Why String Theory?

by Joseph Conlon

  String theory in 1996 was then no longer a theory of strings and no longer a theory in ten dimensions. What had not changed, however, was the view as to what fundamental problem ‘string theory’ was trying to address. String theory might now be called M-theory, and might now appear as a far richer theoretical structure than had previously been dreamed of – but its raison d’être was still as a possible theory of nature on the smallest possible distance scales. The form of the answer had changed: the question it was answering had not. Despite all that had been found, the style of work in this period was fundamentally similar in spirit to that of the late 1970s and early 1980s. It was motivated by and addressed the questions What is this theory and what is its internal structure?’ and not What data does it explain and what does it tell us about experiment?’

  String theory had started in 1968 as a theory of strong interactions and ended up in 1996 as M-theory. If these twenty-eight years had been a journey of abstraction from explaining data concerning the strong force to formal properties of the one true theory of quantum gravity, the period since then has involved travel in the opposite direction. Ideas and techniques that were originally spawned as offshoots are now used in their own right to attack many different problems. Far more people now work on string theory either as a framework for ideas or as a source of calculational techniques than on the apparently canonical question of quantum gravity. Indeed, in an amusing turn of the wheel one application of string theory’ is now, once again, to compute amplitudes for scattering via the strong force.

  We now turn to these more modern attitudes to string theory.

  1Following a conventional academic career, Peter Goddard became Master of St John’s College in Cambridge, before moving to become director of the Institute in Advanced Study in Princeton. He lectured me during my undergraduate studies in Cambridge, and invited the class for a party to the Master’s Lodge – where I learned of his impressive collection of model hippopotami. Charles Thorn is now a senior professor at the University of Florida.

  2The No-Ghost Theorem will reappear again in chapter 9 as part of the discussion of the monstrous moonshine program in mathematics.

  3The language used was a war of the words between Richard Feynman and Murray Gell-Mann. Feynman called them ‘partons’, and pretended not to understand the word ‘quark’; Gell-Mann took the reverse attitude.

  4To quote the Stanford physicist Michael Peskin on this work: ‘The remarkable hubris of this paper makes it required reading for every student.’

  5Ten years of apparent deadend deadbeat research on an area abandoned by the bright young things: it is fortunate that the university administrations of the 1970s were focused more on being universities and less on the outcome-centred impact attainable through leveraging transformative interdisciplinary synergies with stakeholder partners.

  6Those whom the gods love die young – the story of Joel Scherk is, like the story of another great Frenchman Évariste Galois (1811 – 1832), a tale of so much done and with so much still to do. Scherk died tragically young in 1980 at the age of 33 – although not in a duel like Galois!

  7In case anyone is wondering, the Standard Model is free of anomalies. This fact essentially guaranteed the discovery of the top quark, which after many years of searching was found in 1995 at Fermilab outside Chicago. Without this particle, the Standard Model would have been inconsistent – so it was not a shock when it was found.

  8More precisely, the cancellation occurred only for an SO(32) gauge group.

  9In the years leading up to the turn-on of the Large Hadron Collider, exactly the opposite trend was seen. String theory and other more formal topics became unpopular, and the physics of colliders was the hot topic.

  10What does the ‘M’ in M-theory stand for? It is a bit like the ‘S’ in Harry S. Truman – it is polymorphous and can stand for more than one word, for example ‘magic’, ‘mother’, ‘mystery’ or ‘membrane’ according to taste.

  11At first sight this may seem hard to get one’s head around; it remains so at twenty-first sight.

  12When I first read this paper I was quite shocked by its existence; according to the supposed history of string theory that I had ‘learned’, such a paper could not have been written for almost another decade.

  CHAPTER 6

  What Is String Theory?

  Although for many years the story of string theory had been viewed as an attempt to construct a fundamental theory of nature, few now see it solely in this light. The year on which this story pivots is 1997. Up until then, string theory was broadly felt to belong on the top floor of an intellectual tower. Its practitioners regarded it as inhabiting the penthouse suite of ideas: above quantum mechanics, above general relativity and above quantum field theory. Quantum field theory was regarded as something that arose in certain limits of string theory when stringy effects were removed. The converse was not true – strings, and quantum gravity, did not arise from limits of quantum field theory. Indeed, even classical gravity was absent from quantum field theory. The relationship of string theory to quantum field theory was like quantum mechanics to classical mechanics – an upgrade that, while backwards compatible, also added many new features.

  While the change was gradual, a principal source of the transformation in attitude was the discovery in 1997 by Juan Maldacena of what is called the AdS/CFT correspondence.

  6.1 QUANTUM FIELD THEORY BY ANY OTHER NAME

  The AdS/CFT correspondence will be the principal subject of chapter 8, but I shall give here a brief summary. The correspondence is a duality between a gravitational theory and a gauge theory, and it was the last really major result coming from the period around 1995.

  In particular, Maldacena claimed an absolute equivalence between particular quantum field theories and particular solutions of string theory. As a duality, this was in one sense similar to the earlier discoveries of the 1990s. It was a statement that two objects were identical, except expressed in a very different language. However, it was also a very different form of duality.

  In the previous examples, the theories related were theories at the same level. These dualities related quantum field theories to quantum field theories, or string theories to string theories. However different the details appeared, they involved objects on the same footing. Maldacena’s claim involved a string theory on one side and a quantum field theory on the other. The former was a theory of gravity, formulated in terms of extended objects. The latter was a theory of point-like particles. This time, the two sides of the duality involved different types of theory.

  This result certainly offers deep insights into quantum gravity, and so could be seen as part of the ongoing string theory research program at that time. It demonstrates the holographic nature of gravity – the statement that a gravitational theory in D dimensions can be equivalent to a non-gravitational theory in (D − 1) dimensions. While it is true that Maldacena’s correspondence is not universally applicable, and in particular only applies for cases when the energy associated to the vacuum of space is negative, it is general enough to provide powerful conceptual and calculational insights into quantum gravity.

  However, this same correspondence spectacularly muddles up the issues of what is fundamental and what is emergent. While quantum field theory is present within string theory, it turns out that for many interesting cases string theory is also present within quantum field theory. In some cases, string theory just is quantum field theory. Physics that had been thought to be the exclusive preserve of gravity was now arising in a subject that was not meant to have any knowledge of gravity.

  In the winter of 1997, Britain’s popular new Labour prime minister, Tony Blair, had been in power for six months with a landslide majority, while the territories of Hong Kong had just returned to the governance of the People’s Republic of China. In Moscow, one Vladimir Putin had recently joined the junior staff of President Boris Yeltsin, while in Connecticut the hedge fund Long Term Capital Management was on its way to total and
spectacular collapse. In the town of Palo Alto in California, Larry Page and Sergey Brin had just registered the domain name Google.com.

  At this time another answer to the question ‘What is string theory?’ had just emerged, and that answer was ‘quantum field theory’.

  When Maldacena’s original paper appeared on the arXiv, it was unclear how things would turn out. It came at a time when many exciting results had just been published, and there were many promising directions. The paper was rapidly recognised as important though, and large numbers of those who were interested in either string theory or quantum field theory jumped on it. Indeed, it took less than two years from the original paper for the first major review article to appear, an article which itself went on to become a standard reference from which many students would end up learning the subject.

  The early works focused on elucidation, clarification and evaluation of the correspondence. How did the correspondence work in practice? Which calculation in string theory corresponded to which calculation in field theory? Did they indeed match? The performance of such computations is the regular work of theoretical physics, and these checks have been used to provide greater or lesser levels of employment for almost twenty years now. As will be described in chapter 8, these checks now involve exceedingly intricate results, and there can be little serious doubt that the correspondence is correct.

  A large practical change triggered by Maldacena’s discovery arose from the fact that it also opened the door to a more pragmatic role for string theory as a calculational tool. If in certain cases string theory is equivalent to quantum field theory, then for these cases calculations in string theory are also calculations in quantum field theory. On consulting the dictionary provided by the correspondence, any calculation performed in string theory also has a meaning within quantum field theory. If you calculate in one, you calculate in the other. The value of this comes from the fact that under a duality, easy calculations in one theory are related to hard calculations in the other theory. This allowed string theory to be turned into a tool for doing hard computations in quantum field theory.

  During the first decade of the new millennium, this activity was probably larger than any other occurring under the name of string theory. It came in several forms. One part was devoted to understanding model examples of quantum field theories: field theories that did not and could not describe nature, but which served as tractable examples where these techniques could be deployed in all their power. These are typically theories with supersymmetry. For this purpose, some supersymmetry was good; more was better. The more supersymmetry, the more control and the more ability to deploy the correspondence with rapier-like precision.

  Another part was devoted to making AdS/CFT useful for field theories with some applicability to the real world. The best way to define ‘applicable to the real world’ was certainly not clear at first. The most obvious target was the theory of the strong force, quantum chromodynamics. The canonical examples of the AdS/CFT correspondence applied to theories which are cousins of the actual strong force, except with far more supersymmetry. If it were possible to lose the supersymmetry while retaining the correspondence, one might obtain interesting results for the actual strong force. This research, aiming to reproduce the properties of the actual strong force, has now been carried on for over a decade, with more or less degree of rigour and less or more degree of success.

  In the large review of 1999, reproducing the actual strong force was the only real world’ application mentioned. Since then, as we shall see in chapter 8, the number of applications of the AdS/CFT correspondence has proliferated. We briefly mention them here: one application is to understand what happens when heavy nuclei such as gold or lead are collided with one another, while another is to provide a new perspective on systems with both strong interactions and very many atoms. While both applications draw on string theory and explicitly use the AdS/CFT correspondence, it is also clear that they do not rely on quantum gravity. In these applications, string theory may be a tool and it may be a framework – but it is not serving as a fundamental theory of nature.

  One effect of Maldacena’s result was then to make string theory less special. It diluted the attitude that string theory was in some sense better’ than quantum field theory. If quantum field theory could be string theory, and string theory could be quantum field theory, then it was harder to argue with a straight face that string theory was more fundamental. While string theory was still a broader framework than quantum field theory, the sense of difference was reduced. It also greatly increased the number of ways in which string theory could be treated as a tool, compatible with an agnostic attitude to its status as the fundamental theory of this world.

  6.2 THE MISANTHROPIC LANDSCAPE

  In parallel with this, another topic which has seen a significant change over this period is the attitude towards string theory as a theory of the world on the smallest scales. String theory is naturally defined in ten, or possibly eleven, dimensions, which is in both cases a number much larger than four. The only way to obtain a world that looks four-dimensional is by compactification – curling up the extra, unwanted dimensions. The four-dimensional mappa mundi is then determined by the shape and geometry of the extra dimensions. Predictive statements about our world require knowledge of the extra-dimensional geometry.

  What determines this geometry? Broadly, there are two options. Either the dynamics of the theory determine the geometry uniquely, in which case it can take only one form, or they do not, in which case there are many options. If the former case were to hold, only one consistent choice of geometry would exist. This would lead to a uniquely predictive framework for four-dimensional physics. Just as quantum field theory predicts the mass of the positron to be identical to that of the electron, string theory would lead to a unique prediction for not only the electron mass, but all aspects of the Standard Model.

  On the other hand, if the latter case holds, string theory would have no more ability to predict the exact extra-dimensional geometry than Newton’s laws have to predict the exact number of planets in the solar system.

  The general attitude during the 1980s had been that it was the first case that would ultimately hold. String theory was not understood in many ways. We saw in the previous chapter some of the many unknowns then present. At the lowest levels of approximation, there did seem to be many consistent ways of curling up the extra dimensions. However, it was thought that these would melt away once the theory was better understood, leaving only a single allowed option.

  There was certainly no proof that this would happen. However, this attitude would have been moulded by the previous history of string theory, where a plenitude of choices in both spacetime dimensionality and the allowed forces had been shrunk to a single option by subtle and intricate consistency conditions. It was reasonable to believe that as understanding developed further, something similar would happen for compactifications. If it did, then string theory would lead to only one set of forces, one set of particles and one set of interactions – and possibly, an argument for why the Standard Model had to be the way it was.

  Over the last fifteen years this attitude has essentially disappeared. There is now a widespread agreement that string theory provides no unique route from ten to four dimensions. Several factors have influenced this change in sentiment.

  While during the 1980s, there was great uncertainty as to what would happen at strong coupling, now there is a far greater understanding, in particular via branes, of effects in string theory that go beyond weak coupling. For certain special systems, exact solutions have been found for both quantum field theory and string theory. With the domestication of the dragons and sea monsters, the idea that consistency conditions will select a single solution now seems less plausible. For cases with large amounts of supersymmetry, fully exact solutions appear to exist, with nothing wrong with them. Even for more ‘realistic’ solutions with small or zero amounts of supersymmetry, no gaping holes have appeared in
the approximate arguments that suggest the existence of many consistent ways of going from ten to four dimensions.

  As with any broad change in attitude, we should be careful not to attribute it solely to a single result or to the effect of one prominent person. From the very first days of compactified string theory, there have been those who expected string theory to produce a large number of solutions. Today, there remain some who hope that there still exists a unique, so far unidentified, selection principle that will select a single way of going from ten to four dimensions.

  The fact that the fundamental equations of a theory do not have a unique solution is not confined to string theory. Maxwell’s equations of electromagnetism allow for both radio waves and visible light. These equations constrain the allowed form of electric and magnetic fields in a volume of space, but they will not tell you what structure you actually find. In a similar way, Newton’s laws of gravity do not allow us to deduce that the earth has one moon but Jupiter over sixty.

  So, how many correct ways are there to go from ten to four dimensions in string theory? The apparent answer is infinity. There are a large number of exact supersymmetric solutions which have continuous parameters – in particular, type II strings on Calabi-Yau geometries. As these parameters are continuous, they can take an infinite set of values. With an infinite set of choices, there are an infinite number of solutions.1

  For all these riches, not a single one of this infinity of consistent compactifications can ever describe our world. This is because every last one of them preserves exact unbroken supersymmetry, and we know that supersymmetry is not an exact symmetry of our world. Exact supersymmetry would imply that the electron had a bosonic partner with the same mass and the same charge – but no such particle exists.