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The beauty of superstrings
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High energy theoretical physicists, including, of course, string theorists, would hardly disagree with what Godfrey Harold Hardy, one of the finest twentieth-century mathematicians, once said: ‘‘The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.’’[source] (Italics in the original.) Like many mathematicians, theoretical physicists have also tried to convince the outsider that this is exactly what they do in their everyday work: they search for beauty. This idea, as we have seen in this essay, is well disseminated. Popular science writers and historians of science have given into this temptation and assure us that mathematical beauty is the most suitable guide to unveiling the laws of nature. For example, the French historian of science René Taton was convinced by his compatriots Henry Poincaré and Jacques Hadamard that the best guide for making progress in mathematics was that sense of beauty which in diverse degrees every practitioner of the activity possesses.[source] Under the influence of this deeply rooted belief and due in part to the absence of concrete historical analyses of the term, string theorists have been led to employ the word ‘‘beauty’’ in a vague sense; many times with religious connotations. This has given rise to inappropriate metaphysical pronouncements such as the following: ‘‘It’s the kind of beauty that might be hard to explain to a person from a different walk of life who doesn’t deal with science or math professionally.’’[source] According to this belief, mathematical beauty cannot be understood or explained, like a faith it must be felt. Since this ‘‘beauty of superstring theory’’ is a cliché which appears in every single account on the subject, and I could easily resort to hundreds of citations in order to illustrate this, I will instead try to decipher when, and in which circumstances, did string theorists start to talk publicly about it. The idea of this section is then to go through some of the most important articles that string theorists have published since the very beginning of the theory and try to grasp the meaning of the concept of beauty in every specific case. At the same time, I will try to uncover a possible pattern of change in their conception of beauty. The discussion of the previous section will help in this aim.

John Schwarz, considered to be the father of superstrings, has persistently insisted that the strongest motivation for pursuing his research in the early days of the theory was the internal beauty it showed. At a remembrance ceremony for his collaborator Joël Scherk, he said: ‘‘I think we were kind of struck by the mathematical beauty; we found the thing a very compelling structure. I don’t know that we said it explicitly, but we must have both felt that it had to be good for something, since it was just such a beautiful, tight structure.’’[source] And in another place he reiterates: ‘‘We felt that the theory has such a compelling mathematical structure that it ought to be good enough for something.’’[source]

In order to track the origin of this alleged ‘‘beautiful mathematical structure,’’ I have consulted several of the most important early review articles on dual models and string theory. There, I looked for explicit mention of the following words: simplicity, elegance, and beauty. Here it is what I found. (I did not take into consideration references of the kind, ‘‘introducing this parameterization the equation x looks more elegant,’’ or, ‘‘the previous notation allows a simplification of the expression x.” As in the following example: ‘‘Lovelace [25] has introduced a notation in which the results of the above papers can be expressed fairly simply and elegantly.’’[source] In all the cases I detected, the meaning of these uses is simply a desire for a simpler mathematical formulation, more manageable and less troublesome. Here this has little to do with an aesthetical criterion or an ontological conception; the main concern of this essay.)

The first mention of simplicity and elegance appears in Gabriele Veneziano’s review of 1973. This occurred five years after the publishing of his well-known paper on dual models, generally considered the starting point of string theory. In this later review he says:
What makes this approach interesting is that, in spite of looking a priori a desperate problem, construction of dual models has been achieved and the solutions are simple, elegant, and seem to have quite a few properties in common with nature. The problem is not completely solved though, for reasons to be discussed at the end of this review. On the other hand, the number of requirements a dual model has to obey is so large, that it comes as no surprise to find extremely small freedom to manoeuvre. This is of course what makes the dual program most fascinating.[source]

In addition to the standard S-matrix postulates, which Geoffrey Chew introduced in the mid twentieth century, dual models are defined according to an extra set of conditions. The general method is a standard procedure in theoretical physics: all the conditions to be imposed on the model are defined and then the necessary consequences are reached. From the application of the postulates it follows that the symmetries of these quantum relativistic models produce a spectrum that is free of ghosts, negative norm states, and includes low energy masses in accordance with experimental observations (states fitting the linear Regge trajectory). What was then needed was to check if the theoretical model would adjust to a broader range of experimental data yet to be carried out. This conundrum was recognized by Veneziano at the end of the article:
I am now faced with the most embarrassing question of where this duality game is going to take us in the future. Are we really close to a breakthrough explanation of the hadronic spectrum of particles and their interactions, or are we just being carried away from physical reality by some perverse though beautiful mathematical apparatus?[source]

These passages contain two of the three references to elegance and beauty made by Veneziano in his review paper. A third reference can be found in a paragraph going from page 209 to 210: ‘‘On the other hand, when unitarity is enforced on these Regge models, the resulting constraints seem to spoil their original beauty and simplicity.’’ (Italics added.) What he actually means by ‘‘the beauty and simplicity of Regge models’’ does not concern us directly since a few paragraphs above he had noted: ‘‘Hence we shall not need to enter into details of Regge theory.’’ But, what did he mean when he said that ‘‘construction of dual models has been achieved and the solutions are simple, elegant, and seem to have quite a few properties in common with nature’’? He meant that the stringent duality constraint imposed on the model, as expressed in equation (5.20) of the paper, was satisfied by a group of transformations that on the one hand were not very complicated and on the other were rich enough to generate the expected low mass states. This synthesis of an unsophisticated mathematical formulation and successful physical predictions was what Veneziano dubbed ‘‘simple and elegant solutions.’’ There is nothing new about this; it is simply the way physics has been done for centuries. And, what was he suggesting when he talked about the possibility of ‘‘being carried away from physical reality by some perverse though beautiful mathematical apparatus’’? Note that he said ‘‘beautiful mathematical apparatus,’’ implying that mathematics ought to be regarded, for dual model theorists, as a practical tool and nothing more. In this sense, the importance of mathematics for dual models was fairly similar to the role of complex numbers in electrical engineering: complex numbers can be very interesting, beautiful if you wish, however, engineers must restrict their attention to the resolution of real, sometimes highly complicated, physical circuits.

In his 1973 report John Schwarz wrote: ‘‘Since the Veneziano model is especially elegant for a particular ‘critical’ space-time dimensionality, it is natural to ask whether the model can be modified so as to change the critical dimension.’’[source] In fact, during 1971 and 1972 it was proved that the Veneziano model (which Veneziano had named the generalized beta-function model and which we now call the bosonic string) was free of ghosts for spacetime dimensions equal or lesser than twenty-six. The critical dimension Dc = 26 was especially attractive because the cancellation of the quantum anomaly in the algebra of projective transformations was accompanied by the existence of null states, giving rise, among other interesting results, to a complete basis for the construction of physical states. These unexpected properties of the solutions in the D = 26 of the Veneziano model is what Schwarz meant by ‘‘especially elegant.’’

Joël Scherk once used the term elegant in his review paper, though not referring to his own theory but to supersymmetry: ‘‘Section V covers the spinning string (Neveu-Schwarz-Ramond model) beginning with the elegant equations of Wess and Zumino (1973), then solving them and quantizing them. Fermions and mesons are obtained from two opposite boundary conditions of the classical equations.’’[source] It is interesting to note that even though this paper was published in January 1975 – only eight months after Scherk’s and Schwarz’s claim that dual models described ‘‘particles other than hadrons,’’ namely, gravitons − there is no mention of the ‘‘beautiful tight structure’’ of the theory, which Schwarz today tries to persuade us of.

These are the only allusions to simplicity, beauty and elegance I could find in these early articles on dual models and strings. There is again an explicit mention of beauty and elegance in the 1982 Schwarz paper on superstring theory, the new name assigned to dual models. In this new review there is no notable difference in his use of the term “elegant” to his previous paper published ten years earlier:
As already discussed in section 1, any attempt to define the superstring theories for D [less than] 10 is likely to lead to serious problems. Even if this were possible, I would still argue that the D = 10 choice is so much more elegant and beautiful that it may be more fruitful to try to interpret the extra dimensions physically rather than to reject them summarily in favor of less attractive alternatives. This approach — first proposed in the superstring context in ref. (138) and developed further in ref. (139) — requires that six of the spatial dimensions form a compact space small enough to have avoided experimental detection.[source]

During the late eighties and early nineties, and motivated by the relative success of the heterotic superstrings, string theorists were submerged in intricate and endless computations trying to recover the standard model using a ‘‘top-bottom’’ approach. At that time no one was talking publicly about a beautiful construct. In fact, the theory was in an ugly impasse and mathematical consistency was the only remote trace of beauty.

In contrast to string theory, the beauty of supersymmetry, and its local version, supergravity, was recognized very early on. In 1981, talking about supersymmetry, Peter van Nieuwenhuizen wrote metaphorically that: ‘‘It is the most beautiful gauge theory known, so beautiful, in fact, that Nature should be aware of it!’’[source] In a similar vein, Martin Sohnius wrote in a well-known review: ‘‘Supersymmetric theories (the subject of this report) are highly symmetric and very beautiful. They are remarkable in that they unify fermions (matter) with bosons (the carriers of force), either in flat space (supersymmetry) or in curved space-time (supergravity). Supergravity naturally unifies the gravitational with other interactions.’’[source] In fact, for a long time (from the late seventies to the early nineties) the idea of a beautiful unification of all the forces in a single theoretical framework was headed by supergravity theorists. At that time supergravity was the stronger candidate for a unified theory. In Sohnius’ review, the Introduction begin by stating that: ‘‘The aim of theoretical physics is to describe as many phenomena as possible by a simple and natural theory. In elementary particle physics, the hope is that we will eventually achieve a unified scheme which combines all particles and all their interactions into one consistent theory. We wish to make further progress on the path which started with Maxwell's unification of magnetism and electrostatics, and which has more recently led to unified gauge theories of the weak and of the electromagnetic, and perhaps also of the strong interaction.’’[source] At the end of the paper we find a short reference to superstring theory. In popular accounts too supergravity used to receive more attention than it does today. Only in the early nineties, motivated by the finding of mathematical dualities connecting the five different superstring theories and the proposal of the M-theory, did the discourse of a unified theory begin to be led by string theorists. Nowadays, indeed, it is very hard to find an article or a book proclaiming that supergravity is a consistent unified theory. In this respect, supergravity and superstrings have swapped positions. Why superstring theory superseded supergravity is another story.

Let us turn back to string theory. During the same period, late eighties-early nineties, something quite significant was happening. There was a growing public interest in science and in theoretical physics particularly. This move resulted in a considerable amount of popular science books being published and sold. We have already pointed out the books by Paul Davies, Roger Penrose, Kip Thorne, and Murray Gell-Mann. Other best-sellers were written by Stephen Hawking, Dennis Overbye, Steven Weinberg, and, the only string theorist, Michio Kaku. It is not hard to verify that all of them set out to exalt the beauty of the universe and the human quest for these ultimate mathematical truths. It was generally asserted in these accounts that physicists were reaching a theory capable of explaining the birth of our universe as well as its future death, in addition to everything it contained; the so called ‘‘theory of everything.’’ An old dream was reaching an end. As Kaku put it in 1987: ‘‘Over the last two thousand years, we gradually have realized that there are four fundamental forces: gravity, electromagnetism (light), and two types of nuclear forces, the weak and the strong. One of the great scientific puzzles of our universe, however, has been why these four forces seemed so different. For the past fifty years, physicists have grappled with the problem of uniting them into a coherent picture.’’[source]

Although we observe in these popular accounts a clear association between a mystical beauty of the universe and an ontological unity, ‘‘the unity of nature is beautiful,’’ we do not find in most of the string theory literature of the time a straightforward identification of the term beauty with the theory itself. In 1996, one year after the emergence of M-theory, an article entitled ‘‘Explaining Everything’’ appeared in the most popular scientific magazine worldwide. It begins by stating:
The Theory of Everything, or TOE, theorists believe, is hovering right around the corner. When finally grasped — the fantasy goes — the TOE will be simple enough to write down as a single equation and to solve. The solution will describe a universe that is unmistakably ours: with three spatial dimensions and one time dimension; with quarks, electrons and the other particles that make up chairs, magpies and stars; with gravity, nuclear forces and electromagnetism to hold it all together; with even the big bang from which everything began.[source]

In this extract, and throughout the entire article, ‘‘The TOE’’ is deemed to be the mathematical description of the beautiful unity of nature, however, there is still no reference to the beauty of the theory itself. Two years later, again in Scientific American, an article appeared making explicit mention of the M-theory: ‘‘Until recently, the best hope for a theory that would unite gravity with quantum mechanics and describe all physical phenomena was based on strings: one-dimensional objects whose modes of vibration represent the elementary particles. In 1995, however, strings were subsumed by M-theory.’’[source] Once more, there is no comment on the ‘‘beautiful mathematical structure of the theory.’’

In 1999 The Elegant Universe was published. Here, for the first time, string theorists publicly declared that they had found the ‘‘beautiful unified theory’’ grasping the ‘‘beautiful unity of nature.’’ In the first paragraphs of the Preface Brian Greene writes:
Instead, he [Einstein] was driven by a passionate belief that the deepest understanding of the universe would reveal its truest wonder: the simplicity and power of the principles on which it is based. Einstein wanted to illuminate the workings of the universe with a clarity never before achieved, allowing us all to stand in awe of its sheer beauty and elegance. … And now, long after Einstein articulated his quest for a unified theory but came up empty-handed, physicists believe they have finally found a framework for stitching these insights together into a seamless whole—a single theory that, in principle, is capable of describing all phenomena. The theory, superstring theory, is the subject of this book.[source]

Resorting to Einstein’s natural philosophy and “aesthetic criterion,” in addition to, of course, Einstein’s symbolic power, Greene sought to introduce superstring theory to the public at large as the most beautiful theory ever conceived. That this constituted the general opinion within the community, rather than a personal judgement, is confirmed by the fact that The Elegant Universe has been considered since its publication, by string theorists themselves, the essential reference book for every public discussion on the theory (see essay 1). The popularity of the theory during recent years has shown that to some extent they succeeded.

I think that popular science literature and all that it involves (an appropriate social and cultural context, science writers, an interested and stable audience, publishers, an established network of commercialization, among other factors) explains in great part the confidence, first veiled then declared, that many string theorists have had in proclaiming the beauty and simplicity of their theory. For them, as we have seen, it is crucial to know that their theory is a beautiful theory according to the established criteria imposed by other theories. However, the most important criterion is unity. In order to enforce their own criteria string theorists have intelligently interpreted the significance of the other theories and modified their alternative, useless, ‘‘aesthetic criteria.’’ A propitious context favoured them, and the logic for capitalizing on it is straightforward:
1. The general theory of relativity is a beautiful theory The standard model is a beautiful theory
2. If, general relativity + standard model = superstring theory
3. Then, superstring theory is a beautiful theory

A last word on string theorists’ motivations. What brought them to reinterpret the aesthetic criteria of previous theories according to their own needs and to take advantage of the broader context cannot be seen as a wicked practice meriting banishment from science. Firstly, reinterpretations are a common practice in science. Secondly, the degree of acceptance of competing theories by the “scientific” as well as by the “extra-scientific” milieu allow supporters of each camp to gain valuable positions in the field of struggles confronting their intellectual creations. As it is well known, this is one of the major purposes of modern scientific magazines and science books. No doubt, string theorists have been very clever on this sociological issue; much more than its contenders. The ground prepared, the next generations of string theorists are ready to admit and repeat the message: ‘‘String theory dovetails beautifully with the previous ideas for explaining the patterns in the Standard Model, and does so with a structure more elegant and unified than in quantum field theory.’’[source]

In this section we have seen that, in contrast to what is currently claimed, string theory was not always considered to be a beautiful theory. The public recognition of the beauty of the theory is recent, dating from around 1999, and it was due mainly to the convergence of two factors: a favourable context, “internal” and “external,” and an acute sense of opportunism. *******You can read