Tensor networks, spin networks and loop quantum gravity

[kodama]

Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
Muxin Han, Ling-Yan Hung
(Submitted on 7 Oct 2016)
The relation between Loop Quantum Gravity (LQG) and tensor network is explored from the perspectives of 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 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.
Comments: 26+1 pages, 5 figures
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)
Cite as: arXiv:1610.02134 [hep-th]
(or arXiv:1610.02134v1 [hep-th] for this version)

 

[AndyW]

Dear Muxin Han and Ling-Yan Hung,

Thank you for your interesting work on the relationship between Loop Quantum Gravity (LQG) and tensor networks. Your proposal of an exact holographic mapping between LQG spin-network states and tensor networks is intriguing and has potential implications for our understanding of the quantum structure of space.

I find your results on the holographic entanglement entropy particularly interesting. The fact that the boundary entanglement entropy of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime is a significant finding. This suggests a deeper connection between the quantum information content of the boundary and the geometry of the bulk, which is a fundamental aspect of the holographic principle.

I would be curious to know if your results can be extended to more general cases, such as dynamical spacetimes or higher dimensions. It would also be interesting to explore the implications of your work for the black hole information paradox, as the holographic principle has been proposed as a solution to this long-standing problem.

Overall, I believe your work sheds new light on the connection between LQG and tensor networks, and has the potential to advance our understanding of the quantum nature of spacetime. Thank you for sharing your findings with the scientific community.