Entanglement entropy and (non-AdS) holography

On the quest for a comprehensive theory of quantum gravitation physicists like to use simple slogans that are tentatively converted into mathematical formulas so that their consequences can be checked. Everything is geometry was such a slogan that has successfully increased our understanding of black holes in the 1960s. But this slogan can not explain why black holes emit particles - it even seems to forbid it! The slogan Everything is particle could explain in the 1970s and 1980s the evaporation of black holes through Hawking radiation. As satisfying as this realization was, it had a problematic aspect highlighted by Stephen Hawking: Black holes appear to contradict quantum mechanics, because they apparently destroy information. This slogan also seemed to have its limits. To escape this contradiction with quantum mechanics t Hooft and Susskind postulated in the 1990ies the holographic principle, which is inspired by the slogan Everything is information. According to this principle, quantum gravity in three dimensions is equivalent to a quantum theory without gravity in two dimensions. Since the latter does not destroy information also the former cannot destroy any information - if the holographic principle is actually true. The first concrete realization of this idea was the so-called anti-de Sitter / conformal field theory (AdS / CFT) correspondence by Maldacena. Since this pioneering work, which was published in 1998, more than ten thousand papers on AdS / CFT have been published. AdS spacetimes have negative curvature. On the other hand, our universe is essentially flat at intermediate distances and positively curved at very large distances. To find out if the holographic principle applies to our universe, it is necessary to construct quantum gravity theories that do not require AdS. If holography works, any quantity one can calculate on both sides of the holographic correspondence must match. In particular, it is necessary to encounter the key property of quantum mechanics, entanglement, also on the gravity side. The aim of the project is to calculate a certain measure of entanglement, called entanglement entropy, on both sides of novel holographic correspondence (those who do not need AdS). On the quantum mechanical side, certain CFT methods will be generalized for this calculation; that this is possible, is not only plausible, but it has been shown by us explicitly using an example in a recent paper. On the gravity side, a method by Ryu and Takayanagi for calculating the entanglement entropy will be generalized, which amounts essentially to the calculation of a comparatively simple curve (a so-called Wilson line). This project can substantiate the validity of the holographic principle further or disprove it. In either case we hope to gain a deeper understanding and coalescence of the concepts of quantum information and -gravitation.

On the quest for a comprehensive theory of quantum gravitation physicists like to use simple slogans that are tentatively converted into mathematical formulas so that their consequences can be checked. "Everything is geometry" was such a slogan that has successfully increased our understanding of black holes in the 1960s. But this slogan can not explain why black holes emit particles - it even seems to forbid it! The slogan "Everything is particle" could explain in the 1970s and 1980s the evaporation of black holes through Hawking radiation. As satisfying as this realization was, it had a problematic aspect highlighted by Stephen Hawking: Black holes appear to contradict quantum mechanics, because they apparently destroy information. This slogan also seemed to have its limits. To escape this contradiction with quantum mechanics 't Hooft and Susskind postulated in the 1990ies the holographic principle, which is inspired by the slogan "Everything is information." According to this principle, quantum gravity in three dimensions is equivalent to a quantum theory without gravity in two dimensions. Since the latter does not destroy information also the former cannot destroy any information - if the holographic principle is actually true. The first concrete realization of this idea was the so-called anti-de Sitter / conformal field theory (AdS / CFT) correspondence by Maldacena. Since this pioneering work, which was published in 1998, more than fifteen thousand papers on AdS / CFT have been published. AdS spacetimes have negative curvature. On the other hand, our universe is essentially flat at intermediate distances and positively curved at very large distances. To find out if the holographic principle applies to our universe, it is necessary to construct quantum gravity theories that do not require AdS. If holography works, any quantity one can calculate on both sides of the holographic correspondence must match. In particular, it is necessary to encounter the key property of quantum mechanics, entanglement, also on the gravity side. Within the FWF project "Entanglement entropy and (non-AdS) holography" the interplay between quantum mechanical entanglement and holography was studied in great depth in collaboration with two postdocs and three PhD students. One of the key results is the discovery of quantum energy conditions in the context of flat space holography - in plain English this means that we asked (and answered) the question how much energy one can "borrow" from the vacuum and with which "interest rate" one has to pay it back. It turned out that the amount of energy one can borrow is bounded by (certain combinations of derivatives of) entanglement entropy. Information, encoded in entanglement, is indeed a key quantity for energy and holography, even beyond AdS/CFT.