9.2 Entanglement entropy in loop quantum gravity

Another approach to quantum gravity, sometimes considered as competing with string theory, is loop quantum gravity. In this theory one considers polymeric excitations of the gravitational field represented by the states of spin networks. A spin network is a graph, a network of points with links representing the relation between points. Each link is labeled by a half-integer j (the label stands for SU (2) representations). To points, or vertices, of a spin network are attached a SU (2) intertwiner, a SU (2) invariant tensor between the representations attached to all the edges linked to the considered vertex. A simpler and more familiar object in particle physics is the Wilson loop. A surface Σ is represented by vertices (punctures), which divide the spin network into two parts. By tracing over states of just one part of the network, one obtains a density matrix. The entanglement entropy then reduces to a sum over intersections of the spin network with the surface Σ [56, 165, 66],
∑P S(Σ ) = ln (2jp + 1) , (333 ) p=1
where P is the number of punctures representing Σ. This quantity should be compared to eigenvalues of the operator of area,
∑P ∘---------- A (Σ) = 8πG γ jp(jp + 1). (334 ) p=1
Both quantities scale as P for large P, which indicates that the area law is correctly reproduced. However, the exact relation between the two quantities and the classical entropy SBH = A (Σ )∕G is not obvious due to ambiguities present in the formalism. The Immirzi parameter γ is one of them. The question, whether the Bekenstein–Hawking entropy is correctly reproduced in this approach, is eventually related to the continuum limit of the theory [143Jump To The Next Citation Point]. As discussed by Jacobson [143], answering this question may require a certain renormalization of Newton’s constant as well as area renormalization. Indeed, quantity (334View Equation) represents a microscopic area, which may be related to the macroscopic quantity in a non-trivial way. These issues remain open.
  Go to previous page Go up Go to next page