In the best of all worlds, this may result in the equations of general relativity predicting new states of exotic matter that can be observed in the lab. On the other hand, this research may end up only reproducing aspects of condensed matter physics already known for a long time. Which is true? How will this research be viewed in thirty years? We simply do not know. That is why it is called research.
1The use of the word ‘constant’ here is common but unfortunate language, as the value of these ‘constants’ depends, in a calculable way, on the energies of the particles that are interacting.
2One of the best-known examples of such an interesting quantity is called the Witten index, after Edward Witten. Roughly, it counts the difference in the number of bosonic and fermionic states in a quantum field theory.
3It is true that authors in the past cited less frequently and more discriminately. Inflation has applied to citation rates as well as the price of bread. While Weinberg’s paper has fewer absolute citations, its inflation-adjusted number would be higher.
4It is not possible to add any more supersymmetry to the theory without turning it into a gravitational theory. It therefore sits in a unique position as the theory with the largest possible amount of supersymmetry – but which is not yet a gravitational theory.
5‘Anti’ here does not have the same connotations as in anti-particle: there is no sense in which de Sitter space and Anti-de Sitter space can annihilate with each other!
6The statement ‘string theory in five dimensions’ is slightly loose: as alluded to above, is better to say that the string theory is in ten dimensions, and five of those dimensions are curled up into a sphere while the other five dimensions form Anti-de Sitter space.
7There is a technical subtlety here. The quantum field theory actually has two couplings, one related to the interaction strength and one related to the number of particles. In professional language, these are the gauge coupling and the ‘t Hooft coupling. Likewise, the gravitational theory also has two couplings, one relating to the interaction strength of strings, and one to the size of strings relative to the ambient geometry. In the canonical example of the correspondence, both couplings in the gravitational theory are small.
8There exists all kinds of philosophical agonising about the scientific process and the inductive method. One example: define ‘grue’ as the colour ‘green until today; blue from tomorrow’. The statements ‘grass is grue’ and ‘grass is green’ have identical empirical support, although we intuitively think the latter correct and the former unscientific. The jury may be out on whether this represents an intellectually productive train of thought, but for right or wrong physicists note these concerns in the way the Levite on the road to Jericho noted the naked and robbed traveller.
9For experts: this is the anomalous dimension of the Konishi operator in the limit of an infinite number of colours.
10Strictly, the Large Hadron Collider uses lead ions while its American counterpart the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory uses gold.
11The acronyms used in particle physics are not imaginative. ATLAS is A Toroidal LHC AppraratuS, CMS is a Compact Muon Solenoid and ALICE is A Large Ion Collider Experiment.
12Russian theoretical physics developed a distinctive style that produced many first-rate scientists. Its spiritual leader was the scientific polymath Lev Landau, who got to live in interesting times, being born in 1908 and dying in 1968. The holy text of Russian physics was the ten-volume Course in Theoretical Physics by Landau and his student Evgeny Lifshitz. Russian physics is also famous for its seminars, which can last many noisy hours until every claim of the speaker is either satisfactorily settled or refuted.
13Whether the resulting science should be called ‘fundamental’ is the subject of a long and mostly tedious debate. The ‘No’ argument is that many-body physics is only the study of the consequences of existing laws, and so does not by itself make new physical laws. The counter-argument is that the move from few to many is itself moving to a different regime of physics, in which the laws of few-body physics provide no guidance as to behaviour. The ‘no’ and ‘yes’ sides of this debate are associated with the words ‘reductionism’ and ‘emergence’. A cogent and elegant argument for ‘Yes’ can be found in the article ‘More is Different’ by Phil Anderson, one of the great names of condensed matter physics.
14As outlined in this book, the modern view is that the Standard Model is also in fact an emergent theory arising from a deeper underlying layer (such as string theory).
15Other fascinating and more recent examples are those of topological quantum field theories, which were developed by Edward Witten in the 1980s. These arise in condensed matter physics to describe states with effective fractional electric charge, as happens in the fractional quantum Hall effect.
CHAPTER 9
Why Strings? Mathematics
Physicists secretly know their subject to be superior to any other empirical science. It is engrained into the culture of the subject that physics and physicists can bring insights into other sciences, but that the reverse is not true. Physicists become biologists, but biologists do not become physicists. There are master’s degrees available for moving from undergraduate physics into the life sciences, but not for moves in the opposite direction. It is also understood in pectore, even if never to be uttered explicitly, that the reason for this is that physicists are just, well, smarter.1 Why, after all, did the discovery of DNA, the most important discovery of twentieth century biology, take place within a physics laboratory? As physicists, we are the best. We have the right stuff. We are the elect of every nation. We are Sheldon from The Big Bang Theory.
9.1 PEER RESPECT
There is one subject which is exempted from this benign condescension. That subject is mathematics. Mathematics is, together with astronomy, the most ancient of the sciences, with origins that predate recorded history. It is not an empirical science. The truths of its statements are not contingent on the results of any experiment carried out in this world. They are instead accessible to pure thought, starting from well-defined premises and moving to well-defined conclusions. In the language of philosophy, the truths of mathematics are analytic rather than synthetic.
For illustration, each of the following statements can be shown to be correct, without any need either to construct measuring apparatus or to perform observations.
1. There is no largest prime number.
2. Any map of countries can be filled in using four colours only, such that no adjacent countries have the same colour.
3. There is no way with only a straight-edge and compass to construct a square with the same area as a given circle.2
While mathematics does not require the natural world, much of mathematics is still inspired by the natural world. What makes for an interesting mathematical result is a matter of taste, but taste often prefers a starting point with at least a loose connection to the world of experience. The most successful mathematics book ever written, and also the most successful textbook ever written, is Euclid’s Elements. Written by the Greek geometer Euclid in Alexandria around 300 BC, this textbook about classical geometry was in continuous use for over two millennia as a workhorse, torment and inspiration for centuries of schoolboys.
The Elements include as a key postulate the statement that parallel lines do not intersect. This is true for spaces such as the surface of a table, and these flat spaces are sometimes now called Euclidean. It is not true in spaces that are not flat – for example, the surface of the earth considered as a whole, or the curved spaces that appear in general relativity. In these spaces, parallel lines do intersect, and Euclid’s fifth postulate is false – and so then are all the many conclusions in the Elements that rely on this postulate. Euclid’s postulates were chosen as starting points because in many ways they appeal to our natural intuition. This is one reason Euclid’s mathematics has had such lasting value: it is the mathematics suited to spatial geometry as we experience it.<
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The history of mathematics and the history of physics are not stories that can be recounted independently. There has been cross-pollination between the two to the mutual strengthening of both subjects. The problems of one subject can sometimes be fruitfully expressed in the language of the other, acting as a stimulus to development.
Early examples of problems that sit in both mathematics and physics are the problems of infinitesimals. These are often called Zeno’s paradoxes and are the philosophical paradoxes of classical antiquity. One of the best known of these is the question of how Achilles can catch a tortoise. The famously swift Achilles is racing after a tortoise. But how can Achilles ever catch it? For when Achilles has reached where the tortoise once was, the tortoise has moved on. And when Achilles has reached the place the tortoise is now, the tortoise is another small amount further ahead.
This question is a paradox rather than a genuine problem because we all know that in the real world Achilles indeed overtakes the tortoise. The difficulties lie in correctly accounting for infinitely many infinitely small steps occurring in infinitely many infinitely small units of time. A precise resolution of the paradox requires a precise formalism for dealing with these infinitesimals, which had to wait until the development of analysis (‘analysis’ here is the technical name of a branch of mathematics) in the early 19th century by the French mathematician Augustin Cauchy.3 This provides a rigorous treatment of such infinitesimals and how to sum them, thereby dissolving the confusions of Zeno’s paradox.
Another example concerns Newton’s development of calculus. This was undeniably an enormous achievement in mathematics: to this day calculus remains a pons asinorum across which many high school students slowly struggle. However, this development was done not as an abstraction, but as a product of his grappling with the messy problem of the motion and orbits of planets. Newton did not actually include calculus when writing out these results in the Principia. He was writing for a wider audience, and he laboriously expressed his results using the classical tools of Euclid that they would have been familiar with. However underlying all this was the slick and powerful machinery of the calculus, which had made all of Newton’s calculations vastly easier. While the problem of planetary motions is clearly a physics problem, it acted as the trigger to the mathematical development of calculus.
Moving to living history, a more recent example of the interplay between mathematics and physics occurred during the development of the quantum field theories that describe the strong and weak forces. It was realised in the 1970s that the equations being used here had natural interpretations in terms of the geometric structures which were being studied in mathematics at this time. The discoveries in geometry made in the 1960s and 1970s, led by people like Michael Atiyah, had natural applications to the physics of the strong and weak forces. Atiyah is a remarkable Lebanese-Scots product of a vanished society, having grown up under the British empire first in Khartoum and then in Alexandria. Beyond his outstanding research achievements, he was and is a mathematical panjandrum, ending his career with a list of honours that would have intimidated even Pooh-Bah, with the Fields Medal (the highest award in mathematics), the Presidency of the Royal Society and the Mastership of Trinity College, Cambridge being merely the most notable. The work of Atiyah and others merged the mathematics of the solutions of differential equations with the mathematics of topology – the branch of mathematics that explains what makes the surface of a bagel fundamentally different from a sphere, no matter how much you try to squeeze and press the bagel.
What had this to do with physics? As discussed in chapter 5, the 1970s was the decade in which the correct quantum descriptions of the strong and weak forces were determined. During this decade, work in this area transitioned from trying to discover the correct theory of the strong and weak forces to trying to understand the correct theory of the strong and weak forces. As experimental support for these theories grew, so proportionately more effort went into obtaining a deeper understanding of how to calculate in them and of what these calculations meant.
Both the strong and weak forces involve generalisations of the electric and magnetic fields. Any quantum mechanical calculation in these theories is performed, roughly, by adding up contributions from every possible configuration of the fields – genuinely every possible configuration. No matter how bizarre a field arrangement may appear, it still contributes. Certain configurations are more likely, and are weighted more strongly, but every possibility counts. The rules for how to weight different configurations were worked out by Richard Feynman, and this method is called Feynman’s path integral approach to quantum mechanics.
It was realised in the middle of the 1970s that certain rare field configurations caused processes to happen that could not occur in any other way.4 As one example, the proton – the particle out of which we are all made – becomes unstable in the Standard Model. Given enough time, it will decay to other particles. Proton decay is the ultimate ecological catastrophe – without protons, there are no atoms, and without atoms, there is no life. Fortunately, the values of the constants of the Standard Model happen to be such that not a single proton in our bodies would have decayed even after waiting many lifetimes of the universe.
What made these configurations special, and what distinguished them, was their topology. They involved arrangements of fields that, no matter how much you tried, no matter how much you bent them, you could never smoothly deform to a zero configuration. When did these special configurations contribute? What were the conditions? The answer was given precisely by the work of Atiyah and his collaborator Isidore Singer, in particular through the Fields Medal-winning theorem they shared, the Atiyah-Singer Index Theorem.
There is a second example from the same time period of the interplay between maths and physics. It comes from a famously productive 1960s collaboration between the physicist Stephen Hawking and the mathematician Roger Penrose. Hawking was trained as a physicist and has since become an iconic image of science; Penrose was trained as a mathematician and came from a distinguished polymath family. Penrose’s father was a noted psychiatrist and his brother was British chess champion.
Both Hawking and Penrose were interested in geometry, and in particular in the geometries that arose within general relativity. They were interested in questions such as: what happens when a blob of matter starts collapsing under gravity? It had been known since the 1930s that the collapse of a perfectly symmetric shell of matter leads to a black hole. However it was unclear what happened for more general circumstances. Did all the collapsing matter fall into a black hole, or could some of it be expelled via some kind of slingshot mechanism? Similar questions arose in the early history of the universe. If the expanding universe we observe today was extrapolated back in time so that it became progressively smaller and progressively denser, did the equations of general relativity necessarily lead to a singularity where infinities appeared and general relativity broke down?
Together, Hawking and Penrose solved these problems and showed that singularities were unavoidable. They did so using novel techniques that involved looking at the global geometry of spacetime, rather than simply the local behaviour of small regions. By looking at spacetime as a whole, Hawking and Penrose could see what caused what and what followed what. By applying these geometric techniques to general relativity, they were able to map out the entire picture of the geometry – and by doing so they were able to show that singularities were an unavoidable feature of classical general relativity. As the universe was extrapolated back in time, Hawking and Penrose showed that the equations signalled their own demise, and led inexorably to a regime where they ceased to be valid.
These and other applications rekindled the long dormant affaire de coeur between physics and mathematics. The physics that had triumphed between the 1930s and the 1960s was not mathematical in nature: the mathematics used did not come from the front line, and the development of physics and the development of mathematics were proceeding on separate
paths. Richard Feynman may have been a great physicist, but he and his generation left no direct trace on mathematics. They were physicists first and physicists second, and they successfully explained a succession of new data without requiring mathematical sophistication or results that would impress their colleagues in the next department. The 1970s and 1980s were different: for the first time in a generation, modern physics and modern mathematics were sparking off each other and pushing ideas in each other’s direction.
This section has described some examples of the longstanding conversation between mathematics and physics. A good reason for working on physics is because it provides novel ways of looking at mathematical problems and interpreting them. A good reason for working on an area of mathematics is that it involves topics arising in, and required by, modern physics.
These features have been well manifested in string theory. There are many people who work on string theory, or use ideas derived from it, because it can provide either solutions or insights into mathematical problems.
I want to describe two specific examples involving string theory, starting with the wonderful name and wonderful tale of the monstrous moonshine conjecture. This is the story of an interplay between the mathematics of symmetry, geometry, number theory and string theory. It is a story whose first wave lasted for around two decades, and for which smaller wavelets continue to break even now.
9.2 OF MONSTERS AND MOONSHINE
Once upon a time there was a beautiful symmetry that almost no one cared about. As we have seen in chapter 3, the basic notion of a symmetry is an operation that, performed on something, brings it back to itself. We saw there examples of rotational symmetry such as fourfold symmetry or sixfold symmetry. It is easy to see how this can generalise to arbitrarily large examples: hundredfold or thousandfold rotations. There are also other examples of families of symmetries: for example, permutation symmetry, under which groups of objects remain the same when shuffled.
Why String Theory? Page 21