One example of this is the community of those interested in higher-dimensional theories of gravity, and in particular in the supergravity theories with ten or eleven dimensions. Today, these subjects are interwoven with string theory. Starting graduate students confidently describe ten-dimensional supergravity as the low-energy classical limit of string theory, and eleven dimensional supergravity as a classical limit of M-theory. As seen in chapter 5, it was not always so. These subjects have different parentage. Interest in higher-dimensional gravity theories came from the development of classical gravity theories, by a community founded on intricate and difficult computations in supersymmetric theories of gravity. During the 1970s and 1980s they developed these theories of higher-dimensional supergravities and studied many of their properties.
However following the explosion of interest in string theory in 1984, the validity of supergravity came under attack in many ways. Superstring theory showed that the right number of dimensions was ten – what were these people doing studying theories with eleven dimensions? These theories did not exist in quantum gravity. Even ten-dimensional supergravity was only an approximation to string theory. It was not the real deal. Why study the approximation when there was no need to approximate? Instead, it was better to study string theory – these classical computations could have nothing to say about the consistent quantum mechanical theory that was string theory. Instead of studying one-dimensional extended objects – strings – the supergravity folks were studying branes, objects with several spatial dimensions. This was clear further evidence, it was thought, that they did not know what they were doing!
As also discussed in chapter 5, this situation all changed in the middle of the 1990s, as it was realised that the branes that were being studied were an integral part of string theory – and furthermore had a remarkably simple description within string theory that greatly clarified their properties. The study of branes in supergravity and the study of branes in string theory were different sides of the same coin. They were part of the same subject. From the outside, the community of people studying branes in supergravity became ‘string theorists’ – but this occurred not through conversion but rather from a natural merger of the two subjects. If you wanted to understand non-perturbative effects in string theory, it would have been silly to shun the gift horse saddled with diamonds and rubies that was the supergravity literature on branes and eleven dimensions – and in a similar way the string theory picture of D-branes provided a new, simpler perspective on branes in supergravity.
Another example is the case of quantum field theory. Quantum field theory is the great foundational concept of the Standard Model. As seen in chapter 8, the wake of this success contains many groups of scientists who work on understanding quantum field theory better. How does quantum field theory behave when all the interactions are very strong? How does quantum field theory behave when there are lots of additional symmetries? Are there simple, particularly symmetric versions of quantum field theory that can be solved outright? What universal rules can be found for quantum field theory? This busy hive of scholars were and are only tangentially concerned with matching theoretical predictions to experimental data; they instead sought a better understanding of a general type of theory that was used in our accounts of nature. This busy hive of scholars also had no interest in quantum gravity and theories thereof: they wanted to understand quantum field theory in four dimensions and not quantum gravity in ten.
Nowadays, these quantum field theory experts are also experts about string theory. They discuss knowledgeably about higher-dimensional gravity and the different solutions of general relativity in both five and ten dimensions. They talk enthusiastically about strings and branes, and strings ending on branes, and branes intersecting other branes, and strings stretching between branes and the full general theory of branes. They go to string theory conferences, talk about string theory and get called string theorists. However, this is not because they were zombified into the Great Undead String Army. It is not that they came to string theory; string theory came to them and explained that many interesting properties of four-dimensional quantum field theories are best understood via higher-dimensional theories of gravity. As described in chapter 8, this arose through the shocking duality that is the gauge-gravity correspondence – in certain limits, higher-dimensional theories of gravity are one and the same as lower-dimensional quantum field theories.
As we saw there, ‘dictionaries’ were produced that translated between quantities in higher-dimensional gravity theories and quantities in lower-dimensional field theories. It is said that in transnational commerce you can buy from a company speaking your own language, but to sell to them you must speak their language. This is how the influence of string theory has spread – it came to the field theorists and showed them how it could be used to compute quantities of interest to them. As string theory was used to solve problems that were purely problems in quantum field theory and defined only within that framework, string theory grew to encompass the large community of quantum field theorists. This was not because field theorists suddenly became interested in string theory as a solution to quantum gravity, but because string theory was able to show them a way to solve their own problems and on their own terms.
The position of string theory within mathematics has followed a similar path. Various areas of mathematics have natural connections to string theory. One perfectly good reason for working on string theory is to understand mathematics better. This is true both for the overt mathematicians and also the covert ones, who live in physics departments and further their interest in mathematics under the name of physics. Mathematicians do not enter mathematics because they want to understand quantum gravity. Perhaps they want to understand geometry better or to classify algebraic structures, but if their prime interest had been the quantum mechanics of gravity they would have studied physics rather than mathematics as undergraduates.
Deep issues of space, time and quantum mechanics do not get you a hearing in the geometry department. New ideas and methods for counting the number of distinct curves in Calabi-Yau spaces do, and these new methods arose from the study of how the equations of string theory depended on the geometry of the additional dimensions. As we saw in chapter 9, these techniques are now all packaged under the name of mirror symmetry’. These techniques worked and gave correct answers, but they were initially as mysterious to mathematicians as if they had been devised at a witches’ sabbath. The problem of understanding how and why these techniques worked was one of the initial motivations for making many mathematicians interested in string theory. The ideas from string theory were foreign to mathematics, but better than the existing tools. Naturally, these mathematicians wanted to understand both how they worked and how they fitted into what was already known.
Mirror symmetry was simply the first avatar of what has become the sprawling field of physical mathematics. This area is in many ways part of mathematics. It involves the study of geometric structures and the relationships between them. It has the sense and feel of mathematics. It is not about describing the world as observed and is certainly not about predicting the results of laboratory experiments. However, in many ways it is also not traditional mathematics. The ideas that are taken as inspiration are drawn from physics, utilising the properties of quantum field theory (for example). The level of proof is also not at the level of conventional mathematical rigour; agreement in the values of two twelve-digit numbers, computed in entirely different ways, may be taken as convincing evidence for a result even though it is certainly not a proof in the mathematical sense.
We have also seen in chapter 10 the role of string theory as a factory of new ideas for looking for physics beyond that contained in the standard accounts of particle physics and cosmology. If string theory is true, it implies certain facts about the world. For example, the world contains extra dimensions. Even if not immediately observable, these extra dimensions will still leave some traces. One of these traces is the existen
ce of moduli particles with extraordinarily weak interactions, whose couplings to familiar matter are only at gravitational strength. The behaviour of these particles in the early universe can lead to modifications of the background light of the universe, the cosmic microwave background. Another trace are types of particles called axions. The ideas of branes and strings and extra dimensions also provide many other possible scenarios that could be true and have observational consequences – although not discussed here, another example is the possible existence of gigantic strings that are not microscopic but instead stretch across all of the universe.
It would be foolish and false to claim that string theory leads to unique predictions for any experiments that are doable at the current level of technology. This does not prevent it from generating a plenitude of scenarios that are perfectly testable, in the conventional sense, at doable experiments. So far, all proposed extensions for physics beyond the Standard Model have failed. However for this topic, success does not depend only on the scientists involved. The laws of nature are what they are, and we do not get to choose when the next discovery will be. Good ideas may or may not be rewarded. The only approach is to continue looking, and to continue thinking of ideas that may show up in experiment.
It is for reasons such as these, and also many others, that string theory has grown so much. It is not that there are so many more people interested in quantum gravity: it is in many ways due to the applications to areas with no connection to quantum gravity that the crowds came and stayed. As we saw in chapter 11, string theory has had much success in quantum gravity, and there are still people who work on string theory only or solely because they want to understand quantum aspects of the gravitational force. However, to regard these people as a majority would be to paint an inaccurate picture of the subject.
It is in many ways the success of the applications to areas outside quantum gravity that have also caused string theory to be viewed as the standard approach to quantum gravity. It is not true of necessity that the correct theory of quantum gravity should provide deep and profound insights into subjects with little obvious connection to quantum gravity. In fact, there is no logical requirement for this at all. It could be that the correct theory of quantum gravity has nothing interesting to say about mathematics, is disconnected from the Standard Model, tells us nothing new about quantum field theory and offers no additional insights into theories of classical gravity. It is possible in principle that the theory of quantum gravity is a stand-alone entity that is disconnected from the rest of physics. This does not feel correct, but it is not logically excluded. However, to many it seems unlikely, and this feeling explains why string theory is so widely viewed as the best candidate idea here. The presence of so many insights beyond the sticker claim of quantum gravity makes it more likely, it is felt, that the sticker claim is correct.
14.2 RIVALS AND COMPETITORS
The same has not been true for other alternative theories of gravity. Examples are asymptotically safe gravity, causal dynamical triangulations or loop quantum gravity. These theories tend to have larger profiles outside physics than they do inside it. The outer profile is determined by scientists or journalists writing for a general audience; the inner profile is determined by technical results that many scientists care about. One of the reasons many people have heard about loop quantum gravity is that two of the scientists who invented it, Lee Smolin and Carlo Rovelli, have written books explaining their ideas for a popular audience. They are good authors. They write fluently, and by doing so ensure science is present in the public forum. However, it is neither unfair nor unjust to observe that they are not neutral commentators on the ideas for which they are the parents.
Why have these ideas not been so successful among scientists? I am going to focus on loop quantum gravity as one particular example. The positive arguments for loop quantum gravity primarily involve arguments based on various principles that any theory of quantum gravity ‘must’ have. For example, quantum gravity ‘must’ be background independent, quantum gravity ‘must’ respect Leibniz’s principle of the identity of the indiscernible, quantum gravity ‘must’ respect the fact that spacetime has to be relational. On this view, string theory is unsatisfactory because it does not embody these principles. The fact that loop quantum gravity is said to incorporate these principles is claimed as a positive argument for this theory.
However most physicists – even most theoretical physicists – even most theoretical physicists working on ‘fundamental’ physics – are only mildly interested in quantum gravity, and they are even less interested in the philosophical principles it allegedly ‘must’ satisfy. If the discovery of quantum mechanics teaches one lesson, it is to be exceedingly humble about stating any principle nature ‘must’ satisfy. Most physicists are practical, and they like to calculate. Many have invested time learning string theory because it has helped them calculate the solution to a problem that were interested in solving, and they care far more about this than the correct answer to the question ‘Is nature relational?’.
These other theories of quantum gravity also offer little outside quantum gravity. I had personal experience of this when a postdoctoral researcher at Cambridge. A leading advocate for loop quantum gravity came to give a seminar. He was talking about a proposal to realise the Standard Model of particle physics based on octopus diagrams. I am someone who has never been that excited by quantum gravity per se – the problems are too deep and too removed from experiment for my own taste. Each to their own, but I have always been deeply interested in the origins of the Standard Model and for ideas to go beyond it. The talk went on, and the speaker enthused, but both the talk and the questions revealed that the speaker was less than fluent with the intricacies of the Standard Model.
The Standard Model is part of the unquestioned core of particle physics, tested and known to be correct many times over. As said earlier, it is logically possible that someone could think their way to deep principles of quantum gravity even with zero knowledge of the quantum theories describing the other forces of nature. Personally, I do not expect this. This talk made clear to me both that loop quantum gravity was not saying anything interesting about the Standard Model, and also that the leaders of this area did not seem to know in full detail the Standard Model, its problems, and its open issues.
Learning a new area in physics takes time: probably at least a year or so to do well. Speaking for myself, why should I learn any given approach to quantum gravity? How much time to devote to it? I think that quantum gravity as a whole is a much deeper and harder problem than the areas of physics I know well. I am less willing to listen to someone on quantum gravity if their understanding of simpler matters, which I know are correct and I know I understand, is not as high as I would like it to be.
This is a personal response, specific to me. It may be just a function of my own experience and interactions. However, this is what substantially reduces my own motivation to invest significant time in understanding other theories of quantum gravity. The same is not true of string theory; for example one of the most passionate and formidable advocates of string theory, David Gross, has, as we saw in chapter 1, also won the Nobel Prize for foundational contributions to the Standard Model.
However – so what? Has not loop quantum gravity reproduced many of the required features of quantum gravity? Even if it lacks any connection to the Standard Model, there is no reason for every hard problem to be solved. If loop quantum gravity is able to produce some of the key features required of quantum gravity, this surely counts as compelling evidence that the theory is touched with truth and is on the right track.
One claimed example is the entropy of a black hole. As mentioned many times in this book, Stephen Hawking and Jacob Bekenstein showed in 1973 that a black hole has an entropy, given by one quarter of its area when measured in units of the fundamental Planck length. A key test for any theory of quantum gravity is to reproduce that factor of a quarter – after all, in quantum gravity this i
s a computable quantity, and by carefully doing a computation one should be able to derive this number by enumerating all the possible constituents that make up the black hole. It has been claimed that this calculation has been done in loop quantum gravity, and that the answer is indeed a quarter.
At first sight, this sounds extremely promising. However, the situation is actually less appealing than it sounds. On the one hand, reproducing the entropy of black holes is a superb test of quantum gravity – the theory has to reproduce that factor of a quarter, otherwise it is incorrect. On the other hand, this is a dreadful test of quantum gravity. The reason why this is a dreadful test can be understood psychologically. Suppose you have a theory that you really and deeply believe is correct, and you have to do a calculation to test the theory – and you know in advance what the answer must be for your theory to pass the test, and it is just one simple number. This creates all the wrong incentives. There are incentives to fudge the calculation. There are incentives to appeal to physical intuition’ to paper over dubious steps in the calculation. There are incentives not to double-check and triple-check the calculation once you have the right answer. There are incentives to get the right answer because this will lead to a paper with lots of citations in an important journal.
It is important to realise that these incentives require no dishonesty at all, and I am certainly not suggesting any. It is just that tests where the person who knows the (simple) answer required, the person performing the test and the person most emotionally committed to the theory are all one and the same person are far from ideal.1 The traditional check of science on this false incentive is experiment. The scientific gold standard has always been to make predictions in advance of the experiment, and then to see these predictions verified. Nature is gloriously indifferent to human prestige and human desire; it is what it is and its answers are what they are.
Why String Theory? Page 35