Renormalization of Discrete Models without Background
Robert Oeckl
TL;DR
This work develops a background-free renormalization framework by replacing global scale transformations with a renormalization groupoid acting on cellular decompositions, enabling changes of discretization to be treated as renormalization steps. Using circuit diagrams, delta and heat-kernel identities, and local parameter actions, the authors analyze quantum BF theory, generalized lattice gauge theory, and spin-foam gravity models, revealing fixed points and qualitative flows: BF theory acts as a UV fixed point while certain non-topological models flow toward an IR regime with effective trivial representations. Exact renormalization is achieved for BF theory and, in dimension two, for cellular gauge theory, with a clear interpretation of parameters as local metric remnants and a principled method to absorb universal κ-factors arising from moves. The Reisenberger and interpolating spin-foam models illustrate how renormalization concepts extend to quantum gravity contexts, highlighting the essential need for local tunable parameters to realize groupoid-fixed points and exposing the limitations of Barrett–Crane’s move-invariance. Overall, the framework provides a principled pathway to study renormalization in background-free discretizations and offers insights into the potential renormalization behavior of quantum gravity spin-foam 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.