Abstract
We compute the entropy of a closed bounded region of space for pure 3d Riemannian gravity formulated as a topological BF theory for the gauge group SU ( 2 ) and show its holographic behavior. More precisely, we consider a fixed graph embedded in space and study the flat connection spin network state without and with particle-like topological defects. We regularize and compute exactly the entanglement for a bipartite splitting of the graph and show it scales at leading order with the number of vertices on the boundary (or equivalently with the number of loops crossing the boundary). More generally these results apply to BF theory with any compact gauge group in any space–time dimension.
Highlights
Entropy is a key notion in the search for a quantum gravity, both as the thermodynamic quantity useful to probe the physics and potential phenomenology of the theory and as the measure of information useful to identify the physical degrees of freedom and their dynamics
We computed the entropy for a bounded region on a physical state of BF theory
Being solvable and lacking local degrees of freedom, BF theory allows for explicit calculations and a precise analysis of the relationship between boundaries and degrees of freedom
Summary
Entropy is a key notion in the search for a quantum gravity, both as the thermodynamic quantity useful to probe the physics and potential phenomenology of the theory and as the measure of information useful to identify the physical degrees of freedom and their dynamics. The aim is to compute the entanglement between these two parts of the spin network for a physical quantum geometry state solving the Hamiltonian constraint. BF theories are topological field theories lacking local degrees of freedom, allowing for an exact quantization (see e.g [1]). In this context, we know the physical states solving all the constraints and can compute the entanglement explicitly. Focusing on SU(2) BF theory, we explicitly compute the entanglement between the two parts of such a flat spin network states and we show its holographic behavior: it scales with the size of the boundary (more precisely, with the number of boundary vertices). We show that they do only affect the entropy when located on the boundary between the two regions and we compute the finite variation of entanglement that they create
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