Deformation Operators of Spin Networks and Coarse-Graining

TL;DR

The paper develops a coarse-graining framework for spin networks in loop quantum gravity by gauge-fixing a bounded region to a single vertex, revealing a residual closure defect that encodes bulk curvature. It extends the SU(2) deformation/observables from a single vertex to boundary data of regions, preserving a universal U(N) structure on the boundary and introducing SL(2,C) boosts that map boundary data to closure-satisfying configurations. A Lorentz-invariant rest area emerges as a Casimir and potential observable with links to black hole physics, demonstrated concretely in a one-loop example. The work discusses the physical meaning of the closure defect, the tree-dependence of coarse-graining, and potential extensions to tagged/loopy spin networks and twistor formalisms, with implications for holography and quantum gravity dynamics.

Abstract

In the context of loop quantum gravity, quantum states of geometry are mathematically defined as spin networks living on graphs embedded in the canonical space-like hypersurface. In the effort to study the renormalisation flow of loop gravity, a necessary step is to understand the coarse-graining of these states in order to describe their relevant structure at various scales. Using the spinor network formalism to describe the phase space of loop gravity on a given graph, we focus on a bounded (connected) region of the graph and coarse-grain it to a single vertex using a gauge-fixing procedure. We discuss the ambiguities in the gauge-fixing procedure and their consequences for coarse-graining spin(or) networks. This allows to define the boundary deformations of that region in a gauge-invariant fashion and to identify the area preserving deformations as U(N) transformations similarly to the already well-studied case of a single intertwiner. The novelty is that the closure constraint is now relaxed and the closure defect interpreted as a local measure of the curvature inside the coarse-grained region. It is nevertheless possible to cancel the closure defect by a Lorentz boost. We further identify a Lorentz-invariant observable related to the area and closure defect, which we name "rest area". Its physical meaning remains an open issue.

Figures (6)

• Figure 1: A bounded region of a spin network, with its internal graph and its boundary edges labeled $i=1..N$.
• Figure 2: The gauge-fixing of the internal graph to a flower graph: starting with $V=4$ internal vertices linked with $E=6$ internal edges, we choose a reference vertex $v_{0}$ and a tree $T$ in bold; we obtain a flower with $E-V+1=3$ petals and the original 4 boundary edges after contracting the internal graph and setting the holonomies to $\mathbb{I}$ along the tree $T$.
• Figure 3: An elementary change of tree: starting with the tree $T$ going through the vertex $v_{0}$ with $e\notin T$ and $f\in T$, we exchange the roles of the edges $e$ and $f$ to get a new tree $U$ with $e\in U$ and $f\notin U$. The edges belonging to the trees are drawn in bold. The loop holonomy ${\mathcal{G}}_{e}^{T}$ associated to the edge $e\notin T$ and defined by gauge-fixing along the tree $T$ is equal to the loop holonomy ${\mathcal{G}}_{f}^{U}$ associated to the edge $f\notin U$ by gauge-fixing along the tree $U$(up to potentially taking its inverse, depending on the relative orientations of the edges $e$ and $f$ along the loop). It is simply the oriented product of the holonomies along the edges in the loop drawn above.
• Figure 4: The candy graph consists in $V=2$ internal vertices linked by $E=2$ internal edges, with $N=4$ boundary edges. It gets gauge-fixed to a four-valent vertex with one self-loop. 4-valent intertwiners are very interesting since they define dual tetrahedra. Here the internal loop will lead to curvature inside the dual tetrahedron and to an effective closure defect.
• Figure 5: One can simplify the candy graph down to the simplest case with only $N=2$ external edges. This corresponds to a single edge with one loop, similar to the one-loop correction to the propagator in standard quantum field theory. Here we gauge-fix the internal graph choosing the upper edge $\alpha$ as the tree. This leads to one self-loop carrying the holonomy ${\mathcal{G}}=g^-1\tilde{g}$ and two boundary spinors $Z^\alpha_{1}$ and $Z^\alpha_{2}$.