Loop quantum gravity, exact holographic mapping, and holographic entanglement entropy

摘要

The relation between loop quantum gravity (LQG) and tensor networks is explored from the perspectives of the bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin- network states in a space Σ with boundary ∂Σ is an exact holographic mapping similar to the proposal in [X.-L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282]. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse-graining procedure of spin networks. Furthermore, when a region A and its complement A are specified on the boundary ∂Σ, we show that the boundary entanglement entropy S(A) of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e., S(A) is proportional to the minimal area of the bulk surface attached to the boundary of A in ∂Σ.