Abstract
At criticality, discrete quantum-gravity models are expected to give rise to continuum spacetime. Recent progress has established the functional renormalization group method in the context of such models as a practical tool to study their critical properties and to chart their phase diagrams. Here, we apply these techniques to the multi-matrix model with ABAB interaction potentially relevant for Lorentzian quantum gravity in 3 dimensions. We characterize the fixed-point structure and phase diagram of this model, paving the way for functional RG studies of more general multi-matrix or tensor models encoding causality and subjecting the technique to another strong test of its performance in discrete quantum gravity by comparing to known results.
Highlights
JHEP12(2020)131 where the integration runs over all field histories, given by spacetime metrics g of the d-dimensional manifold M up to diffeomorphisms thereof at fixed spacetime topology
A much used method pioneered for gravity by Reuter [43] is provided by the functional Renormalization Group (FRG) [44, 45], see [46] for a review
It is argued that these pathological configurations are the result of topology change leading to spaces called branched polymers which are built from one-dimensional branched-out filaments [82, 84, 89, 90]
Summary
Spacetime is rather distinct from space, both at the conceptual as well as mathematical level. In the t + a/2 spatial planes, quadrangulations are formed as, e.g., in figure 3, the duals of which are the ribbon graphs of the matrix model This already suggests that any CDT configuration can be encoded in terms of a Feynman diagram of the matrix model, which would motivate setting the CDT partition function equal to the free energy of the matrix model, as usual in the correspondence between triangulations and matrix models. Such pathologies of the local geometry are disallowed by construction in CDT [11] Despite these differences, the similarities between the ABAB matrix model and CDTs reinforce the more general notion that causality can be imposed on matrix and tensor models by enlarging the field content of the model and introducing a second matrix/tensor that enables a distinction of spacelike and timelike edges in the dual triangulation. This will pave the way for future studies of multi-field models that encode causality in the interaction structure
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