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Spacetime and Quantum Mechanics

By Juan Maldacena

Carl P. Feinberg Professor, School of Natural Sciences
Institute for Advanced Study, Princeton NJ

Photo: Jean-Francois Dars

Juan Maldacena is Carl P. Feinberg Professor at the Institute for Advanced Study in Princeton. After receiving his Ph.D. from Princeton in 1996, Juan Maldacena became associate professor of physics at Harvard in 1997. In November of that year he published his most renowned work, “The Large-N Limit of Superconformal Field Theories and Supergravity.” Maldacena was promoted to Professor at Harvard in 1999 and moved to the Institute for Advanced Study in Princeton in 2001. Among his awards are the MacArthur Fellowship (1999), the APS Bouchet Award (2004), and the Dannie Heineman Prize for Mathematical Physics (2007). 

There are few things more basic than geometry: the basic arrangement of points in space, the lines that connect them, etc. Geometry was one of the first subjects that rose from its empirical roots to its formal Euclidean formulation. The notion of geometry is central to physics. In fact, according to Einstein’s theory of relativity, we live in a curved spacetime. Curved and dynamical, spacetime responds to the motion of matter. This was spectacularly confirmed by the discovery, via the LIGO/VIRGO collaboration, of gravitational waves coming from black hole collisions. This discovery also provides the best evidence for the existence of black holes, which are one of the most dramatic consequences of general relativity. 

Black holes are regions where the universe is collapsing. Our universe is generally expanding, however in the neighborhood of bound massive objects, our Earth, the galaxy, etc., we can say that it is not expanding. In the interior of black holes, it is collapsing. The process that gave birth to the universe at the beginning of the Big Bang is reversed in the black hole interior. The boundary of the collapsing region is called a horizon. A horizon is just an imaginary surface in the sense that an observer that crosses it does not notice anything peculiar.  According to classical general relativity, black holes can only grow. Matter can fall in, but once the black hole forms it cannot disappear. 

But nature is quantum mechanical, and, as Hawking famously showed, black holes emit thermal radiation once quantum effects are taken into account. This is surprising because, after matter falls in, a black hole is just made out of empty space. Nevertheless, empty space itself can have a temperature. The thermal nature of black holes is related to the fact that black holes have a horizon. The horizon divides the vacuum into two regions: the exterior and the interior. An observer who remains outside has only access to the outside. An interesting property of the vacuum in quantum field theory is that, though it is a single definite state, it cannot be split into single definite states locally. In quantum physics, we call this ‘entanglement’. If you zoom into a small piece of space, the quantum state looks very random, but this randomness is apparent in the sense that random results in that region are correlated with random results somewhere else so that they all fit into what we usually call a ‘pure’ state, or a minimum uncertainty state. 

When an observer stays outside a horizon, she is only observing part of the vacuum and that is the reason for its thermal nature. It is possible to quantify how much randomness, or entropy, is generated by the presence of the horizon. It turns out that this is equal to the area of the horizon in Planck units. A Planck unit of distance is the distance at which quantum gravity effects become very important. In nature, it is about 10¯­­³⁵ meters. With this notion of entropy, and Hawking’s notion of temperature, black holes obey the standard laws of thermodynamics, taking the energy to be equal to its mass (or more precisely E= mc2). In classical general relativity, with ordinary matter, the area of black holes always increases. This is related to the second law of thermodynamics, which says that entropy always increases. These results inspired Hawking to suggest that black holes destroy quantum information. Other researchers suspected that black holes, when they are observed from the outside, behave as ordinary quantum systems that preserve information. In an ordinary quantum system, the increase of entropy is only an approximate notion; it only arises because we are limited to doing only coarse measurements. The system evolves in a complicated way and we lose our ability to predict its behavior because of our limitations on the measurements we do. Through some computations in string theory, there is very convincing evidence this is true. This is based on some computations of black hole entropy from a microscopic point of view, which were done by Strominger and Vafa in 1996. Also, the conjectured duality between gravity in anti-de Sitter space and conformal quantum theories on the boundary implies that black holes behave as relatively standard quantum systems. 

In quantum physics there are two types of entropy. One of them is the precise ignorance that we have about the system. For example, we could have a qubit that entangled with another one outside our control. In this case we have some amount of ignorance that is irreducible. No matter how well we measure this qubit, we will not be able to determine its state. This is what we call ‘fine grained’ entropy. On the other hand, we could imagine that we have some set of qubits and we only decide to make coarse measurements. In this case there will be many states that can potentially reproduce our measurements. The number of such states is called the ‘coarse grained’ entropy. This is the entropy that increases under the second law. The fine grained entropy remains constant under evolution. So, the area of the horizon should be interpreted as a coarse grained entropy.

One notable development in black hole physics was a proposal, initially by Ryu and Takayanagi, and then by Hubeny, Rangamani and Takayanagi, of a geometric formula for the fine grained entropy. They proposed that this entropy is computed by the area of an extremal surface in spacetime. As expected, this area is smaller than the area of the horizon and it is a surface that typically lies behind the horizon of the black hole. This is non-zero in cases where the interior of the black hole is geometrically connected to other universes. It is zero in cases where the black hole is produced by gravitational collapse. 

This has divided the interior of a black hole into
further regions. The region outside the horizon is the region where we can do measurements in an easy way. The region behind the horizon but outside the extremal surface can be probed by doing more complicated measurements, assuming we have complete control over the black hole degrees of freedom. 

An interesting geometry that has been studied intensively in recent years is the full two-sided Schwarzschild geometry. This is actually the solution that Schwarzschild originally found more than a hundred years ago, but maximally analytically extended beyond its coordinate singularities. This was fully done only about fifty years later. But it was understood that it describes two black holes rather than one black hole. In other words, the solution contains two spacetime regions that can be viewed as black hole exteriors and one region that is the interior region, which is common to both black holes. This is a surprising solution in general relativity. If we had such a solution, we would have two objects looking like ordinary black holes from the outside, but they share a single interior. If you and your friend jump into the two distant black holes you could meet immediately in the interior. The spatial distance between the two could be very large in the ambient space, but is very short through the interior. This type of geometry is sometimes called a ‘wormhole’. But it is a wormhole that cannot be crossed. It is a time dependent solution, describing a collapsing region of the universe. You can enter it but, once you enter, you cannot exit it on either of the two sides. 

Now, we had said that we can think of a black hole as a quantum system with many degrees of freedom. How should we think about the full Schwarzschild wormhole? In 1976, Werner Israel suggested that it should be viewed as a kind of entangled state, one that arises naturally from a mathematical construction designed to simplify computations in thermal systems. In such cases, it is convenient to double the thermal system, introducing two copies of the original system and setting up an entangled state such that, when we restrict to measurements on one of the systems, then, from the point of view of that system, we have just a thermal state. These states are called ‘thermofield doubles’. This type of state can, in principle, be constructed for any system. In particular, it can be constructed for the quantum systems that describe black holes. In that case, we expect that the geometry would become connected. 

One can wonder how difficult it is to produce such a state.  For some systems it does not appear complicated to find a state that is close enough to the thermofield double state. This can be done relatively easily for strongly interacting quantum systems that have some features in common with certain black holes. For this reason, we expect that perhaps there is, at least in principle, some relatively easy way to do it with actual blackholes too. 

The interest in this wormhole geometry arises because, in this system, the interior spacetime seems to be arising from entanglement. The most solid and fundamental of notions, spacetime itself, is determined by the ghostly pattern of entanglement in the quantum state. We hope that by studying these issues further we will understand how to think about spacetime in general. In the meantime, this avenue of research has given us some surprising connections. One of my favorites is the connection between quantum teleportation and travel through wormholes, which was shown by Gao, Jafferis and Wall, after some more qualitative suggestions by Susskind. To explain what quantum teleportation is, let us first give a simple classical analog. 

Imagine that Alice and Bob want to communicate over radio, with everyone listening, but they do not want other people to understand what they are saying. One way they could do it is to share a secret key. For example, they have two identical copies of a very large random sequence of numbers. So, if Alice wants to send a message, she could convert it into a sequence of numbers and then she can add the numbers (modulo ten) to her copy of the secret key and then send the resulting number over the radio. The eavesdroppers would only get a random number and cannot extract the message. On the other hand, when Bob hears the message, he can subtract the numbers from his copy of the key and recover Alice’s original message by taking the received number and subtracting the number in the key (modulo ten).  

Quantum teleportation is a quantum version of this. The key is now an entangled state made out of two quantum systems.  Alice has one of them and Bob has the other. Now Alice wants to send Bob a quantum state. So, she mixes it very thoroughly with her share of the entangled state and then performs a measurement on the combined system. She sends the result of the measurement over the radio, this part continues to be classical. Then Bob extracts the quantum state by performing an operation on his share of the entangled state. His operation depends on the result of Alice’s measurement. 

One technological application of this process is to send secret messages. One advantage of using quantum mechanics is one cannot copy quantum systems. So, an eavesdropper cannot copy the key without Alice and Bob noticing. On the other hand, for a classical key, there can be many copies so the eavesdropper could have one. This cannot happen in quantum physics because quantum states cannot be copied. 

If we take the entangled state to be a wormhole, then one can send a message into the first black hole. By measuring some of the Hawking radiation of this first black hole and sending it to somebody near the second black hole, the second person can send a signal into the second black hole that extracts the message. 

Just to explain this in an alternative way, let us imagine a science fiction story. This story takes place sometime in the far future where people can manipulate black holes in a controlled way. In this story, there is a princess and prince Charming. The princess is held hostage in a high tower by her evil stepmother. She allows her to use the internet, even the quantum internet, and she can send quantum messages. The prince is outside and wants to rescue the princess. They read the quantum teleportation paper and come up with an idea to teleport the princess. First, they create lots of entangled qubits and share one of the members with the other person. At the end of the process they have a huge number of entangled qubits. They are getting ready to teleport the princess, and she is setting up the machinery that will mix her with the large number of entangled qubits. But she has second thoughts. She worries that when she gets mixed with all the qubits she will die and it will be painful. She is assured that at the end of the process she will be reconstituted on the other side. However, the process would be painful. But then she reads the paper by Gao, Jafferis and Wall. Thus, she convinces prince Charming to create a black hole with his qubits and she does the same with hers. They further agree to create the black holes in precisely the thermofield double. Now she can jump into her black hole and have the machine collect the Hawking radiation. The evil stepmother thinks she has died in the blackhole. However, the prince receives the result of the measurement and can send the appropriate signal into his black hole. Then something amazing happens. His black hole, instead of emitting Hawking radiation, suddenly spits out the princess, whom he gallantly helps out of the black hole. 

The interesting thing is that the princess feels as if she travelled through empty space, and she did not feel anything unpleasant in the process. 

From the point of view of the evil stepmother, who stays out of her black hole, she died in the black hole and was mixed very thoroughly with the black hole degrees of freedom. 

Notice that the pleasant experience of the princess is due to the emergence of spacetime from the precise entanglement pattern of the two black holes. The idea that spacetime can be manipulated in this way highlights how deep the connection between quantum mechanics and geometry is. The pattern of entanglement generates a particular geometry. We are not studying these questions to rescue princesses, but to understand better the nature of spacetime. Eventually, we would like to betterunderstand the collapsing geometries inside black holes and their big crunch singularities. If that were thoroughly understood, maybe, via an intellectual wormhole, we would also understand the Big Bang singularity… •