Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy

Tuesday, February 7th
Muxin Han, Florida Atlantic University
Loop Quantum Gravity, Tensor Network, and Holographic Entanglement Entropy 
PDF of the talk (2M)
Audio+Slides [.mp4 18MB]

by Jorge Pullin, Louisiana State University


The cosmological constant is an extra term that was introduced into the equations of General Relativity by Einstein himself. At the time he was trying to show that if one applied the equations to the universe as a whole, they had static solutions. People did not know in those days that the universe expanded. Some say that Einstein called the introduction of this extra term his “biggest blunder” since it prevented him from predicting the expansion of the universe which was observed experimentally by Hubble a few years later. In spite of its origin, the term is allowed in the equations and the space-times that arise when one includes the term are known as de Sitter space-times in honor of the Dutch physicist who first found some of these solutions. Depending on the sign of the cosmological constant chosen, one could have de Sitter or anti-de Sitter (AdS) space-times.


It was observed in the context of string theory that if one considered quantum gravity in anti-de Sitter space-times, the theory was equivalent to a certain class of field theories known as conformal field theories (CFT) living on the boundary of the space-time. The result is not a theorem but a conjecture, known as AdS/CFT or Maldacena conjecture. It has been verified in a variety of examples. It is a remarkable result. Gravity and conformal field theories are very different in many aspects and the fact that they could be mapped to each other opens many possibilities for new insights. For instance, an important open problem in gravity is the evaporation of black holes. Although nothing can escape a black hole classically, Hawking showed that if quantum effects are taken into account, black holes radiate particles like a black body at a given temperature. The particles take away energy and the black hole shrinks, eventually evaporating completely. This raises the question of what happened to matter that went into the black hole. Quantum mechanics has a property named unitarity that states that ordinary matter cannot turn into incoherent radiation, so this raises the question of how it could happen in an evaporating black hole. In the AdS/CFT picture, since the evaporating black hole would be mapped to a conformal field theory that is unitary, that would provide a way to study quantum mechanically how matter turns into incoherent radiation.


Several authors have connected the AdS/CFT conjecture to a mathematical construction known as tensor networks that is commonly used in quantum information theory. Tensor networks have several points in common with the spin networks that are the quantum states of gravity in loop quantum gravity. This talk spells out in detail how to make a correspondence between the states of loop quantum gravity and the tensor networks, basically corresponding to a coarse graining or averaging at certain scales of the states of quantum gravity. This opens the possibility of connecting results from AdS/CFT with results in loop quantum gravity. In particular the so-called Ryu-Takahashi formula for the entropy of a region can be arrived from in the context of loop quantum gravity.