The technical argument for the finiteness of string theory is that the structure of the theory always allows potentially dangerous high-energy divergences to be reinterpreted as harmless low-energy divergences. In any place where it can be tested – including for far more complicated perturbative calculations than could be performed in 1985 – this principle has continued to hold over the last thirty years.
We still do not know fully how string theory works or what its most fundamental principles are. However, it by now requires something approaching dishonesty for a professional to doubt that it exists as a consistent theory of something. A Victorian engineer confronted with the latest mobile phone would be totally baffled as to how it works. He would have no possible conception at all as to the nature of the internal circuitry – the transistor would not be invented until long after his death – but he would also have no doubt that this circuitry worked.
It may still be said: that may be so, but why cannot someone still just take a few months to write down a proof? The answer is that physics is not easily amenable to proofs, and proofs cannot be found even for topics far simpler than string theory. As mentioned in chapter 8, there is a one million dollar prize available from the Clay Mathematics Foundation for proving one of the basic features of the Standard Model: the presence of a mass gap in the strong force. This is the statement that there are no massless particles charged under the strong force – there is a ‘gap’ to the first allowed mass. Compared to questions involving quantum gravity, this is a baby problem. The Standard Model is much simpler than quantum gravity. The techniques are far more understood. This question is also accessible to experimental study. There is also a one million dollar incentive – and yet there is still no proof.
CRITICISM: String theory comes in so many forms that it is impossible to make any predictions. There are an almost infinite number of ways to compactify down from ten dimensions to four. Each way represents a different string theory, and each will lead to entirely different physics. String theorists themselves say that there are 10500 such possibilities, and so if you can get 10500 different theories you can get anything you want out. A theory that can predict anything is a theory that predicts nothing. A theory that makes no predictions and is not falsifiable is not science.
This criticism contains several errors and exaggerations, which I will address below. The criticism also contains an attitude to falsifiability characteristic of Popperians of the strict observance, which I note but shall not challenge.
The first main error is that it conflates the questions of ‘What is science?’ and ‘What is the state of current technology?’. It is clear that the ability to test any idea experimentally is a function of the technology of the time. Nuclear physics was just as true in the stone age as it is today,2 and it will remain just as true if we are returned thither through some war or catastrophe. Today, nuclear physics and the associated quantum mechanics is testable. In the past they were not, and in the future they may not be either. Their scientific truth, however, endures.
That said, it is always better – not least for the scientists involved! – when ideas can be tested within a few years of their proposal, or at most within the lifetimes of the scientists. It would have taken the heart of a Vulcan not to rejoice in Peter Higgs’s pleasure in living to see the discovery of the Higgs boson in 2012, at the age of eighty-three and almost fifty years after his paper on the topic. No one enters science for the money, but that does not confer immunity from the human desires for recognition and acclaim.
Science is also healthiest when the interchange between theory and experiment is rapid. Wrong ideas, like aggressive weeds, are best killed quickly, and experiment is the best killer of them. Science moves fastest when theoretical ideas are closely coupled to experiment. However, sub specie aeternitatis it is ultimately irrelevant whether bridging the technological gap required to test a theory takes ten years, a hundred years – or longer. Democritus was no less right that the world is made from atoms for having died over two thousand years before the construction of the periodic table.
It is clear that the natural scale of string theory is not the scale of atoms and is not the scale of the Large Hadron Collider. It is the scale of quantum gravity, and whatever that may be precisely, we certainly know it is far smaller than any distance scale we can currently access. Our inability to access this scale is technological, but not a question of principle. Given magnets large enough and long enough, we know how to accelerate protons to quantum gravity energies.
However, the Large Hadron Collider currently represents the best that we can do. If money were no object, we could do better; but as seen in chapter 11 even then there is no open path to studying physics directly at the Planck scale. All current technologies fail long before we reach these scales. While history teaches us to be exceedingly modest when attempting to constrain future ingenuity, it is clear that predictions for the Planck scale are for the moment a question of principle rather than practice.
Nonetheless, what are the predictions of string theory at these quantum gravity scales? In brief, they are extra dimensions, extended objects and soft scattering. As we have seen in chapter 10, from a four-dimensional perspective extra dimensions manifest themselves as additional particles: ten-dimensional gravity has many more internal degrees of freedom than four-dimensional gravity. This statement remains true whether the extra dimensions are classical geometric dimensions or quantum stringy dimensions with no easy classical interpretation.
Likewise, strings are characterised by an enormously rapid – an exponentially rapid – growth of the number of harmonics with energy, corresponding to the many possible directions in which a string can vibrate. As we have also seen, the scattering of strings (or any other extended object) at high energy furthermore has the distinctive feature of soft scattering – colliding objects have minimal tendency to go off at right angles from the collision axis.
These predictions are not hard to test. Once you have a microscope that is capable of resolving sufficiently small lengths, there is no mystery about how to test the relative claims that the electron is a particle or the electron is a string. You use the microscope, and you go and look. Indeed, no philosophical agonising about falsifiability occurred when string theory in its original incarnation was proposed as a theory of the strong force, and the characteristic length of strings was thought to be a femtometre. The reason string theory was originally ruled out as an account of the strong force was precisely because, as more experimental data arrived, its predictions totally and spectacularly failed to accord with this data.
If you can look at the quantum gravity scale, string theory is then not hard to test. At this point a rider is sometimes added to this objection: what about M-theory? The different string theories are all meant to be different limits of M-theory, but the equations of M-theory are unknown. If you cannot say fully what string theory really is, how can you say it is testable? How can you make any statements about predictions without a full definition of what is meant by the theory?
There are two answers to this. The first, conservative, one is to say simply that the above statements about testability apply only to all the work done on string theory in the last thirty years. In that string theory is a topic that has absorbed real people’s time, it is testable in this sense, and these statements certainly apply to all the work that caused anyone to be interested in string theory in the first case.
The stronger, but still reasonable, response is that extra dimensions and extended objects are always present in string theory, and extra dimensions do carry physical meaning. As we go to higher energies, these extra dimensions become apparent and the number of particle-like states grow enormously. Likewise, the presence of spatially extended objects – whether strings or the branes of M-theory – is something that one can always look for once sufficiently high energies are attained.
Returning to the original criticism, the second inaccuracy is that it is not true that an almost infinite nu
mber of ways to go from ten dimensions to four dimensions implies an almost infinite number of possibilities for four dimensional physics. The number 10500 sounds large – and it is. However, as discussed in chapter 6 it is dwarfed by the number of genetic permutations that can arise when mummy and daddy get jiggy and make a unique human being. Despite this, observation of siblings and their parents belie the notion that infinite variation is therefore possible – and while there may be far more than 10500 possible human genomes, we can predict with good accuracy the number of fingers someone has.
Specifically, we have seen in chapter 10 that theories with extra dimensions leave characteristic legacies in lower dimensions. Almost always, there are additional light particles with very weak interactions: moduli, additional hidden forces or axion-like particles. These particles are simply a feature of extra dimensions and are present in any theory with extra dimensions. Their existence is therefore insensitive to the many different ways of moving from ten to four dimensions. It is not a theorem, but I am trying and failing to think of any counterexamples.
As we have also seen in chapter 10, there are many ways to look for such particles experimentally. These searches are not easy, and success is not guaranteed, but this situation is hardly unique to string theory.
It is certainly true that, as a fundamental theory of nature, string theory is hard to test. Of course, it would be undeniably nice to have an experiment with existing technology that was capable of giving a definitive answer about whether string theory – or anything else – was a correct description of physics at scales fifteen orders of magnitude smaller than those we are able to probe directly. To which the only response is: yes, it would be nice.
CRITICISM: Modern physics, of which string theory is an example, ignores philosophy and does so at its peril. It is not reflective, but instead attempts to develop the subject following the ‘shut up and calculate’ tradition. In doing so it cuts off the hand that feeds it; it believes it can answer foundational questions while ignoring foundational thinking. The development of relativity required input of philosophical ideas such as Mach’s principle; there is no reason to suppose the much harder problem of quantum gravity should be any different.
The essence of this criticism is that many of the deepest problems in physics are philosophical in nature. What is the nature of space? What is the nature of time? What are the basic principles that any quantum theory of gravity must satisfy? The argument made is that blind calculation is not enough – these questions cannot be answered without philosophical reflection, and that this process has been systematically rejected. The particle physicists of the 1960s and 1970s, flush with data, could get away with rejecting philosophy. However for problems without abundant data, this attitude is presumptuous at best and idiotic at worse.
Where this objection chiefly fails is in a conflation between the concept of ‘philosophy’ and ‘what those calling themselves philosophers do in the philosophy department’. Nature does not divide itself by university department. Up until the nineteenth century, what we now call science used to be called natural philosophy. Isaac Newton’s most famous work is called ‘Mathematical Principles of Natural Philosophy’. In the title, he makes the statement that natural philosophy is best done with the language of mathematics – while also gently alluding to Descartes’ non-mathematical 1644 work Principles of Philosophy. Science’ at that time was just natural philosophy – the philosophy of nature.
While the name has changed, the essence of the subject has not. For example, Richard Feynman was famously disparaging about philosophy – ‘low level baloney’ was one of his more polite comments. But, Feynman was also the person who reformulated quantum mechanics as a sum over all possible histories of a system. If you want to know what is the quantum mechanical probability for a particle to go from A to B, Feynman said, then you can do it by adding up contributions from all the possible paths there are from A to B.3 Which way did the particle go? It went every which way. All paths contribute, and we do not and cannot say more. This is a deep truth about nature, and it a deep truth that deals with the same branch of knowledge that Aristotle’s Physics did.
An example more relevant to string theory is the case of the holographic principle. This is the statement that the physics of a gravitational system in D dimensions can be captured by the physics of a non-gravitational system in (D − 1) dimensions. This is a statement that is made sharp in the AdS/CFT correspondence, which gives a precise mathematical formulation of it. This is one of the major components of string theory research in the last twenty years, and no criticism of string theory can simply excise this topic from consideration. But – how can the holographic principle not be regarded as philosophy? In any way that philosophy is worthy of the name, how can such a deep statement about nature not be called philosophical? It is every bit as deep as any of the ideas that fed into the development of relativity, and the sharpness of the calculational tests of it can only be a virtue and not a vice.
AdS/CFT is an example of a duality. There are other dualities that provide similar examples. In chapter 5 we encountered T-duality, which is essentially the statement that in string theory very small spaces are indistinguishable from very large spaces. Despite its surprising identification between two very different geometries, T-duality is still one of the best understood dualities in string theory. The mathematical subject of mirror symmetry that we encountered in chapter 9 can be seen as a generalisation of it. How can T-duality not be regarded as a philosophical statement about what space really is? Furthermore, it is a result backed by precise calculations. Just as in the time of Newton, a statement should not be seen as less philosophical merely because it is backed by mathematical evidence.
My general response to this criticism is then that on any historic reading of what counts as philosophy, or on any self-respecting notion of what philosophy encompasses, string theory does not ignore philosophy. It is instead part of (natural) philosophy.
The narrower statement that string theory is deficient as a theory of quantum gravity because it pays insufficient attention to what is going on in the philosophy department is simply weak (or, as Feynman might say, baloney). It is the same sort of baloney as the argument that your plumber might not be able to fix the drains because she is not an expert on the Victorian novel. It may be true that a plumber would be a better plumber for a wider knowledge of literature, but it is hardly the crucial aspect of the job. One part of my employment involves teaching physics at New College in Oxford, and one of the many pleasures of working in an Oxford college is a greater-than-average exposure to professional philosophers – who are intelligent and sensible people who do not make this sort of silly argument.
CRITICISM: String theory is too mathematical and has lost touch with actual physics. Physics advances through experiment, and it is extremely dangerous to believe it is possible just to think one’s way to the answer without any input from observation. Practitioners of string theory have become obsessed with mathematical beauty and regard it as a reliable guide to truth. However their idea of ‘beauty’ may be false, and other people may find different ideas beautiful. Furthermore, it is not mathematical beauty that is relevant in evaluating a physical theory, but success in explaining experimental data. The ‘beauty’ beloved of string theorists leads them to ten spacetime dimensions; this is in manifest contradiction with observation.
This criticism certainly contains elements of truth. There are many who work on string theory who are entirely uninterested in either observational input or output. It is not what motivates them. They are interested in the formal structure of theories or in mathematical applications of them. The prospect of explaining experimental data is not what gets them out of bed in the morning.
There is nothing wrong with this. Mathematics is a worthy subject, and it is not less important because it does not involve experiment. Studying string theory for its mathematical applications is an entirely sensible reason to study it. There is a valid question, w
hich I have sympathy with, as to whether too many people are currently working on the subject for reasons only tangentially related to physics. This is a legitimate question about distribution of funding, effort and resources, but it is a question of a different kind.
What is not defensible is the idea that string theory cannot be relevant for physics because many aspects of it involve advanced mathematics – where ‘advanced’ means significantly more mathematics than was needed for the formulation of either the Standard Model or general relativity. Mathematics is certainly not the only guide to truth, but it is historically true that advances in mathematics and advances in physics have fitted together hand in glove.
Furthermore, what precisely is meant by ‘too mathematical’? Difficult mathematics has been encountered in physics before. This is what Max Born, one of the founders of quantum mechanics and winner of the 1954 Nobel Prize, had to say about the start of quantum mechanics:
By observation of known examples solved by guess-work [Heisenberg] found this rule and applied it successfully to simple examples …
I could not take my mind off Heisenberg’s multiplication rule, and after a week of intensive thought and trial I suddenly remembered an algebraic theory which I had learned from my teacher, Professor Rosanes, in Breslau. Such square arrays are well known to mathematicians and, in conjunction with a specific rule for multiplication, are called matrices.
Matrices are now the type of diddy topic that are taught in school and professional physicists end up unable to remember not knowing. Looking further back, Cartesian coordinates – labelling graphs with an x and a y axis – were also at their time a shocking innovation. However useful, obvious and natural they may seem to us, the hard truth is that their discovery eluded Greek, Arabic, Chinese and mediaeval science and mathematics.
Why String Theory? Page 33