The Standard Model represents one particular example of a quantum field theory. Of all possible quantum field theories that one can write down, the Standard Model is the only one nature is known to use as a fundamental theory. However, large numbers of physical systems are, in practice, described to very high accuracy by quantum field theories. The behaviour of electrons in metals is described by quantum field theory. The phenomenon of superconductivity is described by quantum field theory. Many, many complicated systems involving vast numbers of interacting particles are best described using quantum field theories.
The Standard Model, in principle, involves an infinite number of components: one for each point in space. Systems involving atoms may have only a million billion billion such degrees of freedom, one for each particle in the system. To a mathematician, a million billion billion is closer to one than infinity – two finite numbers are like any other when compared to infinity. To a physicist, a million billion billion is at least approximately infinite, and quantum field theory can be, and is, used with great success to describe such systems.
The exploration of quantum field theories in their many guises is therefore an active part of modern theoretical physics. It cuts across subject boundaries and is part of the toolbox of many. While the main rationale for this study is that quantum field theories do appear in physics in so many ways, it would be a mistake to think that all this research is performed with the goal of a rapid comparison of theoretical prediction and observational phenomenon. Much research on quantum field theory is instead carried out simply to advance the better understanding of quantum field theory itself, and this certainly includes the study of quantum field theories that are not relevant to describing nature. This may sound paradoxical – why study theories that are definitely known not to describe nature? However, it is often the case that the particular is illuminated by placing it in a broader context. You understand a child better by knowing his or her parents and relatives; you understand humanity better by knowing the evolutionary tree of other closely connected mammals.
Quantum field theories often belong to continuous families, labelled by the values of the parameters that enter them. For example, in the quantum theory of electromagnetism one such continuous parameter is the mass of the electron. If this were different, the quantum mechanics of atoms would also be different – and so also would be the chemistry and biology derived from them. In principle, every possible value for the mass of the electron corresponds to a different quantum field theory. In the real world, only one of these quantum field theories is realised – the electron mass is what it is, and theories with the wrong’ electron mass are not correct and do not describe nature. However, it is more than useful to regard the ‘right’ theory as part of a family of different theories with different values for the electron mass.
Understanding quantum field theory as a whole is also important because the individual quantum field theories that are most important – for example, the Standard Model – may be quite complex. It is hard to calculate in them, and easy to make mistakes. There are many difficulties and subtleties, and while some of these difficulties may be inseparable from the required calculation, others may be only present due to the baroque nature of the Standard Model.
It is helpful to distinguish unavoidable complexities from avoidable ones. The same problematic question may arise both in a theory with two particles and also in a theory with thirty-seven particles, and it is perhaps better and easier to understand it first in the former theory rather than the latter. Simplified toy models have a honoured place in physics, and one common approach in quantum field theory research has always been to understand its hard questions within controlled sandbox environments where calculations are tractable.
8.1 THE PROBLEM OF STRONG COUPLING
Of all the confusing but fun subtleties of quantum field theory, I will concentrate here on one ubiquitous and hard problem. This is the problem of understanding quantum field theories at strong coupling. What does this mean?
Many quantum field theories – for example, the Standard Model – are used to describe interactions between particles. If you fire beams of electrons at beams of positrons, quantum field theory tells you the ways these particles can interact and the probability of the different interactions occurring. These calculations are most easily done when all interactions between particles are extremely weak, and so there is little to no chance of double interactions. In chapter 3 we described this by using the metaphor of Arabella and Bert walking towards each other on the street. The case of very weak interaction is the case where Arabella and Bert are almost total strangers to one another – almost certainly they will carry on past one another to their respective destinations, and the chance of something happening is small.
The measure of interaction strength between particles is called a coupling constant.1 For the quantum theory of electromagnetism, this constant is called the fine structure constant and has a numerical value of 0.0073 – approximately . Different values for these coupling constants correspond to different quantum field theories. One can visualise these as being on a sliding scale, where we are free to slide the value of the constant up and down.
At one end of the scale is the limit of no interactions at all. This theory is called a free theory. It is the easiest limit of all. In this limit particles have no potential for interaction and pass each other as ghosts. In our analogy, Arabella and Bert are total strangers and walk straight past each other. As the constant is increased, we come to the point of weak interactions. There is a non-zero but small chance that particles will interact – but most of the time they still just carry on. Here Arabella and Bert are distant acquaintances, who nod, say hello, and go on their way. For this case of weak interactions, there is a stack of well-sharpened tools that can be used to compute the scattering probabilities. These tools are based on successive approximations: first the probability of no interaction at all, then for a single interaction, then two interactions, then three, and so on.
This procedure of successive approximations works well when the likelihood of any interaction is small. The higher interactions refine the answers, but do not change them qualitatively. The greater the required accuracy, the more interactions need to be included – but these only matter far to the right of the decimal point. The view obtained by including only the first term is fuzzier, but does not mislead.
For the case of strong coupling, this approach breaks down. The stratified hierarchy of calculational significance disappears. It is no longer possible just to compute the first terms of an approximation to obtain a rough picture of the final answer. Every term matters as much as any other – the case of sixteen interactions is on an equal footing with the case of a single interaction. To obtain the correct answer, we now need to find all the terms – an infinite number – and sum them up. Arabella and Bert are first cousins who have not seen each other for years: who can predict what will happen when they bump into one another?
It is important to say that being at strong coupling does not signal any intrinsic fault with the theory, which still makes perfect sense. The value of a parameter has been changed, but the theory is as well defined as it ever was. The problems lie with us, and our desire to calculate in the theory. The old tools have gone, and it is totally unclear what should replace them.
How to proceed? How to calculate? This problem of understanding quantum field theories at strong coupling is an old one that extends back fifty years. Its original motivation was the historic purpose of physics: to understand the natural world. For almost all purposes, one of the four fundamental forces – unsurprisingly, the strong force! – is strongly coupled. At any energy that was accessible prior to the start of the 1970s, the strong force lies in a regime where it is strongly coupled. At the time, this caused enormous bafflement. No one knew the defining equations of the strong force, or even had a clue where to start looking for them.
Ultimately, the crucial simplifying feature of the strong force w
as the fact that it is only strongly coupled at low’ energies. As colliders became larger, the energies of the colliding particles increased and, mirabile dictu, the strong force ceased to be strong. It became weak. It became calculable. It became accessible. All that had been needed was an increase of the kinetic energy of the colliding particles to a level much greater than the rest mass energy of protons and neutrons – and then suddenly the strong force could be analysed with familiar tools.
This does not make it any easier to understand the strong force at low energies, but it does give some clarity to the question. The fundamental equations for the strong force do not have to involve strong coupling. There are energies at which the strong force is strongly coupled, and there are energies at which the strong force is weakly coupled. While it remains difficult to compute in the strong coupling regime, one can also in a certain sense quarantine these regions and hive them off as a ‘bad’ part of the theory where calculations are difficult. The hard problem remains hard, but its boundaries are now clearly marked.
Once upon a time, the motivation to study quantum field theory at strong coupling was the desire to understand the behaviour of the strong nuclear force at low energies. This historic rationale is now less pressing, and this particular task is no longer considered an important frontier problem in theoretical physics. The reasons for this are several.
First, the strong force really does describe nature. As a result, vast computing power has been thrown at it. By formulating the problem carefully and throwing enough processors at it, many aspects of the behaviour of the strong force at strong coupling can be found. This is not ideal. It cannot address every aspect, it is not always elegant and it is not always insightful: but it does give correct answers. Computers calculate rapidly, do not get bored and do not need sleep.
The second reason is that, again given the importance of the strong force, every possible cheap trick that can be used on it has been exploited. For example, the masses of the up and down quarks are far lighter than the masses of all other quarks. This did not have to be true and we do not know why it is true. It is a particular, unexplained feature of the quark masses in the Standard Model. However, it does lead to a series of calculational simplifications that apply for the actual strong force but cannot be extended to more general examples of strongly coupled quantum field theories. Whatever the restrictions on their broader use, these little dodges provide information about the actual strong nuclear force that arises in nature.
The final reason is that physics moves on. Not all problems are equally important. Effort requires motivation, and attaining funding requires a good answer to the ‘Why are you working on this?’ question posed by both fellow scientists and research councils. When there was no idea what the correct theory of the strong nuclear force was, it was unclear what the angle of attack should be – any direction may have produced the key that turns the lock. However, once the correct theory is established, it is not so important to compute everything that could ever possibly be computed. Complicated problems that require time and effort to solve require a justification for the investment of time and resources. The full gravitational dynamics of the solar system, including all the minute little wobbles from the other planets and their moons, is studied today to investigate its stability and to ensure that rockets reach their destinations, not because we want to test Newtonian mechanics.
Today, the chief motivation for understanding strongly coupled quantum field theory is not primarily to discover new laws of physics. New physics is generally discovered first through weakly coupled effects rather than through strongly coupled effects. This was true for all three forces of the Standard Model and one may wonder why. It is not that the complications of strongly interacting physics are never relevant for nature. It is more that the first signs of new strongly coupled physics can be described as new weakly coupled physics. New physics almost always appears as small deviations from existing known laws – and small deviations can be described through weak coupling approximations. The reason for this is as follows. If a deviation is large, under less sensitive instruments the deviation would still be present, but small. Given incremental progress in technology, any new physics will appear first as simply a small deviation from existing laws, and such small deviations can always be parameterised through weak coupling techniques.
As an analogy, we could ask what sound is made by a stampeding herd of rhinoceroses. The sensory experience at the centre of the charge of the pachyderms would indeed be striking. However, the first answer would always be as a soft, distant patter on the ground. The soft patter will turn to a quiet rumble, and the quiet rumble will turn to full thunder: but the patter will come first.
However it arose historically, the upshot is that there are a large body of physicists whose research is aimed at understanding quantum field theory at strong coupling, and who are interested in any tools that may be used to calculate in this region.
Based on what I have said above, it may seem hard to conceive how any calculations can be performed in this area, even in principle. This book is not a textbook and is almost free of equations. Nonetheless, I wish to try and illustrate how this can work, by describing one method of computation that can be, and is, used to give results in this regime. It is certainly not the only method, but it is a real method. I describe it to illustrate how, despite all difficulties, controlled computations can actually be performed within a strongly coupled theory.
This method is based on the fact that the strong coupling regime is obtained by sliding the value of a coupling constant from weak values to strong values. We suppose the theory starts in the weakly coupled regime, within which it is ‘easy’ to calculate. As the coupling is smoothly varied, the theory continuously moves into the incalculable strong coupling regime.
The key word here is ‘continuously’. Whatever the change may be, it is continuous, and so whatever quantities we compute must also vary continuously as we go from weak to strong coupling. In particular, jumps are forbidden. However, suppose we consider a quantity that can both only take integer values – nought, one, two, three … – and is also forbidden to jump. Such a quantity must take the same value irrespective of whether it is evaluated at either weak or strong coupling, since the change has to be continuous and the quantity cannot jump. If this quantity is computed in a theory with weak or even no interactions, it will have precisely the same value as in the strongly interacting theory. If the quantity is chosen carefully, this will give interesting information about the behaviour of the strongly interacting theory.2
Many people find strongly coupled quantum field theory interesting. However, none of what has been said so far has anything to do with either string theory or quantum gravity. The merry community of quantum field theorists could have happily continued on their merry way, innocent of gravity in two, four, ten or twenty-six dimensions. However they did not, and this merry community is now infested with ideas from string theory. This is because it turned out that certain problems in strongly coupled quantum field theory could be best attacked, and indeed solved, using higher-dimensional theories of gravity. Even if your sole concern in life was an understanding of quantum field theory, it became apparent that one of the most powerful tools was string theory – and as the millennium approached, even the most staid of quantum field theorists became drunk on the new wine of string theory.
8.2 THE ADS/CFT CORRESPONDENCE
The date of this wine’s bottling was Thursday the 27th of November 1997 – Thanksgiving Day. On that day the Argentine physicist Juan Maldacena, then at Harvard, uploaded to the electronic preprint archive a single, short paper with the unprepossessing title ‘The Large N Limit of Superconformal Field Theories and Gravity’. Maldacena is a quiet and softly spoken man not given to large demonstrative gestures. The paper is technical, and on first glance might appear to be dealing only with minutiae. However on careful reading one realises that the paper makes a bold claim: certain quantum field theories are one and the
same as certain gravitational theories, specifically certain string theories. ‘One and the same’ means exactly that – they are completely identical objects written in different notations. Any calculation in one can also be done in the other. In the language of chapter 3, there is a duality between the two formulations – gravitational theories can be dual to quantum field theories.
One way to quantify the impact of Maldacena’s paper is through numbers. Since it first appeared, it has been cited by more than ten thousand other papers. This is a lot. To put this figure in perspective, every year around six thousand papers are published within theoretical high-energy physics. These figures imply that, over the eighteen years since it appeared, Maldacena’s paper has been cited by around one in every ten of the hundred thousand or so papers that have appeared. Another, maybe slightly embarrassing, perspective on this is the fact that this adds up to more citations than any other paper in high-energy physics ever – including all the foundational papers of the Standard Model, such as Steven Weinberg’s 1967 classic ‘A Model of Leptons’ with ‘only’ nine thousand citations.3
Why String Theory? Page 17