Bulk Entropy in Loop Quantum Gravity

TL;DR

This work defines bulk entropy for bounded regions in loop quantum gravity by counting spin-network states on a fixed internal graph with fixed boundary data. After gauge-fixing holonomies on internal loops to avoid divergences, the authors derive finite expressions for the entropy that depend on the boundary size $n$ and the graph loop number $L$. They identify three regimes: a logarithmic regime with fixed $L$, a holographic regime with $L$ proportional to the boundary and a tunable $S/A$ still compatible with an area law, and a non-linear regime where entropy grows faster than linearly with the boundary. The results suggest the LQG dynamics should select the holographic regime to realize holography and allow the area-entropy relation to hold without fixing the Immirzi parameter, highlighting open questions about the gauge-fixing interpretation and its connection to Hamiltonian constraint gauging.

Abstract

In the framework of loop quantum gravity (LQG), having quantum black holes in mind, we generalize the previous boundary state counting (gr-qc/0508085) to a full bulk state counting. After a suitable gauge fixing we are able to compute the bulk entropy of a bounded region (the "black hole") with fixed boundary. This allows us to study the relationship between the entropy and the boundary area in details and we identify a holographic regime of LQG where the leading order of the entropy scales with the area. We show that in this regime we can fine tune the factor between entropy and area without changing the Immirzi parameter.

Figures (6)

• Figure 1: The internal edges are shown as regular lines, the boundary edges are dashed, and the selected exterior edges are rendered in bald.
• Figure 2: One-loop spin networks with the $2n$ boundary links: the internal loop carries the spin $j$.
• Figure 3: Two-loop graphs with boundary links
• Figure 4: Fully gauge-fixed two-loop spin network: we fix the spins carried by the three edges on the loops.
• Figure 5: Gauge-fixed spin networks with $L$ loops: the $L+1$ edges of $\Gamma$ carry the labels $j,j,2j,\ldots,2^{L-1}j$.