigate the critical behavior of different group field theory models in the Gaussian approximation. Applying the Ginzburg criterion to quantify field fluctuations, we find that this approximation breaks down in the case of three-dimensional Euclidean quantum gravity as described by the dynamical Boulatov model on the compact group $\text{SU}(2)$. This result is independent of the peculiar gauge symmetry and specific form of nonlocality of the model. On the contrary, we find that the Gaussian approximation is valid for a rank-$1$ GFT on the noncompact sector of fields on $\text{SL}(2,\mathbb{R})$ related to Lorentzian models. Though a nonperturbative analysis is needed to settle the question of phase transitions for compact groups, the results may also indicate the necessity to consider group field theory on noncompact domains for phase transitions to occur.