Why String Theory?

by Joseph Conlon

  The problem with too much mathematics’ as an objection is that it appears to be shorthand for there is too much unfamiliar mathematics compared to the mathematics I learnt as a student’. It is clear from history that advances in physics have very often required mathematics that was unfamiliar and that initially appeared bizarre. Mathematics that is necessary becomes familiar, and mathematics that becomes familiar becomes easier.

  The complaint that whatever progress has been made in string theory has not been through explaining experimental data is a true one. It is also a slightly unfair one. The book started with an account of the unreasonable success of the Standard Model, a theory that has been far more successful in explaining experimental data than it ever deserved to be. All data in particle physics is consistent with the Standard Model. So far, all searches for qualitatively new physics have been without success – and if anything is to blame’ for this fact, it is the laws of nature.

  Null results do give (some) information, but they are nowhere near as informative as discoveries. It is almost a tautology that if there is any progress that can be currently made about physics at quantum gravity scales, this progress will require more than just experimental data – and mathematics will play some role in it.

  CRITICISM: One of the major features of Einstein’s theory of general relativity is that it is background independent. Its formulation does not depend on a choice of coordinates. All that really exists are relations between objects, and any fundamental formulation of physics must be done in a way that does not depend on any particular choice of coordinates. In particular, any correct theory of quantum gravity must be background independent.

  However, string theory is not background independent. The standard formulation of string theory is in terms of an expansion in terms of small perturbations about a particular spatial background. String theory therefore always depends on a choice of a background. Its physics is not background independent, and consequently string theory is not a theory of quantum gravity.

  To my mind, the problem with this view is that it is based on a rather fixed ideological belief concerning what quantum gravity must be. The criticism is founded on the notion that one can first guess or deduce the principles underlying fundamental physics, and then construct the theory according to the principles. It expresses an overconfidence in the ability to know how everything will turn out, independent of input from either experiment or calculation. I am reminded of the (possibly apocryphal) response of Niels Bohr to Albert Einstein when he expressed his doubts about quantum mechanics:

  Einstein: God does not play dice.

  Bohr: Don’t tell God what to do!

  Let me make two more detailed responses. First, geometry is not fixed but manifestly dynamical in string theory. The fields that describe spacetime are not static. They have equations of motion, and these equations of motion cause them to change. In that string theory is an expansion about a fixed background, it is also a background that changes dynamically according to Einstein’s equations. Small changes build up to large changes, and large changes can be as large as one wishes.

  There is also an important distinction between the statement that the physics must be background independent, and the statement that the formulation of the physics must be background dependent. This may seem unclear. While quantum gravity may be esoteric, there is a more familiar topic in which one can re-express this same issue: cartography and the making of maps.

  How do we describe the geometry of the earth? There are two ways, one ‘background independent’ and one ‘background dependent’. The background dependent way is through an atlas of charts. If you purchase an atlas, on every page you will find a map of a different part of the earth. Depending on the purpose of the atlas, these maps can be of varied quality with varying levels of detail. They contain cities, towns and villages. They contains the contours of the land and the depths of the sea. They contain ship wrecks and sandbanks, castles and churches. Each chart only describes a small part of the overall picture: a ship sailing to Archangel will have little use for a map of Cape Horn. The charts also depend entirely on coordinates, as they have latitude and longitude lines stretched across them. Patched together however, the charts describe the surface of the entire earth: they are good for any purpose.

  There is also a ‘background independent’ way of describing the earth. This is through a globe. A globe provides a visualisation of the full geometry of the earth. With a globe there is no requirement of labels for latitude and longitude, or indeed any other choice of coordinates. Globes preserve perspective and area in a way that is not possible to do with an atlas, and they are excellent educational tools.4 However – it is not reasonable to argue that globes are ‘right’ and atlases are ‘wrong’. An atlas – which uses particular choices and charts to label every part of the globe – is a precise and correct description of the globe. In fact, the mathematical definition of any geometric space, technically called a manifold, is done precisely in terms of an atlas of charts. So there is nothing ‘wrong’ with the choice of coordinates – it is the choice to use an atlas of charts rather than a globe.

  The second objection is that ‘background independent’ is a slogan, and a hollow slogan without deep content unless further accompanied by a notion of what a background is. To claim to be independent of all backgrounds, it is first necessary to say what these individual backgrounds actually are.

  The simplest possible background is flat spacetime: a background that was already present in special relativity. Slightly more complicated backgrounds are the curved but classical geometries that arise in general relativity. These solutions are still, however, well approximated by (generalisations of) Einstein’s theory of gravity.

  What about more complex backgrounds than these classical, weakly curved spaces? In string theory, it has required many centuries of work to determine the large variety of possible different backgrounds that are permitted in the subject, and ‘background’ in string theory is a much richer concept than in general relativity. As seen in chapter 11, it must enlarge to include geometries that are of different topology. ‘Topology’ refers to the properties of objects that remain unaltered under any smooth change. The shape of a bagel cannot be deformed into the shape of a tennis ball no matter how much you knead it – you have to tear it. Any such geometric transition is entirely impossible in Einstein’s theory of gravity, as you cannot tear space. As also seen in chapter 11, string theory contains controlled examples in which the space changes topology. You can smoothly change the topology of spacetime in string theory without anything funny happening.

  Secondly, the concept of a background must also include examples where the background smoothly deforms, in a calculable fashion, from the classical picture of Einstein into a form of quantum geometry’. In quantum geometry the background no longer admits an interpretation in terms of classical notions of space. Coordinates no longer have any meaning – these represent an idea that sensibly applies only for the classical geometries your grandparents grew up with.

  As a final illustration, the range of backgrounds must also include geometric spaces of different dimensionality. As we saw in chapter 5, one of the most surprising results from the mid-1990s was that string theory taken as a whole has limits in which it is either a ten-dimensional theory or an eleven-dimensional theory – and it is possible to interpolate between the two.

  While ‘background’ in string theory is mostly a richer concept than in general relativity, it is also in some ways poorer. There are backgrounds that look very different in a classical theory of gravity, but that are identical in string theory. In string theory, T-duality implies very big spaces and very small spaces are the same. As backgrounds, they are absolutely identical. They are one and the same. This is not at all obvious at the outset, and it can only be seen by looking at the actual equations of string theory on an actual background.

  All these results were found through hard calculation, by looking at particular backgrounds in
detail and understanding what happens as small changes are made near those backgrounds. None of these results would have been easy to guess in advance.

  The danger with the assertion that background independence is a guiding principle of quantum gravity is that it tends towards an impoverished view of what is possible. Real content comes from knowing what all the possible ‘backgrounds’ can be. Once you know what all the possible backgrounds can be, you are a long way towards knowing what quantum gravity is.

  What the criticism does correctly capture is the fact that in an ideal world, you would have a formulation of string theory that gave you a view from which all these surprising results become ‘obvious’. From the right perspective, crazy relationships just become simple consequences of general principles. When this is attained, it produces one of these glorious moments of scientific ecstasy when understanding brushes aside confusion.

  This perspective does not yet fully exist for string theory. However the fact that such a formulation does not yet fully exist does not make string theory wrong – it just makes it a topic of research. It is like saying that because no-one can yet prove the elementary (and apparently true) statement that all even numbers can be expressed as the sum of two primes, we have no theory of prime numbers.

  In the end, ‘background independence’ is a rallying call. If someone believes that the only way to make progress is by following this principle and writing down, in one go, the full theory of quantum gravity, then that is what they believe, and no amount of result or calculation will convince them otherwise. In this respect, extensive argument with proponents of this view becomes like a discussion with either committed Marxists or members of the Chicago school of economics, where independent of the question the answer is either ‘dialectical materialism’ or ‘monetarism and free trade’.

  CRITICISM: String theory receives too high a fraction of the available resources for fundamental physics. String theory has promise, and it is reasonable that some people are interested in the subject and work on it. However, there are many approaches to quantum gravity and this should not be to the exclusion of other methods. In the same way that retirement savings should not be entirely invested in a single stock, resources in physics should be far more equitably distributed so that similar levels of attention can be paid to each of the different approaches to quantum gravity.

  While it superficially sounds entirely reasonable, this criticism contains two implicit assumptions. First, it assumes that there is actually a lot of money spent on string theory. Second, it assumes that this money comes from a large pot of soft goodies, which is jealously guarded to prevent it being shared out. In this world, there should be more than enough money to go round, with some to spare. The only reason it is not is because of bad behaviour by string theorists, who hoard these resources, keeping them for themselves and their friends.

  The fault with this criticism is that it supposes an idyllic world entirely disconnected from the practicalities of funding. In the real world, scientists almost entirely get money to do research by asking funding bodies for grants. If I want to get money to do research on string theory (as I do, and as I have done), I do not do so by asking a committee full of my chums. Instead, I have to make my case to a panel from many different specialisms. The vast majority of this panel will have never worked on anything even tangentially related to either string theory or quantum gravity, and indeed may not even be working on particle physics. I have to convince this panel both that I, as an individual, am worth funding and also that the topic I propose to work on deserves public money. For the largest grant, by cash terms, that I have been awarded, the relevant committee involved sixteen people, of whom a total of one – precisely one – was in even the broadest and most generous interpretation of ‘my area’.

  Success in such grant applications depends on both tangible and intangible factors. The tangible factors involve both past history and publication record: the papers you have written and the number of times they have been cited. Past performance is no guarantee of future success, but it certainly helps in a grant application. There are also the intangibles – the fluency of a presentation and the ability to make a research proposal convincing and comprehensible to those outside the subject, all mixed with the individual perversities and predilections of the interview panel and its members.

  The people making the decisions to spend money on string theory, then, are not string theorists. Grants are hard to get and grant applications are competitive. The success rate for the major grants that launch independent careers can be smaller than ten per cent, and the large majority of applications fail. Scientific funding is not a Care Bears’ tea party. Panels have to decide where limited resources can be most productively spent, and every penny obtained is obtained by convincing those outside your field that what you do is worthwhile and deserves to be funded. There is no soft pot of money available for those who would like a good salary to develop their own theory of quantum gravity at the taxpayers’ expense.

  Why has string theory been successful in this endeavour? The long answer has been given throughout this book. The short answer is that, as seen in chapters 8 to 11, string theory has proven to be so much more than just quantum gravity – and by doing so it has become attractive to large numbers of scientists.

  In the next and final chapter I summarise this positive case for string theory, and also explain why string theory has in fact been preferred over other alternative theories of quantum gravity.

  1This passion for abstract topics cannot however compete with fourth-century Byzantium under the emperor Theodosius, when fishmonger and carpenter would passionately debate in the marketplace the relative merits of the homoi-ousian and the homo-ousian nature of Christ.

  2And earlier – around two billion years ago, a natural nuclear reactor operated at Oklo in Gabon in central Africa, fissioning much of the uranium present via a chain reaction.

  3More precisely, probability in quantum mechanics is the square of an amplitude. Feynman’s prescription was to first sum the individual amplitudes for each path and then to square this sum.

  4Maps in an atlas can either preserve area and violate angles, or preserve angles and violate areas. The most familiar atlas projection of the globe is the Mercator projection. This preserves angles but does not preserve area – making the relative size of Europe compared to Africa appear far larger than it actually is.

  CHAPTER 14

  Why String Theory?

  14.1 REASONS FOR SUCCESS

  Why have so many chosen to study string theory? It is an esoteric theory that requires many years of study to appreciate. It is not a commercial subject offering large financial rewards at the end. It is also not experimentally validated. Unlike the Standard Model, which has run the gauntlet of data so many times, there is no direct evidence that string theory is a correct theory of nature.

  What is the reason for the success of string theory as an idea? As we saw in the previous chapter, there is a negative account of this success which goes something like the following. String theory is an approach to quantum gravity, one of many. All have virtues and all have problems. As it is extraordinarily difficult to find direct experimental probes of quantum gravity, we cannot know which of these many approaches is the correct one. Given this uncertainty, it would be best to pursue research in quantum gravity through a happily diverse community working on a variety of areas. Instead, powerful people working on string theory are only willing to hire other string theorists. These people in turn then hire more string theorists, leading to a situation where the only route to a professional career is through working on string theory. String theory then became the dominant approach to quantum gravity not through any merit of its own, but rather through the Cosa Nostra method of protecting family and destroying competitors.

  While this argument is basically false, it does contain a grain of truth. The grain of truth is that it is simply true that the quality standard required for the student of Professor Bigshot La
rgecheese to get a job is lower than that required for the student of Professor Podunk Smalltown. Senior people with power and influence do try and lean jobs towards their friends and connections, and this certainly exists as a factor in determining who gets hired to which jobs at which universities. While this nepotistic habit of favouring friends and mentees is not to be commended, most would also regard the suggestion that this illness is particular to the groves of academe as reflecting a touching naivety about the way the world works.

  What then lies behind the success of string theory? A wise and wealthy entrepreneur once observed that the route to riches in a gold rush is not through discovering nuggets but instead through selling shovels and pans. In this analogy, the large nugget of pure gold corresponds to the true and experimentally validated theory of quantum gravity. The shovels and pans are calculational techniques, mathematical insights and applications to other parts of theoretical physics.

  One of the major themes of this book has been that while there are many people who can loosely be called string theorists, very few of them actually work on quantum gravity or have their main interests in quantum gravity. The above complaint is that nefarious trickery lies behind the large numbers of people working on string-theoretic approaches to quantum gravity. The more accurate statement is that string theory has been found useful by many people who are in no way interested in quantum gravity. The enormous growth and professional success of string theory is because so many physicists with no a priori interest in it found that it had interesting things to say about topics that they cared about.