Renormalization Group Approach to the Continuum Limit of Matrix Models of Quantum Gravity With Preferred Foliation

Abstract

This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson [1], which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions.

Editorial: Coarse graining in quantum gravity -- Bridging the gap between microscopic models and spacetime physics

“…A different solution to the coarse-graining problem in gravity is implemented in the discrete, pregeometric setting of matrix models by T. Koslowski and A. Castro. In [52], they apply discrete functional RG techniques to matrix models which encode a preferred foliation. They discover fixed points which are compatible with known results from causal dynamical triangulations in two dimensions and thereby establish the applicability of these techniques in a causal setting.…”

Status of Background-Independent Coarse Graining in Tensor Models for Quantum Gravity

“…Accordingly, a more direct search for a universal continuum limit for Lorentzian models could become possible in this setup. This includes applications of the FRG to tensor models dual to causal dynamical triangulations [73] as in [74], as well as the application of coarse-graining techniques to the link matrix in causal sets [75].…”

The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization

“…In the matrix and tensor model approach, the additional structure imposed on discrete configurations by causality can be implemented through additional degrees of freedom: In Refs. [103,112], this is done by an external matrix, whereas the ABAB model uses two dynamical matrices to generate configurations which carry imprints of causality. This strongly motivates the further development of FRG techniques for multi-matrix/tensor settings, such that similar developments can be made possible in higher dimensions.…”

“…Related developments for non-commutative geometry [110,111] and tensorial group field theories [18][19][20][21][22][23][24][25][26][27][28][29][30] exist. Recently, a first FRG analysis of the above-mentioned causal matrix model for CDT in 1 + 1 dimensions has been completed [112].…”