Why String Theory?

by Joseph Conlon

  Alchemy fails for reasons that were not known, and could not have been known, at the time of Newton. It fails because the transmutation of base metals to gold is not a chemical change but a nuclear change. In a chemical change, the bonds between different atoms and molecules are reordered and rearranged. In a nuclear change, it is the bonds between the protons and neutrons that make up nuclei that are broken and forged anew. It is an empirical fact that chemical changes need energy inputs that can be obtained from domestic sources of temperature and pressure, while nuclear changes require inputs of energy going far beyond those domestically accessible.

  Given the amount of scientific effort that went into attempts to make gold, it is somewhat ironic that experiments now exist that are dedicated to the destruction of gold and the dispatch of its constituents whence they came. These experiments take atoms of gold and strip them of all their electrons. This leaves a large denuded gold nucleus of around two hundred protons and neutrons, carrying seventy-nine units of positive charge. With so much charge, the nucleus is easily accelerated within a magnetic field. Indeed, these gold nuclei can be accelerated by the same magnets that are used to accelerate protons, which hold one unit of positive charge, and in the same accelerators – such as the Large Hadron Collider.10 In precisely the same way that the Large Hadron Collider accelerates two counter-rotating beams of protons and collides them into each other, it can also accelerate and then collide two counter-rotating beams of nuclei. The effect of the collision is exactly what you would expect: the two nuclei and their constituents are utterly destroyed. It is a nuclear Götterdammerung of the highest order. Each gold nucleus starts with a total of around two hundred nucleons – and in the collision they are entirely destroyed and broken into their constituent quarks and gluons.

  What happens next? And why do it in the first place? The reason for performing these collisions is not to enhance through scarcity the remaining value of the world’s gold supply: even at continuous operation it would take longer than the age of the universe to do so. Neither is it to satiate an infantile urge of destruction for the sake of destruction. The purpose of these collisions is to create a state of matter that last occurred naturally only a fraction of a second after the Big Bang. This state of matter is called the quark-gluon plasma, and it consists of a hot thermal bath of quarks and gluons. This bath can be formed only in temperatures that are hot enough to melt nuclei and dissolve the strong force bonds that normally bind the quarks and gluons.

  In this respect, the formation of the quark-gluon plasma resembles the formation of an ordinary bath of water starting from a tubful of ice cubes. If the bath is placed outside on a summer’s day, the ambient temperature is large enough to break the chemical bonds that hold the water molecules together as ice, and produce water. The quark-gluon plasma is formed in a similar fashion, except it is nuclear bonds that are melted by high temperatures, and the temperatures required to sustain this state are now a hundred thousand times greater than those present at the centre of the sun.

  Once upon a time in the early universe, these temperatures were everywhere. The quark-gluon plasma remained in existence during this period, as there was no need to worry about a hot patch being cooled by contact with a cold patch. It is never fully accurate to say that the Large Hadron Collider recreates the Big Bang, but this is the least inaccurate sense of the statement. Through forming the quark-gluon plasma, the Large Hadron Collider replicates for a brief instant the conditions that held in the very early universe.

  That was then. As described in chapter 2, as the universe grew it cooled, and in the here and now such temperatures can no longer be sustained.

  However, just for a brief moment, one can imagine it is the early universe again. When two high-energy gold nuclei are smashed into each other, for a short period after the collision all its energy goes into the quarks and gluons that are liberated. It heats them up. Quark hits quark that hits gluon that produces anti-quark that hits quark that makes gluon that hits antiquark. All the produced particles collide rapidly with each other and thermalise. There are around four hundred nucleons initially, and immediately after the collision there may be many thousands of particles present. The concept of temperature requires many particles. Ten thousand is not a million billion billion, but it is morally far closer to it than it is to two or three. Indeed, ten thousand is certainly enough to briefly form a little region of hot quark-gluon plasma.

  This little region does not survive very long. Like high-pressure superheated steam emerging into an Arctic winter, it rapidly expands, dilutes and cools down. As it cools, the quarks and gluons recombine to form protons, neutrons, pions, kaons and anything-you-likeons. However, deep inside the accelerator, for a twinkling the quark-gluon plasma was re-made, and its empirical properties can then be inferred from the debris recorded by the surrounding detectors.

  The purpose of these experiments is to study the quark-gluon plasma. Two such experiments exist. There is one dedicated experiment, the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory, and one part-timer, the ALICE experiment at CERN’s Large Hadron Collider. ALICE is the grubby night-shifter of the four big CERN experiments, allowed to sneak one month of heavy ion data a year away from the glitterati of the two general purpose experiments ATLAS and CMS. 11 And the purposes of studying the quark-gluon plasma? These are twofold. The first reason was just described: the quark-gluon plasma is a phase that the universe passed through very early in its history. By re-creating this phase, we can study its properties experimentally and thereby better understand the physics of some of the earliest moments of our universe.

  It is the second reason that fits into the story of this chapter. This is the fact that the quark-gluon plasma is another phase of the strong force, quantum chromodynamics, in the same way that ice and steam are other phases of water. For those who are interested in the quantum field theory of the strong force, the quark-gluon plasma gives them a chance to study it experimentally in a novel setting. The quark-gluon plasma fills in part of the map of quantum field theory and part of the map of the strong force, making it interesting to those who care about understanding these matters.

  These experiments are then interesting to the community of people who want to understand what happens in collisions of heavy nuclei at high energies, and also to the overlapping but not identical community who want to understand quantum field theory at strong coupling and finite temperature. String theory has become interesting to them because it provides a novel perspective on the quark-gluon plasma that has had some calculational success in describing real-world experimental phenomena.

  String theory does so by providing, through the AdS/CFT correspondence, a dual description of the hot mess that is produced in the collisions of heavy nuclei. The dual description provides novel calculational techniques for the hot thermalised fireball of thousands of quarks and gluons that make up the quark-gluon plasma.

  As we have seen, the basic mantra of AdS/CFT is that the dual of a strongly coupled field theories is a weakly coupled gravity theory. One may wonder what corresponds to temperature – what object is the gravitational dual of a hot field theory? We mentioned the (initially surprising) answer a few pages ago. The answer is a black hole. The fact that the field theory is hot, and at a constant temperature everywhere, is dual to the statement that a black hole is present within the gravitational solution. This sounds odd, but there are sanity checks that show that this is, if not obviously correct, not obviously wrong either.

  First, both a black hole and a completely thermalised plasma are endpoints. For both, the gate of entry is tall and wide, and the gate of exit short and narrow. Suppose you start with one region of the plasma at slightly higher temperatures, another at slightly lower, and a third superheated bubble in the middle. The laws of thermodynamics tell you that this inexorably leads to the hotter regions transferring heat to the cooler ones, such that in the end all are at the same temperature. As with two taps feeding into a sink,
the combination of an influx of hot and an influx of cold results in everywhere warm. Whatever the initial temperature variations are, given time you end up with a uniform temperature bath.

  In a similar way, a black hole is the final destination for many gravitational problems. Matter attracts via gravity. The more matter you have, the more it attracts. As matter comes closer to other matter, the gravitational attraction becomes larger and larger. Eventually, when there is sufficiently much stuff in sufficiently small a region, it collapses to form a black hole. At this point all details of the original configuration disappear and what remains is a featureless endpoint with no trace of whether we started with dust or diamonds. Both a uniform thermal bath and a black hole therefore share the property of being an end rather than a beginning, an omega rather than an alpha.

  There is a second reason to motivate why a black hole might sensibly be the dual description of a field theory at a finite temperature. This is the fact that black holes are not actually black. In classical physics, black holes only swallow. In quantum physics, they also spit. This shocking result is due to Stephen Hawking, who in addition showed that the spectrum of the light they emit is that from an oven at a fixed temperature: the Hawking temperature. The ‘temperature’ property of the field theory is mirrored by the ‘temperature’ property of the black hole. Small black holes have big temperatures and big black holes have small temperatures.

  None of this makes a proof. However, it may provide more motivation for the idea that, in the AdS/CFT correspondence, the dual to a field theory at a finite temperature is a gravitational system with a black hole. This is conceptually interesting – but so what? What gives this relationship teeth? The teeth come from asking the question: what interesting properties can be computed about black holes? What do these properties correspond to on the field theory side?

  One happy benefit of the duality is that black holes come with a totally different set of intuitions as to what is easy, what is hard, and what ‘must’ happen than do lumps of hot plasma. Indeed, black holes are rather universal objects. Unlike fingerprints, faces and DNA, it is not the case that every black hole is unique and with a large number of distinguishing features. While humans may be unique, black holes are identical up to a relatively small number of parameters: their mass, their rate of spin and their electric charge. This is a famous result of black hole physics, laconically expressed as black holes have no hair’ and known as the no-hair theorem. It is as if all human faces could be labelled by age, skin colour and eye colour – and these alone and nothing else. This universality result implies that, for many properties, if you calculate for one black hole, you calculate for all black holes. There is no need for a separate calculation for each new black hole.

  Why is this interesting? The AdS/CFT correspondence is best defined for theories that have no precise realisation in the real world. These are theories with an infinite number of particles and special symmetries that restrain the form of allowed interactions. These theories are related to the real-world strong force that appears in the quark-gluon plasma, but only as second or third cousins. The advantage of computing ‘universal’ quantities is that it offers a hope of finding the quantities least sensitive to the difference between the theory you want to calculate in and the theory you can calculate in. All our results about black holes are obtained for classical Einstein gravity. This is not dual to the real strong force. However, the hope is that perhaps the difference is small enough that one can use Einstein gravity as an approximation to the real theory.

  What universal quantity is most interesting? The answer to this question involves a story that starts back in 1985 within the student dormitories of Moscow State University. Moscow was the political, cultural and intellectual centre of the Soviet Union. From all over both the USSR and the allied socialist countries, the best and brightest of young high schoolers came to study at Moscow State to be taught physics the Russian way.12 Two of these were Dam Thanh Son from Vietnam and Andrei Olegovich Starinets. The students were grouped in dorm rooms by subject, and Son and Starinets were two of four physicists sharing a room. The assumption was that the Soviet students would assist their foreign colleagues to get better acquainted with the country and the language. Son was quiet and unassuming, but soon revealed his intellectual mettle. At one point in the first week, he was sitting quietly in his corner, reading a Russian-Vietnamese dictionary while the other students discussed a problem in complex analysis. He gently intervened, spent a few seconds explaining the solution, and then retreated back to his corner.

  Son and Starinets remained roommates for the next three years. As they advanced in their studies they began to go their separate ways. Son became a graduate student at the Institute for Nuclear Research, and he and Starinets would now only occasionally meet in the dining hall. Then came the dissolution of the Soviet Union, and the collapse of the Soviet scientific culture. Many scientists emigrated to ensure a continued paycheck, as well as for other reasons, resulting in a diaspora of Soviet-trained scientists across the globe. Both Son and Starinets would end up in the United States, Son working on nuclear theory and Starinets working on string theory.

  Many student friendships reduce with the passage of time. Moving forward to the turn of the millennium, Son and Starinets had not seen each other for many years. However they were now once again in the same city: Starinets was coming towards the end of a PhD at New York University, while Son had just started as an assistant professor at Columbia University several miles away. Starinets was working with his fellow student Giuseppe Policastro on the probability that gravitons – elementary quanta of the gravitational force – would be absorbed or scattered by black branes, string theory relatives of black holes. This represented a generalisation of earlier work by Igor Klebanov of Princeton University to the case of a finite temperature. By March 2001 they had finished the calculation: the question was, what could they do with it? They knew that under the AdS/CFT correspondence it said something about quantum field theory, but what was it?

  Starinets suggested consulting Son, who he knew had already worked on finite-temperature quantum field theory. Policastro and Starinets headed uptown to Columbia on March 23rd, 2001 as one of their colleagues was also giving a seminar there. After the seminar, they found Son’s office, knocked on the door, and told Son about the calculation. Son had been working on nuclear theory and had little prior knowledge of the AdS/CFT correspondence. The physicists talked together for several hours about the results and what they might mean. No definite conclusion was reached, but the ideas were buzzing. Policastro and Starinets returned home. That same evening, Starinets received a short email from his old friend:

  Andrei,

  I think I can relate < F2 F2 > correlator with the viscosity.

  Also, the bulk viscosity of the N=4 SYM is probably 0 because of scale invariance.

  Is there a good review of AdS/CFT correspondence that you would recommend?

  Son

  Within a week the three physicists had a draft of a paper, which was then submitted to the preprint archive on 6th April, 2001. Why did they move so quickly and what were they excited about?

  Son saw that the equations that Policastro and Starinets were using were not just calculating the absorption of gravitons in the gravity theory. They were also calculating something else: the shear viscosity of the field theory plasma it was dual to. For any kind of fluid, shear viscosity is a measure of how easy it is to slide the fluid past itself, or how easy it is for the fluid to flow. All fluids have some level of viscosity. If we treat water as a standard and familiar example, honey, tar and Marmite have high levels of viscosity, while air has a low level of viscosity. What was being computed here was the shear viscosity of the maximally supersymmetric field theory at high temperature.

  The three physicists found that the viscosity of the field theory was proportional to the area of the black hole the field theory was dual to. All black holes have a horizon: the region of space which, once cr
ossed, leads inevitably to a fall into the singularity at the centre of the black hole. The area of the black hole is simply the direct measure of how big the horizon is, and calculations show that the area of the horizon is set by the mass, or equivalently the temperature, of the black hole.

  An important quantity associated to any black hole is its entropy. The entropy is, roughly, a measure of the total number of internal configurations that can possibly make up the black hole. The entropy was computed by Jakob Bekenstein and Stephen Hawking in 1973 and is also given by the area of the black hole. In the AdS/CFT dictionary, entropy on the gravity side goes across to entropy density on the field theory side, and so shear viscosity over entropy on the gravity side goes across to shear viscosity over entropy density on the field theory side. So, ultimately, what Policastro, Son and Starinets were calculating was the ratio of the shear viscosity of the field theory – its resistance to deformation – to its entropy density, roughly its level of disorder.

  While these original calculations applied only to the special, maximally supersymmetric theory, it would subsequently be realised that the calculations applied not just there, but also for any quantum field theory whose dual gravitational description was a classical one involving classical black holes. Any field theory that was related by the AdS/CFT correspondence to such a gravity theory gave the same result. The result was applicable not just to one exceedingly special theory, but to an entire class – probably an entire infinite class – of field theories.

  However, I would be fibbing if I said that these results were greeted by the world with rapturous acclaim. The result of Policastro, Son and Starinets was initially almost entirely ignored. For the eighteen months following their paper’s appearance, not one single paper made any reference to it, except for those that had been written by the authors themselves. If anyone else cared, they were doing a good job of hiding it.