A more graphic illustration of the pentecostal impact of this paper is through dance. Physicists do not, pace Feynman, have a reputation as party animals. In most people’s estimation, an assembly of theoretical physicists would rank slightly below a gathering of IT database experts in the ordering of places to let your hair down. And yet at the Strings 1998 conference at Santa Barbara in California, the after-dinner speech at the conference banquet saw four hundred physicists dancing in unison as the speaker Jeff Harvey from the University of Chicago orchestrated a twist on the hit pop song ‘Macarena’:
You start with the brane
and the brane is BPS.
Then you go near the brane
and the space is AdS.
Who knows what it means
I don’t, I confess.
Ehhh! Maldacena!
Super Yang-Mills
with very large N.
Gravity on a sphere
flux without end.
Who says they’re the same
holographic he contends.
Ehhh! Maldacena!
Black holes used to be
a great mystery.
Now we use D-brane
to compute D-entropy.
And when D-brane is hot
D-free energy.
Ehhh! Maldacena!
M-theory is finished
Juan has great repute.
The black hole we have mastered
QCD we can compute.
Too bad the glueball spectrum
is still in some dispute.
Ehhh! Maldacena!
It was not quite the chairman of the Federal Reserve doing the Can-Can in stockings and suspenders, but it has never been seen again.
What precisely is this correspondence? As said above, it is the claim that certain gauge theories – quantum field theories of the kind that are used to describe the strong, weak and electromagnetic forces – are exactly the same theory as certain gravitational theories, just written in a different language.
The equations used to write down the two theories are totally different. The types of interactions that are present are totally different. Even the number of dimensions in the two theories are different. In this correspondence, it is always the case that the gravitational theory has one extra dimension than the quantum field theory it is equivalent to. The claim that the theories are exactly the same is therefore shocking and audacious. Although I have just used the word ‘exactly’ twice already, it is good to repeat it a third time. This is not a statement that one theory is a metaphor to describe the other, or that sectors of one theory provide a useful approximation to the other. It is a statement that the two theories are completely and utterly equivalent. They are exactly the same theory, simply re-expressed in different terms.
In Maldacena’s original paper, this correspondence was first applied to one particularly special quantum field theory, called four-dimensional super Yang-Mills theory. This theory does not itself describe the real world, but is a generalisation of the theories that do. It is a souped-up version of the theories that describe the strong, weak and electromagnetic interactions, and what makes it special is the large amount of symmetry it has. It has both the maximal amount of symmetry and the maximal amount of supersymmetry possible.4 It is to the Standard Model what a flawless cut diamond is to rock pulled fresh from a mine. The enormous amounts of symmetry highly constrain its structure and allowed interactions, and it had already been singled out as a ‘special’ quantum field theory worthy of particular study even when Juan Maldacena was a young boy in short trousers in Buenos Aires trying to understand how the radio worked and helping his father fix the family car.
This special theory also comes with two parameters. The first is an integer imaginatively denoted by N, which relates to the number of force carriers’. In electromagnetism there is one force carrier, the photon. In the weak force there are three, the W+, W– and Z0 bosons. For the strong force, there are eight force carriers – the different kinds of gluon. In our special theory of interest, a value of N implies N2 – 1 force carriers. The second parameter is a continuous coupling constant, which determines the strength of the interactions.
In its simplest form, the correspondence says that this special theory – maximally supersymmetric four-dimensional Yang-Mills theory – is exactly the same as one of the string theories, type IIB, living on a ten-dimensional space in which five dimensions make up a sphere and the remaining five dimensions form a particular geometry called Anti-de Sitter space.5 The presence of two parameters, one continuous and one an integer, is mirrored in the string theory. The continuous parameter this time is the string coupling constant, which describe the likelihood that two strings will interact with each other. The integer parameter also exists in the string theory – but where it comes from and what is means is far too technical to discuss here.
These last few paragraphs have been dense. If they have been too dense, it is important to return to the basic claim: a quantum field theory in four dimensions with no gravitational interactions is entirely equivalent to a gravitational string theory in five dimensions.6 The claim is that any result present in one theory can also be obtained in the other, once re-expressed in the right language.
Having the same information expressed in different languages is of course familiar to scholars of the humanities, and in certain cases exceptionally useful – as for the Rosetta stone, where the presence of the same text in Demotic, Ancient Greek and Egyptian hieroglyphics enabled the last to be deciphered in the early 19th century and thereby read for the first time in fifteen hundred years. Both linguists and scientists need a dictionary to translate from one language to another. In the physics case of the AdS/CFT correspondence, this requires an ability to say ‘Quantity A in the gravity theory corresponds to quantity B in the gauge theory. To calculate quantity A in the gravity theory, it is sufficient to calculate quantity B in the gauge theory, for they are one and the same.’
If Maldacena’s paper served the role of the Rosetta stone, two papers in February 1998, one by Steven Gubser, Igor Klebanov and Alexander Polyakov, and another by Edward Witten, all at Princeton, started to fill in the entries of the dictionary. While Maldacena’s paper had claimed the equivalence of the two theories, these papers gave calculational prescriptions for checking this equivalence, with statements of precisely which calculation in the gravity theory was equivalent to precisely which calculation in quantum field theory. A further paper by Witten the following month filled in some more of the dictionary. In particular – and this will be important later – he filled in the entry for ‘black hole’. A black hole in the gravity theory, Witten said, is equivalent to heating up the quantum field theory so that it is at a finite temperature.
In many cases, the prospect of curling up in an armchair to read a dictionary is less than enthralling. What made this dictionary a scientific bestseller was the fact that it identified strongly coupled quantum field theory with a weakly coupled gravitational theory.7
This makes the duality not just of conceptual interest, but also useful. The duality identifies strongly coupled quantum field theories, in which it is hard to compute, with classical gravitational theories, in which it is easy to compute. This allows the hard problem of computing in quantum field theory at strong coupling to be transmuted into the easy’ problem of computing in classical general relativity at weak coupling.
It is this fact that is of most relevance to this chapter. Even if you cared not a whit about gravity, even if you are cold to any and all ideas about extra dimensions or black holes or mathematics: if you are interested in calculating in general quantum field theories, you should be interested in string theory. This was the carrot that attracted many workers in quantum field theory to learn about string theory and spend time on it, all for the furtherance of their own interests.
All very important – if correct. But how do we know, and why should we believe, that the correspondence is correct? There are many, many ideas i
n science whose path to importance and influence is blocked only by their falsehood. What makes AdS/CFT different?
This is an important question which deserves an answer, to be given over the next few pages. The first point of clarity is that it will not be a philosophical answer. The question of how ideas become accepted, and how we can acquire justified true belief, is an interesting question within philosophy and epistemology – but the issues raised there apply equally well to the statements ‘The sun will rise tomorrow’ and ‘I am the son of Tom and Theresa Conlon’.8
The second point of clarity is that it will also not be a mathematical answer. Mathematicians work with precise and well-defined objects. The AdS/CFT correspondence involves on the one hand string theory and on the other hand quantum field theory. Neither side of the correspondence involves an object with a rigorous mathematical definition. Forty years after the correct theory of the strong force was established, there remains a one million dollar prize from the Clay Mathematics Foundation for showing that this theory exists mathematically and determining some of its (well-established) properties. The mere existence of this prize illustrates the difficulties in giving a mathematical definition of a theory as well established physically as quantum field theory. The difficulties for string theory are proportionately harder; there is not even a starting point for writing down rigorously a mathematical object that represents ‘type IIB string theory on the product of a five-dimensional sphere and five-dimensional Anti-de Sitter space’.
A mathematical proof of the correspondence must involve a construction of mathematical structures representing both sides of the correspondence – one structure for quantum field theory and one structure for string theory – and then a formal proof that these two structures are the same, or in the language of mathematics, isomorphic. As no one knows how to construct either mathematical structure, no such mathematical proof exists, or indeed will appear anytime soon.
The answer I will give is instead that of a physicist. It is that you check the correspondence by calculation, you carry on checking the correspondence by calculation, and you continue checking by looking for as many ways as possible of breaking the correspondence. At some point – and this point will vary from person to person – sufficiently many checks have been performed that it becomes hard to think of any way the correspondence could both pass all these tests and still not yet be true. This is not proof. No number of checks ever make a proof. However physics is not mathematics, and those with scruples on this matter can be well advised that the math department on campus is generally in the next building down the street.
The tests build up in complexity. The simplest version of the correspondence involves on one side an entirely classical gravity theory called ten-dimensional supergravity, and on the other side the maximally special Yang-Mills field theory in the limit of an infinite number of particles. The calculational content of the correspondence relates on the gravity side masses of particles in the gravitational theory, and on the field theory side the way certain quantities change on moving from large to small distances. This last expression sounds somewhat vague, but it does have a precise technical definition as what are called anomalous dimensions of operators. The check in this limit has many moving parts, and it is by no means trivial – but it works.
These simpler tests can be extended. The gravitational theory moves away from being merely a supergravity theory, with equations coming from a glitzed-up version of Einstein’s theory of general relativity, to being a full string theory. The simplest version of string theory is a classical string theory – an extension from general relativity to string theory, but where all the quantum effects have been switched off. The correspondence works. The tests can then be extended again to quantum string theory – and again the correspondence works.
These progressively more complicated tests give progressively more impressive evidence that the AdS/CFT correspondence really is true, and that the two objects it claims to be identical are indeed identical. Of course, it is true that these calculations are performed using a standard of rigour common in theoretical physics rather than in mathematics. It is also true, as said before, that the accumulation of any number of successful tests of the correspondence is mere inductive evidence, offering no logical proof of correctness – as with the examples of ‘grue’ and ‘green’ in the previous footnote.
Everything I have just said is words. However I do want to try and illustrate just why physicists find these tests so compelling, by moving beyond words to include an equation. I do so in the hope of providing a sniff of what is so powerful about this correspondence, by showing just how nontrivial this agreement is. I will write below an equation representing the result of a calculational tour-de-force: a five-loop calculation in super Yang-Mills theory. To professionals in the field, ‘five loop’ is to calculational difficulty what the Strongest Man in the World competition is to shifting some Christmas flab from your tummy. No higher level of precision has been attained for quantum electrodynamics, the oldest and simplest element of the Standard Model. The equation is9
Here ‘ζ’ refers to a particular mathematical function called the Riemann zeta function, named after the 19th century German mathematician Bernhard Riemann, and ζ(3), ζ(5), and ζ(7) are the particular values this function takes for values of 3, 5, and 7.
What matters here is not so much what the equation means as the rather intricate form it has. Look how complicated it is! This equation has been computed in two different ways, once using the techniques of quantum field theory and once using the techniques of string theory – obtaining perfect agreement. This is almost immediate evidence that something deep is happening – it is not possible to write a term like by chance, and it is clear the agreement between these calculations is no mere accident.
This is not a solitary example. There are many similar cases where the same quantity has been computed in two different fashions and in two very different theories, both on the field theory side and on the gravitational side, and complete agreement has been attained. As with this example, what is psychologically convincing is the deeply non-obvious form of the expressions encountered. These are not factors of two or minus signs. The expressions are the product of arduous calculation and involve results one could never guess in advance. It is striking that two different calculations, done in different theories and using different methods, give the same answer. This test of veracity is the theoretical equivalent of when two experiments on two continents use two techniques to measure the same physical quantity and get the same answer. When this happens many times, it does not prove a thing – but it does convince.
In fact, life is even better than this. The special theory referred to above is indeed special. Many of its properties have been studied extensively. This means that in some cases it is possible to obtain exact results for this theory. This is not possible in general quantum field theories and is only possible here because this theory has many more symmetries than a normal quantum field theory. Symmetries constrain. They enforce relations between quantities with no other connection. The more symmetries, the more constraints. For this particular theory, the symmetries can sometimes be enough to constrain everything so well that in the end there can only be one result: the exact answer. An exact answer in quantum field theory is normally an unattainable dream, and this is why all the many approximation schemes have been developed. However, the specialness of this special theory does in some cases allow for exact solutions, to which the five-loop quantity just described is ‘merely’ a leading approximation. For such exact quantities, it is possible to interpolate results smoothly between the regions where the gravity description is ‘easy’ and where the quantum field theory description is ‘easy’.
This section has described checks of the correspondence for its canonical avatar, between the maximally symmetric quantum field theory in four dimensions and a particular string theory in five dimensions. However, while this case offers the most precise arena for a critical tes
t, there are many other examples as well. These include cases of quantum field theories in two or three dimensions that are equivalent to gravitational theories in three or four dimensions. It also includes many cases of ‘less symmetric’ quantum field theories in four dimensions. These theories have more in common with a theory like the Standard Model but also have extra symmetries that constrain the results and make calculations more tractable. Again, the correspondence has been tested, and has not been found wanting.
To recap: while there is no formal proof of the AdS/CFT correspondence, most physicists are happy with its correctness. This is because it works. There are solid physical arguments for why it should work, and its implications have been validated in numerous disparate environments. The tests are not trivial, and involve the reproduction of intricate formulae with no obvious structure. At some point, and after many many successful tests, the question shifts from ‘Is there any interesting way this idea could possibly be correct?’ to ‘Is there any interesting way this idea could possibly be wrong?’.
Given it is correct, what should we use it for? One application is to understand quantum field theory for the sake of understanding quantum field theory. While this is a worthy goal with many papers written on the topic, it would be nice to apply the AdS/CFT correspondence to real systems in the real world. This is the focus of the next two sections.
8.3 APPLICATIONS TO COLLISIONS OF HEAVY IONS
According to legend, Hermes Trismegistus managed it. Isaac Newton tried – and failed, as did his contemporary Robert Boyle. What is ‘it’? ‘It’ is the fabled goal of alchemy – the ability to turn base metals into gold.
Newton and Boyle did not fail in their alchemical experiments because there was something intrinsically stupid about their attempts. It is only heat and pressure that divides pencil shavings from diamonds, and both cooks and cement manufacturers know well how easy it is to change the texture, colour and hardness of an object. It is true that the alchemical quest is one marked by many failures, and it is true that the sane and reasonable give up after decades of trying and not succeeding. However it is also true that the sane and reasonable are often not the ones who make major discoveries.
Why String Theory? Page 18