Coarse graining methods for spin net and spin foam models

TL;DR

The paper tackles extracting large-scale physics from spin foam and spin net models by applying real-space coarse-graining methods to finite-group realizations. It develops and compares Migdal–Kadanoff (MK) and Tensor Network Renormalization (TNR) schemes, introducing a Gauß-constraint preserving TNR algorithm and analyzing Abelian cutoff models to probe BF symmetry restoration and fixed-point structure. A central result is the identification of conditions under which BF/diffeomorphism-like translation symmetry re-emerges under renormalization, plus the demonstration of equivalences between certain models within the TNW framework. Overall, the work provides a quantitative bridge between quantum gravity spin foams and statistical-physics renormalization techniques, offering insights into phase structure, symmetry restoration, and the viability of continuum limits for discretized gravity models.

Abstract

We undertake first steps in making a class of discrete models of quantum gravity, spin foams, accessible to a large scale analysis by numerical and computational methods. In particular, we apply Migdal-Kadanoff and Tensor Network Renormalization schemes to spin net and spin foam models based on finite Abelian groups and introduce `cutoff models' to probe the fate of gauge symmetries under various such approximated renormalization group flows. For the Tensor Network Renormalization analysis, a new Gauss constraint preserving algorithm is introduced to improve numerical stability and aid physical interpretation. We also describe the fixed point structure and establish an equivalence of certain models.

Figures (13)

• Figure 1: On the left: the three objects associated to every edge. On the right: their schematic representation. Every straight line joining two objects means a contraction of indices.
• Figure 2: Four-valent vertex with two outgoing edges, $e_1$ and $e_2$, and with two incoming edges, $e_3$ and $e_4$. On the left: representations meeting in the vertex. On the right: schematic representation of the resulting vertex weight.
• Figure 3: Employing the schematic representation of figures \ref{['fig:edge']} and \ref{['fig:vertex']} it is straightforward to realize that the partition function of spin net models can be written in the form of a tensor-trace over a network of tensors, being the tensor-trace the sum over representations.
• Figure 4: Schematic definition of the vertex tensor $\tilde{T}^v_{\{k_e\}_{e\supset v}}$ in a square lattice.
• Figure 5: ($a$) A face $f$ bound by the edges $e_1,\cdots,e_5$. The curvature is $h_f=g_{e_1}g_{e_2}g_{e_3}^{-1}g_{e_4}g_{e_5}^{-1}$. ($b$) An edge $e$ with four faces attached, $f_1$ and $f_4$ with positive relative orientation with respect to $e$, and $f_2$ and $f_3$ with negative relative orientation. The corresponding edge-weight is denoted by $\tilde{P}^e_{a^{f^1}_{v},a^{f^4}_{v},a^{f^2}_{v},a^{f^3}_{v}; a^{f^1}_{\tilde{v}},a^{f^4}_{\tilde{v}},a^{f^2}_{\tilde{v}}, a^{f^3}_{\tilde{v}}}$.