Renormalization of Discrete Models without Background

TL;DR

This paper generalizes renormalization methods to discretized models without a background metric, using cellular decompositions and a renormalization groupoid, with applications to quantum gravity models.

Contribution

It introduces a novel renormalization framework for background-independent models using cellular moves and groupoids, extending traditional methods.

Findings

Quantum BF theory is nearly topological with trivial renormalization.

Generalized lattice gauge theory shows a clear renormalization flow.

Quantum BF theory acts as the UV fixed point in studied models.

Abstract

Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The group of scale transformations is replaced by the groupoid of changes of cellular decompositions. We introduce cellular moves which generate this groupoid and allow to define a renormalization groupoid flow. We proceed to test our approach on several models. Quantum BF theory is the simplest example as it is almost topological and the renormalization almost trivial. More interesting is generalized lattice gauge theory for which a qualitative picture of the renormalization groupoid flow can be given. This is confirmed by the exact renormalization in dimension two. A main motivation for our approach are discrete models of quantum gravity. We investigate both the Reisenberger and the Barrett-Crane spin foam model in view of their amenability to a renormalization treatment. In the second case a lack of tunable local parameters prompts us to introduce a new model. For the Reisenberger and the new model we discuss qualitative aspects of the renormalization groupoid flow. In both cases quantum BF theory is the UV fixed point.