sign in sign up
<<
Home , Quantum gravity, Strong gravity

F EATURE Kavli IPMU Principal Investigator Yasunori Nomura Research Area： Theoretical

Quantum and

Understanding the of space and time has the Sun because of this bending of . brought many revolutions to physics. In old days, This revolutionary of spacetime makes the introduction of a coordinate system̶which is an important prediction: because spacetime often attributed to Descartes̶led to Newtonian is dynamical, its distortions can propagate through contributions by many scientists, independently of the existence of . For including Galileo. At the beginning of the 20th example, when heavy bodies such as black holes century, proposed , move violently, they disturb spacetime greatly. which, in a sense, uni es space and time. According This effect can then propagate to places far to this theory, space and time are closely related, away from the location of the original bodies as causing phenomena such as a contraction of wave of spacetime distortions. This wave is called length and a delay of time when an object moves , and it was rst detected directly at a speed close to that of . These effects are in 2016, almost 100 years after the formulation of precisely measured in present-day experiments, . This was an extremely important leaving no room to doubt the theory. The resulting experimental discovery, and for this achievement the concept of spacetime, which treats space and time 2017 Nobel Prize in Physics was awarded. in a uni ed manner, has been playing a fundamental role in contemporary physics until today. Theory of Quantum Gravity One of the most important properties of spacetime was revealed by Einstein himself about 10 As we have seen, our understanding of spacetime years after the announcement of special relativity. has dramatically evolved over the last few hundred According to this new theory called general years. However, we know that this is not the end. relativity, spacetime is not a xed entity, as one The reason is that our world obeys quantum might imagine from the term“ coordinate system”; mechanics. rather, it is a dynamical object that can be bent was discovered at the or distorted. General relativity says that it is these beginning of the 20th century in the process of bending and distortion of spacetime that we call studying microscopic objects such as atoms, but gravity. For example, gravity between the Sun and later it was found to be a more fundamental the Earth arises because the existence of the Sun principle governing the world. Quantum mechanics bends spacetime around it. The Earth moves around makes a number of strange predictions. For

4 Kavli IPMU News No. 43 September 2018 example, an object such as an electron cannot length̶where quantum effects on spacetime and exist in one place; rather, its location spreads gravity become important̶leads to an innumerable probabilistically. (Since this spread is usually so small amount of in nities, destroying the predictivity that we cannot perceive it directly in our daily life.) of the theory. This means that in order to obtain It is also possible that two worlds exist parallelly in a complete theory including quantum effects on probability space, the state we call a superposition. spacetime and gravity, it is not enough to simply While these phenomena sound extremely strange consider general relativity and quantum mechanics from the sense of everyday life, the correctness together. The currently most promising theory to of the theory has been repeatedly demonstrated solve this problem is theory, whose structure experimentally. In fact, the accuracy of these and consequences are vigorously studied by many experiments has dramatically improved in recent researchers. years, so that predictions of quantum mechanics Another reason to study quantum mechanical have now been con rmed precisely at a detailed theory of spacetime and gravity̶commonly level. Furthermore, scientists are advancing the referred to as quantum gravity̶is that it brings development of a computer̶quantum computer̶ about a revolutionary picture of spacetime. As we which has capability completely different from the will see in this article, it is not true that the effect conventional one and which positively utilizes the of quantum gravity becomes important only at the principle of quantum mechanics. ; on the contrary, it has increasingly Let us return the story to spacetime. General become clear that it also plays a decisive role in relativity explains all gravitational phenomena physics at long distances such as the physics of that have been observed until now. This includes black holes and cosmology. Moreover, minuscule deviations from the predictions of this relatively recent progress is creating a new Newtonian gravity, the existence of gravitational research area that uni es two elds of quantum wave, and the expansion of the . There gravity and quantum information sciences, which is almost no doubt about the correctness of the have been developed separately so far. Below I will theory at scales we can currently observe. General give a brief overview of this new development. relativity, however, does not contain the effect of quantum mechanics, which is a basic principle of Quantum Mechanics of Spacetime and the world we live in. This is not an experimentally the urgent problem. The scale at which the quantum The beginning was the physics of black holes. In effect of spacetime becomes important directly can a theory with gravity, when the density of matter be estimated theoretically, which is about 10-33 cm, becomes larger than a certain limit, it produces a the distance called the Planck length. This is about region around it from which nothing can escape 16 orders of magnitude smaller than the smallest due to its strong gravity. The boundary of this Feature length probed so far by the world’s most powerful region is called a horizon, and the inside is a black accelerator. Does this mean that it is meaningless to hole. investigate the quantum effect on spacetime? Black holes had been known to have the There are a number of reasons, at least following intriguing property: in general relativity, theoretically, why this is not the case. One is the the total area of horizons does not decrease. For problem of in nity. Quantum mechanics and general example, when a swallows an object, its relativity do not get along very well. Assuming area increases. (Here and below, when we refer to that the theory of gravity at our scales is general the area of a black hole, it means the area of the relativity, its simple extrapolation below the Planck horizon of the black hole.) Also, when two black

5 A2

A3

A1

A3 ≥ A1 + A2

Figure 1: The area, A3, of the black hole formed after the merging of two black holes is equal to or larger than the sum of the areas, A1 and A2, of the original black holes.

holes merge into one̶ this phenomenon was absorbed is always equal to or larger than the sum actually observed by the detection of gravity wave of the entropies of the initial black hole and the in 2016!̶the area of the produced black hole is thrown object. equal to or larger than the sum of the areas of the The idea that a black hole has entropy was initial two black holes; see Figure 1. initially received with skepticism. If a black hole The fundamental importance of this fact was has entropy, the rst law of thermodynamics realized by Jacob Bekenstein in his revolutionary requires it to also have temperature. However, paper in 1973. Bekenstein asked what happens to if it has temperature, then it must be radiating. physical quantities when an object is thrown into Isn’t it the case that nothing can come out of a a black hole. The answer is clear for energy. The black hole? This apparent contradiction, however, energy of the thrown object becomes (a part of) was resolved in 1974 when the mass of the black hole, so that the total energy performed a calculation which included the effect is conserved. But what about entropy? According of quantum mechanics into general relativity. While to the second law of thermodynamics, entropy his calculation was not based on a complete theory should not decrease. However, the entropy of the of quantum gravity, he could still show that a black thrown object seems to simply disappear when it is hole indeed emits radiation because of quantum absorbed into the black hole. Does this mean that effects. This established the thermodynamics of the second law of thermodynamics is violated? black holes. What Bekenstein suggested in his 1973 paper is A strange thing here is that the entropy of a black that the entropy of the thrown object is converted hole is given by its area. Since a black hole is the into the entropy of the black hole. What is the nal state of the evolution of any initial state, the entropy of the black hole? He proposed that the entropy of the black hole must be indicating“ the area of the horizon is the one. Recall that the largest possible entropy that the region can have.” In total sum of horizon areas does not decrease in modern days, we know that entropy is given by the general relativity. This is exactly the property that logarithm of the number of possible quantum states entropy must have. Indeed, by de ning the entropy that the system can take. If space is composed of of a black hole in this way, it can be shown that some simple constituents as in ordinary materials the entropy of the black hole after the object is (e.g., if space can be approximated by a lattice with

6 Kavli IPMU News No. 43 September 2018 a spacing of order the Planck length), then the was made by studying structures of . largest entropy of a region must be proportional In 1997, Juan Maldacena proposed, based on to its volume. However, the discovery of Bekenstein numerous evidences, that quantum gravity A2 and Hawking says that it is proportional to the describing physics in that asymptotically surface area. approach a certain space called Anti-de Sitter A3 This implies that in a with gravity, (AdS) space is equivalent to conformal eld theory the number of fundamental degrees of freedom is (CFT) formulated in non-gravitational spacetime given by that of spacetime with one less that has one less than the gravitational A1 than that of the original, dynamical spacetime. For asymptotically AdS space. This relationship is called example, while spacetime we live in seems to have the AdS/CFT correspondence. A3 ≥ A1 + A2 three spatial dimensions (ignoring possible small The AdS/CFT correspondence is an extremely extra dimensions), the“ true theory” describing it powerful mechanism despite the fact that it applies must be formulated in spacetime with two spatial only to special spacetimes. First, CFT is a class dimensions and one time dimension. This is possible of quantum eld theory and is mathematically because if you try to ll matter at each point in well de ned. This implies that quantum gravity in space (e.g., at each site of the Planck- lattice), spacetimes that are not so distant from the one we black holes form long before it is completely lled, live in (asymptotically AdS space) is de ned for the and putting further matter only increases the size rst time without relying on . of the black holes. Namely, it is merely a ction Also, many describing the nature (e.g., that the space we live in has full three dimensional QCD describing nuclear and theories used in degrees of freedom. condensed matter physics) are well approximated The idea that the theory of quantum gravity is by CFT at strong coupling. In general, it is extremely formulated in spacetime that has lower dimensions dif cult to solve such a theory with strong coupling, than the apparent, dynamical spacetime is called the but by the AdS/CFT correspondence we can solve it holographic principle. The lower dimensional theory approximately using general relativity in one higher formulated in this way̶called the holographic dimensions. And above all, this correspondence theory̶does not have the discrepancy between the gives a concrete example of the holographic number of true and apparent degrees of freedom, principle, showing how dynamical spacetime with so it does not have gravity: the dynamical spacetime gravity is generated from non-gravitational theory in higher dimensions and its associated gravity are in lower dimensions. only emergent. The fact that the holographic theory The mechanism of generating dynamical does not contain gravity at the fundamental level spacetimes has been studied by many researchers, means that it can be treated as a regular quantum and in this process a very important discovery system, and hence can give a rigorous de nition of was made by Shinsei Ryu and Tadashi Takayanagi. Feature quantum gravity. This seems to be a pretty crazy What they found is the following relation between conclusion. It would be natural if one cannot believe the geometry of AdS and quantum states in CFT; such a thing. see Figure 2. Consider a on an equal time slice in CFT. Let us divide this slice into – AdS/CFT Correspondence a subregion A and its complement A. Then, the entanglement entropy＊ between A and A– is given However, it has been shown̶though only in by the area of the surface in AdS that is anchored to some special cases̶that quantum gravity indeed satis es the holographic principle! This discovery ＊See, Kavli IPMU News No. 31, p. 28.

7 time holographic dimension space

A

Figure 2: Entanglement entropy of a state in CFT between a – spatial region A and its complement A is given by the area of the surface in AdS which is anchored to the boundary of A and whose area takes an extremal value.

the boundary of A and has the minimal area (more equation in higher dimensional AdS background is generally the surface which extremizes the area an automatic consequence of the relation between in AdS). Here, the entanglement entropy between the geometry of emergent spacetime and quantum – A and A is a quantity representing the amount of information in the holographic theory. between the degrees of The AdS/CFT correspondence gives us a basic – freedom in A and A. picture of how dynamical spacetime may be The importance of this relationship, called the generated. We could draw the following analogy. Ryu-Takayanagi formula, is not just that it relates Consider liquid water in place of spacetime. We two quantities. First, the fact that a geometric want to know its microstructure such as water object in the gravitational bulk̶a minimal area molecules and acting between them. For surface in AdS̶is related to a quantum information this purpose, it is barely useful to study phenomena theoretic quantity in the holographic theory̶an that occur as perturbation in the limit that liquid entanglement entropy̶implies that the mechanism water is regarded as continuum, e.g., water waves. of generating dynamical spacetime is intrinsically These are described by the Navier-Stokes equation quantum mechanical. Also, taking the view that of uid dynamics, which corresponds to the Einstein the theory of quantum gravity is rigorously de ned equation in our context, regarding spacetime as only holographically, we may say that what we continuum. In contrast, AdS/CFT can tell us the perceive as gravitational spacetime is nothing other identity of water molecules and forces between than quantum information theoretic properties them, i.e., CFT. A big difference between water and of quantum states in a more fundamental, non- spacetime, however, is that in the case of spacetime gravitational theory. In fact, by using the Ryu- its constituents somehow live in one lower Takayanagi formula, one can show that the Einstein dimensions.

8 Kavli IPMU News No. 43 September 2018 gravity, however, there is no corresponding direct Beyond AdS/CFT experimental result to which we can compare As we have seen above, AdS/CFT is extremely theoretical results. Then how can we test the validity powerful, but the actual spacetime we live in is not of the hypothesis? an asymptotically AdS space. On the other hand, Our idea is to use general relativity as the holographic principle is expected to apply “experiment.” Speci cally, under the assumption more generally in quantum gravity as can be seen that the holographic relationship between quantum from the consideration of black holes. How then information and generated geometries is valid in does holography work in general spacetimes? This general spacetimes, we have derived the properties question has been investigated by several research that the dynamical spacetimes must satisfy in order groups. Below, I will briey describe the work we for the quantum information theoretic quantities have done. to be consistently interpreted holographically. The The starting point is to identify which of the result was a series of monotonicity properties and properties of AdS/CFT are speci c to AdS space and inequalities, which were not known before. If the which are more generally applicable in quantum original assumption is correct, then these properties gravity. For example, we expect that, from various must be satis ed for general spacetimes, and indeed considerations, the fact that the holographic we were able to prove them using general relativity. theory is local quantum eld theory like CFT is due An important point is that if general relativity did to a special feature of asymptotically AdS space. not possess features that are needed to prove the Then, which of the properties of AdS/CFT are properties we found, then the original assumption manifestations of the more general holographic could have been rejected. In other words, the principle? hypothesis that quantum information theoretic Based on a number of suggestions, we have properties of states in the holographic theory are hypothesized that the relationship between the origin of dynamical spacetime in cases beyond quantum information in a lower dimensional AdS/CFT has passed a nontrivial test. This increases holographic theory and the geometry of higher our con dence about the holographic principle dimensional quantum gravity, as represented by and relation between quantum information and the Ryu-Takayanagi formula, is general. Since the quantum gravity. holographic theory for general spacetimes is not yet known, this requires us to extract essential Summary elements of the theory using entropy bounds obtained by extending the result of Bekenstein and This article has described the relationship based Hawking. I omit the details of this procedure due on the holographic principle between quantum to the limitation of pages. In any case, the idea is to information and dynamical spacetime, including Feature examine the validity of the hypothesis by analyzing AdS/CFT. This has been investigated not only what consequences it leads to. by our group but also by other researchers, and In a sense, this strategy corresponds, e.g., to important discoveries have been made day-to-day. the one Max Planck took when he examined These discoveries are providing fundamentally new the consequences of his quantum hypothesis answers to the question of what spacetime really upon the creation of quantum mechanics. At is. Hopefully, this new research area emerging at that time, there was an experimental result̶the the intersection of quantum gravity and quantum black body spectrum̶to which he compared information will reveal a deep and fundamental the consequences. In the case of quantum picture of spacetime.

9