*******You can read this blog for free! Please, do not copy its content.*******
*******You can read this blog for free! Please, do not copy its content.*******
Beauty in twentieth-century theories
*******You can read this blog for free! Please, do not copy its content.*******
In this section, the essence of the discussion concerning beauty and physics during the last century is exemplified by four cases: Maxwell’s equations, quantum mechanics, particle physics, and the general theory of relativity. I also assess how string theorists have incorporated interpretations of these theories into their own and in turn how this process has affected the way in which standard physics is now viewed and taught.

A. Maxwell’s equations:

When Maxwell began his studies on electric and magnetic phenomena, in the 1850’s, four laws were known to be of proven validity: Coulomb’s law, accounting for the interaction between static electric charges; Ampère’s law, describing the orientation in space of the magnetic field created by an electric current; Faraday’s law, quantifying the electromotive force produced by a varying magnetic flow; and, finally, the equation imposing the absence of magnetic monopoles. As the historical evidence shows, the problem confronted by Maxwell was the incompatibility of Ampère’s law with the local conservation of electric charge; the latter expressed in the continuity equation of charge and current. Maxwell’s solution, the introduction of the displacement current, has been the subject of much concern among historians of physics, especially due to its significance in contemporary teaching of electromagnetism. This has been particularly emphasized by the historian of electromagnetism Daniel Siegel. In his thorough study of Maxwell’s theory he warns: “Each year many thousands of students in physics courses through the world learn that Maxwell, on the basis of the theoretical considerations, modified Ampère’s law, through the introduction of a new term called the displacement current, and thereby perfected the enduring foundation for modern electromagnetic theory. The centrality of this episode in the history of physics, its paradigmatic status as an example of theoretically motivated innovation, and its prominence in the pedagogy of physics have all contributed to making it a topic of prime concern for historians of physics.”[source] That is, instead of going through an historical account of the early reception and adaptation by others of Maxwell’s findings (which would divert our attention away from our main aim: contemporary interpretations by string theorists), I will obviate physical considerations of the time and concentrate more on the nature of the inconsistency of the set of equations Maxwell was confronted with and its subsequent reinterpretations. The “paradigmatic status” of Maxwell’s equations and its symbolic function in string theory is the subject of this subsection.

That Maxwell’s investigation was conducted by a ‘‘physical way of thinking,’’ with mathematics as a useful tool, is a fact according to historians of science. However, under the strain of subsequent theoretical and experimental achievements, we observe the temptation by many active physicists and contemporary science writers to overestimate the role of mathematical consistency in the elaboration of the electromagnetic theory. In the same book mentioned above, Paul Davies also writes: ‘‘Maxwell found initially that the equations looked unbalanced; the electric and magnetic parts did not come in quite symmetrically. He therefore added an extra term to make the equation look more pleasing and symmetric. […] Nature obviously agreed with Maxwell's aesthetic sense!’’[source] This temptation to reify Maxwell’s equations and to play with them looking for consistency is very normal in theoretical treatises on electromagnetism.

Take, for example, Jackson’s Classical Electrodynamics.[source] This textbook has been acclaimed since its first edition in 1962 as the definitive reference on the subject; taught for decades to advanced undergraduate and graduate students across the globe. As the author emphasizes in the Preface, the theoretical approach is favoured: “My choice of topics is governed by what I feel is important and useful for students interested in theoretical physics, experimental nuclear and high-energy physics, and that as yet ill-defined field of plasma physics.”[source] Even though he is here talking about the last part of the book, containing new topics, the advanced mathematical level of the whole material presupposes a theoretical inclination from the reader; a feeling for mathematical consistency, so to say. In the section “Maxwell’s Displacement Current, Maxwell’s Equations,” after a thorough presentation of electrostatics and magnetostatics, Jackson starts by writing down “Maxwell’s equations” without the displacement current (equations 6.22 in the text); then, he writes:
In fact, the equations in set (6.22) are inconsistent as they stand.
It required the genius of J. C. Maxwell, spurred on by Faraday’s observations, to see the inconsistency in equations (6.22) and to modify them into a consistent set which implied new physical phenomena, at that time unknown but subsequently verified in all details by experiment. For this brilliant stroke in 1865, the modified set of equations is justly known as Maxwell’s equations.[source]

At first, it is not clear what he means by “inconsistent as they stand,” because, in fact, nothing, from the mathematical point of view, prevents them from being correct. But turning the page we get the answer: they are inconsistent because they do not agree with the continuity equation for charge and current. Afterwards, in two simple steps Jackson is able to tell the student how to reproduce the right equations; written at the bottom of the page. Now they are “mathematically consistent with the continuity equations.” In this simple form ― of course, also motivated by pedagogical reasons ― Maxwell’s contribution is reduced to physical ingenuity and “mathematical consistency.”

Jackson’s presentation is an example of what the historian of physics Daniel Siegel has pointed out to be the paradigmatic status of the displacement current episode. Just one case, though an important one, among the many prevalent theory-laden readings of the history of physics. As we shall see, for string theorists Maxwell’s case is an exemplary one. For them, this is one of the best examples of what mathematical consistency can lead to, as Jackson says, “new physical phenomena, at that time unknown but subsequently verified in all details by experiment.”

Let us illustrate this well-established interpretation of Maxwell’s equations with another widely used textbook: The Feynman Lectures on Physics.[source] This set of lectures, delivered by one of the most creative and influential theoretical physicists of the twentieth century, has been the most important contribution to physics undergraduate education in fifty years. Thousands of physicists were, and some still are, trained according to the idea that ‘‘if we take away the scaffolding [the mechanistic ether model] he used to build it, we find that Maxwell’s beautiful edifice stands on its own. He brought together all of the laws of electricity and magnetism and made one complete and beautiful theory.’’[source] (Italics added.) In his lectures, Feynman approached Maxwell’s equations in the same way as Jackson does in his book; that is, prioritizing mathematical consistency. However, there is an ingredient in Feynman’s passage that I would like to stress here. I mean, the completeness of the unification of electricity and magnetism ― something that seems to be absent in Jackson’s presentation. Mathematical consistency, according to this interpretation, does not only provide us with new discoveries, it can also give us a unified and complete description of apparently diverse phenomena. In crude terms, the discussion so far can be summarized as follows: if you take a set of equations and play with them intelligently, that is, fit them in a consistent way, you will obtain a beautiful theoretical edifice. Future experiments will prove that the procedure was correct.

Mathematical consistency and unification are two aesthetic attributes that string theorists, following the paradigmatic interpretation of Maxwell’s equations, ascribe to their model. In order to see how string theorists have exploited in their favour this widespread interpretation of Maxwell’s legacy, let us consider once again Barton Zwiebach’s textbook for novices. Since string theory is a theory of unification, we shall first examine what he says about this attribute and how it is related to Maxwell’s theory of electromagnetism. At the very beginning, in the first paragraph of the Introduction, he narrates why he thinks string theory “fits into the historical development of physics”: “Over the course of time, the development of physics has been marked by unifications: events when different phenomena were recognized to be related and theories were adjusted to reflect such recognition. One of the most significant of these unifications occurred in the nineteenth century.”[source] Of course, he means Maxwell’s equations. And, after a brief summary of the main discoveries of the nineteenth century in the realm of electricity and magnetism, he concludes saying: “These equations [Maxwell’s] unify electricity and magnetism into a consistent whole. This elegant and aesthetically pleasing unification was not optional. Separate theories of electricity and magnetism would be inconsistent.”[source] As we shall see in the rest of this essay, the unification program is one of the strongest motivations for pursuing research in string theory. And, in this sense, Maxwell’s exemplary case is a source of inspiration.

But Zwiebach also refers to the other aesthetic attribute currently associated with Maxwell’s equations: experimental consequences from mathematical consistency. In between the two previous quotes, Zwiebach notes that before Maxwell’s discovery there were several laws describing all the electric and magnetic experiments of the time (he is certainly thinking about the set of equations written down in 6.22 in Jackson’s book). However, Zwiebach notes that “they were, in fact, inconsistent. It was James Clerk Maxwell (1865) who constructed a consistent set of equations by adding a new term to one of the equations. Not only did this term remove inconsistencies, but it also resulted in the prediction of electromagnetic waves.”[source]

It is worth pausing for a moment in order to repeat Zwiebach’s arguments once more; for they sum up very well the assessment that string theorists usually make of the introduction of the displacement current in particular, and Maxwell’s theory in general. Firstly, Zwiebach, interpreting the historical event, accentuates the role of Maxwell’s work in unifying electric and magnetic phenomena. Whatever the historical accuracy of his account, as I have argued, he relies on the mainstream interpretation of the episode. In this sense, we must recognize that he is very lucid with respect to the message he wants to pass on to his audience: string theorists, following Maxwell’s example, look for unity, for a ‘‘consistent whole.’’ Moreover, string theory tries to solve, as Maxwell’s theory did, the oldest and most crucial problem of physics: unification. Secondly, string theory is, or should be, as Maxwell’s equations are, an ‘‘elegant and aesthetically pleasing unification.’’ However, it must be remembered that beauty in its own equations is “not optional,” it is mandatory. Thus, this implies that the final unification can be reached by pure aesthetic judgements.

Let us illustrate how string experts have incorporated Maxwell’s discoveries into their theory. Electromagnetism tells us that electricity and magnetism are of similar nature: varying magnetic fields produce electrical phenomena and vice versa. Some have suggested, and not only string theorists, that this co-dependent relationship is confirmed by the fact that under the switch E B and B → − E, Maxwell’s equations in vacuum remain unchanged. Edward Witten, in an article intended for physicists working in various areas, affirms that this duality ‘‘has been known for nearly as long as the Maxwell equations themselves.’’[source] Saying this Witten intends to persuade the reader that string theorists’ use of the duality EB does not rely on an elaborate and desperate reasoning, but an analogous procedure has always been at the root of Maxwell’s theory itself.

The limitation of the duality EB to the vacuum condition has been superseded by contemporary theoretical physicists who have constructed models where the electric/magnetic duality is accomplished even in the presence of both electric charges and magnetic monopoles. A construct of this sort was built in 1978 by Georgi and Glashow; soon after, monopole solutions were found by ’t Hooft and Polyakov. Interesting extensions to supersymmetric models, such as N = 2 and N = 4, were subsequently devised and monopole and dyon solutions obtained.

Theoretical physicists have also generalized these results to dimensions higher than four. In technical terms, an electric gauge field Ae(p+1) in D dimensions ¬is identified with a (p+1)-form whose dual is the (D – 3 – p)-form associated with the magnetic dual field Am(D-3-p). Notice that with p = 0 and D = 4, Maxwell’s duality is recovered: Ae1 = Am1. That is, in four dimensions magnetic and electric fields are interchangeable. The physical interpretation of these mathematical properties is as follows: as well as a point particle generates a 1-form electric field, and a (p+1)-form is attached to a p-dimensional object, the corresponding dual fields are produced by dimensionless magnetic monopoles and (D – 4 – p) dimensional objects. Multidimensional objects of this sort have dramatically changed the way string theorists conceive their theory: ‘‘Researchers have gradually realized that string theory is not a theory that contains only strings. A crucial observation, central to the second superstring revolution initiated by Witten and others in 1995, is that string theory actually includes ingredients with a variety of different dimensions: two-dimensional Frisbee-like constituents, three-dimensional blob-like constituents, and even more exotic possibilities to boot.’’[source] These ‘‘p-branes,’’ known to be solutions of the supergravity equations, are essential objects for connecting and assembling the “final theory of physics”: the M-Theory.

The previous discussion suggests that string theorists have taken the latest developments of Maxwell’s theory and incorporated them successfully into their own theory. In addition to these concrete achievements, the historical interpretations of string theorists, and string theory enthusiasts, on the implications that Maxwell’s equations have had on the construction of modern unified models, are diffusing into many different areas of physics. Consequently, this influence is modifying the mode classical electrodynamics is currently being conceived of and taught. In the introduction of an intermediate textbook on electrodynamics, a textbook published a decade ago, the impact contemporary research in unified models has had on the teaching of Maxwell’s equation is made explicit.
Einstein dreamed of a further unification, which would combine gravity and electrodynamics, in much the same way as electricity and magnetism had been combined a century earlier. His unified field theory was not particularly successful, but in recent years the same impulse has spawned a hierarchy of increasingly ambitious (and speculative) unification schemes, beginning in the 1960s with the electroweak theory of Glashow, Weinberg, and Salam (which joins the weak and electromagnetic forces), and culminating in the 1980s with the superstring theory (which, according to its proponents, incorporates all four forces in a single ‘‘theory of everything’’). At each step in this hierarchy the mathematical difficulties mount, and the gap between inspired conjecture and experimental test widens; nevertheless, it is clear that the unification of forces initiated by electrodynamics has become a major theme in the progress of physics.[source] (Italics added.)

The introductory section is titled “Advertisement: What is electrodynamics, and how does it fit into the general scheme of physics?” It is remarkable that Zwiebach also opens his introduction with an historical assessment of its subject: “We see how it [string theory] fits into the historical development of physics, and how it aims to provide a unified description of all fundamental interactions.” The author of the textbook on electrodynamics concludes by confirming what we have been saying until now: ‘‘So electrodynamics, a beautifully complete and successful theory, has become a kind of paradigm for physicists: an ideal model that other theories strive to emulate.’’[source] From the point of view of string theory, we cannot but agree with this statement.

B. Quantum mechanics:

Quantum mechanics was conceived alongside experimental results and grew strong on the solid soil of a massive amount of physical data. This well-known experimental backdrop of quantum mechanics has compelled string theorists to associate with it an aesthetic discourse that differs substantially from that constructed around electromagnetism. To examine how string theorists evaluate the aesthetic attributes of quantum mechanics is the goal of this subsection.

Spurred on by Einstein’s philosophy and under the influence of the authoritative mathematical teaching at the University of Göttingen, the chief creators of quantum mechanics had a high regard for “beauty” in the physical sciences. On his early contribution to quantum mechanics, de Broglie recalled that his 1923 revolutionary proposal affirming that measurable particles also behave like waves was based entirely on its intellectual beauty.[source] Dirac also underlined the unusual beauty of Schrödinger’s equation: “I found myself getting into agreement with Schrödinger more readily than with anyone else. I believe the reason for this is that Schrödinger and I both had very strong appreciation of mathematical beauty, and this appreciation of mathematical beauty dominated all our work. It was a sort of faith with us that any equations which describe fundamental laws of nature must have great mathematical beauty in them. It was like a religion with us.”[source] It must be said, however, that despite their yearning for order and beauty in the realm of microphysics, the founders of quantum mechanics never attained it satisfactorily. It was never clear what they meant by beauty. The historian of science Arthur I. Miller has for years insisted on the importance of the role of aesthetic considerations in the early development of quantum mechanics. Miller has devoted special attention to the dispute between Schrödinger, with his wave mechanics, and Heisenberg, with his preference for, according to Schrödinger, complicated mathematics and-non-visual representations of atomic processes. Nevertheless, his evaluation is not often brought up in present discussions among theoretical physicists interested in the aesthetics of science. Indeed, and contrary to Miller’s historical interpretation, quantum mechanics is generally not considered to be a beautiful theory.[source] Only particle physicists (to be discussed in the next subsection) satisfied their longing with the extensive exploitation of the concept of symmetry first introduced in the microphysical world by Eugene Wigner.

Dirac’s equation, the first to combine quantum mechanics and relativity, deserves a special mention. Let us give some examples as how Dirac’s equation and personal conception and attitude on beauty and science has influenced contemporary appraisements of the subject. Two historians of science are emphatic on Dirac’s momentous contribution to the discussion: “Inspired by the views of Albert Einstein and Hermann Weyl, Dirac, more than any modern physicist, became preoccupied with the concept of ‘mathematical beauty’ as an intrinsic feature of nature and as a methodological guide for its scientific investigation.”[source] (Italics added.) Steven Weinberg opens his chapter “Beautiful Theories,” contained in his popular book Dreams of a Final Theory, with a significant story: “In 1974 Paul Dirac came to Harvard to speak about his historic work as one of the founders of modern quantum electrodynamics. Toward the end of his talk he addressed himself to our graduate students and advised them to be concerned only with the beauty of their equations, not with what the equations mean. It was not good advice for students, but the search for beauty in physics was a theme that ran through Dirac’s work and indeed through much of the history of physics.”[source] (Italics added.) Something similar was sustained by Chen Ning Yang: “To many physicists today what Dirac said contains a great truth. It is astonishing that sometimes if you follow the guidance toward beauty that your instincts provide, you arrive at the profound truth, even though contradictory to experiments. Dirac himself was led in this way to the theory of anti-matter.” As a final example, consider what John Barrow, a professional cosmologist and science writer, says about Dirac. According to him, most theoretical physicists, like Dirac, “make aesthetic quality a guide or even a prerequisite for the formulation of correct mathematical theories of nature.”[source] Concerning Dirac’s equation, Frank Wilczek claims that “of all the equations of physics, perhaps the most ‘magical’ is the Dirac equation. It is the most freely invented, the least conditioned by experiment, the one with the strangest and most startling consequences.”[source]

Hence, it should come as no surprise if we found out that many physicists are familiar with the following anecdote: “When someone asked him (as many must have done before) ‘How did you find the Dirac equation?’ he is said to have replied: ‘I found it beautiful.’’’[source] Or with the following passage by Dirac, published in a popular scientific magazine thirty five years after his revolutionary paper:
It is more important to have beauty in one’s equations than to have them fit the experiment … . It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one’s work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory.

Echoing Dirac’s innumerable declarations like these where he affirmed that beauty was a requisite and a guide to theoretical progress, modern theoreticians have assumed that beauty can show them the road to truth. Like Maxwell’s equations, the aesthetic criterion that supposedly led to Dirac’s equation has become an exemplary case.

The originality of Dirac’s philosophy cannot be fully understood without referring to his interpretations of electromagnetism and general relativity (remember that, in addition to his great devotion to general relativity, he did pioneering work on magnetic monopoles). According to him and his followers, what he did was to seize, bring up to date, and apply ‘‘Maxwell’s legacy’’: take two physical theories, combine them in a consistent manner, and get something unexpected. This interpretation of Dirac’s contribution has been determinant in the constitution of quantum field theory, as stated firmly by Weinberg: ‘‘The point of view of this book is that quantum field theory is the way it is because (with certain qualifications) this is the only way to reconcile quantum mechanics with special relativity.’’[source]

The distinguished theoretician Joseph Polchinski, author of a classic textbook on superstrings, in a commemorative conference honouring Dirac said:
Dirac comes across in many ways as the first modern theoretical physicist. Many of his statements illustrate this, but the following strikes me as particularly apt:

One must be prepared to follow up the consequences of theory, and feel
that one just has to accept the consequences no matter where they lead.

Dirac is often quoted on the importance of mathematical beauty in one’s equations; I did not choose one of these quotations because beauty is so difficult to define. He also made various statements that one should not being distracted by experiment; I did not choose one of these because they are inflammatory.

The reason that I find the chosen quotation so striking is that it is not supposed to be possible to follow theory alone. Without experimental guidance, it is said, one will quickly become lost. But of course today in high energy theory we are to a large extent following theory where it leads us, and we are rather confident that this is a correct and fruitful path. Why this approach can work is illustrated by Dirac’s great discovery:

quantum mechanics + special relativity → antiparticles .

This was not a direct deduction (though in the framework of quantum field theory, one can show that antiparticles are necessary for causality.). Rather, when Dirac tried to find a consistent framework that combined quantum theory and special relativity, he found it very difficult — so much so that when he did find one he had great confidence in its inevitability, and was prepared to take its other consequences seriously.[source]

The reasoning of string theorists, here uttered by Polchinski, is very simple: if quantum mechanics + special relativity → quantum field theory, then, quantum mechanics + general relativity → M-theory. As for electromagnetism, we note the tendency to stress the beauty of the unification rather than the beauty of mathematical consistency: ‘‘The existence of a single structure that unifies such a broad range of physical and mathematical ideas, and many others as well, is unexpected and remarkable. Earlier I declined to define beauty, but one can recognize it when one sees it, and here it is. This is one illustration of why the scientific path that Dirac laid out has been such a fruitful one in recent times.’’[source] (Italics added.)

C. Particle physics:

In his Nobel Lecture, given on 11 December 1957, Chen Ning Yang underlined the ontological base which drove the investigation of the majority of the particle physicists at that time: ‘‘One learns to hope that Nature possesses an order that one may aspire to comprehend [mathematically].’’[source] This belief (a learnt stance as Yang correctly says), was reinforced a few years later with the confirmation of Murray Gell-Mann’s ‘‘Eightfold Way.’’ Gell-Mann, dubbed by some the ‘‘Mendeleev of elementary particle physics,’’ arranged in simple geometrical structures all the baryons and mesons that were known. The conclusive experimental evidence for his predicted Ω− arrived in early 1964 (see Figure 1). After this astonishing success, in addition to new experimental discoveries and the theoretical introduction of quarks, including the b (b for beauty, or bottom) and the t (t for truth, or top), particle physicists started to ideate more fascinating geometries in order to understand the most fundamental structure of matter. For particle physicists, these geometrical structures have become ever since an indubitable sign that the microphysical world is beautifully organized. Needless to say this disposition reflects the strong Platonic preconception in which most theoretical physicists are being educated.

1. The baryon decuplet showing the ‘‘missing’’ Ω- and a supermultiplet for baryons of spin ½.

Nowadays, almost half a century after Murray Gell-Mann introduced his revolutionary ideas, some of these proposed independently by Yuval Ne’eman, nobody doubts that it is a ‘‘fact that Lie groups have become as essential to modern theoretical physics as complex analysis and partial differential equations.’’ This assessment was made by Sheldon Glashow, another influential particle physicist, in his Introduction to Howard Georgi’s widely taught textbook on Lie Algebras.[source] In the same Introduction, Glashow brilliantly summarized the basic function assigned to symmetry principles in the study of particle physics. His interest concerning what he thought were the most pressing questions of particle physics, adopted the form of a recommendation:
We cannot yet answer these questions, but it is clear that a command of the simple and beautiful theory of Lie groups will be needed. For the future, we can only repeat the remarks of an earlier Harvard colleague, P.W. Bridgman, who wrote in 1927 that ‘‘whatever may be one’s opinion as to the simplicity of either the laws or the material structure of nature, there can be no question that the possessors of some such conviction have a real advantage in the race for physical discovery. Doubtless there are many simple connections still to be discovered, and he who has a strong conviction of the existence of these connections is much more likely to find them than he who is not at all sure they are there.’’

Therefore, for Glashow, a truly modern theoretical physicist who tries to understand the fundamental structure of nature, ‘‘the simple connections,’’ must be guided by the ‘‘simple and beautiful theory of Lie groups.’’ That is, by the belief that fundamental matter is geometrically organized and its dynamics respect such simple symmetries. Only by following this attested procedure can one organize matter and, consequently, foresee new elementary particles to discover in the laboratory. From this perspective, theoretical particle physicists read Gell-Mann’s episode as follows: they assume that Gell-Mann did not introduce any ad hoc assumption into his geometrical structures in order to predict new particles, but it was only the simplicity of the mathematics used which prompted particle physicists to realize the predictive power of the model. And, if someone did not recognize at first this power, it was due to ignorance or lack of self-confidence. Today, no theoretical particle physicist, according to this accepted viewpoint, can commit this mistake.

In his address delivered before the Nobel Foundation in December 2004, David Gross expressed his gratitude in the following terms: ‘‘As I end I would like to thank not only the Nobel Foundation, but nature itself, who has given us the opportunity to explore her secrets and the fortune to have revealed one of her most mysterious and beautiful aspects – the strong force.’’[source] (Italics added.) This declaration is very significant, and not only because Gross is one of the most influential theoretical physicists of our time, both within and without the community of experts, but also because he has been one of the leading theorists to have built the ‘‘beautiful edifice’’ of superstrings.

So far I have argued that the beauty physicists find most appealing in particle physics is its simple geometries allowing them to organize the fundamental building blocks of nature. This contrasts somehow with the two previous paradigmatic cases discussed above. In all three cases, Maxwell’s equations, quantum mechanics, and particle physics, mathematical consistency is considered crucial; however, it must be noted that the visual symmetrical forms of group representations give it extra credit.

Like other theoretical particle physicists, string theorists assume that symmetry principles play a crucial role in microphysics and must be considered the supreme guide in the search for reality. Moreover, they have drawn on previous results in order to support their theory. As is the case for the grand unification models:
The second motivation for supersymmetry in the TeV range comes from the idea of gauge unification. Recent experiments have yielded precise determinations of the strengths of the SU(3) × SU(2) × U(1) gauge interactions – the analogs of the fine structure constant for these interactions. They are usually denoted by α3, α2 and α1 for the three factors in SU(3) × SU(2) × U(1). In quantum field theory these values depend on the energy at which they are measured in a way that depends on the particle content of the theory. Using the measured values of the coupling constants and the particle content of the standard model, one can extrapolate to higher energies and determine the coupling constants there. The result is that the three coupling constants do not meet at the same point. However, repeating this extrapolation with the particles belonging to the minimal supersymmetric extension of the standard model, the three gauge coupling constants meet at a point, MGUT, as sketched in Fig. 2 [see Figure 1]. At that point the strengths of the various gauge interactions become equal and the interactions can be unified into a grand unified theory. Possible grand unified theories embed the known SU(3) × SU(2) × U(1) gauge group into SU(5) or SO(10).[source]

2. ‘‘Coupling constant unification in supersymmetric theories.’’[source]

More precisely: including the particles of the Minimal Supersymmetric Standard Model (MSSM), the gauge couplings unify at MGUT ≈ 2 × 10^16 GeV, taking the approximate value 5/3 α1 = α2 = α3 ≈ 1/25. The grand unification group SO(10) contains the standard model symmetries according to the decomposition SO(10) → SU(5) × U(1)’ → SU(3) × SU(2) × U(1) × U(1)’. It is worth noting that these conclusions are currently upheld independently of superstring theory arguments. There are several ways in which string theorists integrate these results into their explanatory framework.

A standard ‘‘top-bottom’’ approach is that of the heterotic string E8 × E8 formulated by Gross and collaborators more than twenty years ago. Since the gauge group of this heterotic string decomposes as E8 × E8’ → SU(3) × E6 × E8’ → SU(3) × SU(3) × SU(3)L × SU(3)R × E8’, where the last two SU(3)s contain the SU(2) and the hypercharge U(1) of the electroweak force, the system guarantees, at least in principle, that all the implications of the standard model are deducible from it. (Of course, things are much more complicated than this rough analysis suggests. A string phenomenologist advises us: ‘‘But what for this theoreticians is a nice classification, for the phenomenologist is a horrible nightmare. He has a incredible mess, a jungle, in his hands.’’[source])

This argument seems too simple for some people, however, I consider it very powerful: if symmetries are, as particle physicists have taught us, the supreme guide in the search for the most fundamental truths of nature, then string theory has satisfied one of the most stringent conditions required to any theoretical model. For string theorists, the beauty of nature, characterized by the application of symmetry principles to the microphysical world, has been realized in the unification of all the forces. Once again, as in the case of electromagnetism and Dirac’s equation, we observe that string theorists have brought the discussion on beauty and simplicity to their own ground: the unity of the physical laws.

More than thirty years after the introduction of the Eightfold Way, in The Quark and the Jaguar, Murray Gell-Mann explained his reasons for thinking that the standard model could not be the ultimate theory of elementary particles. There, he advanced that ‘‘Il se peut que ce rêve ait été realisé. Une théorie d’un nouveau genre, nommée théorie des ‘‘supercordes’’, semble posséder les bonnes propriétés pour accomplir l’unification.’’[source] This type of support from authoritative theoretical particle physicists has strongly contributed to the acceptance of superstrings as the simplest, most beautiful, organization and description of matter.

This general discourse has already penetrated some standard texts on particle physics and quantum field theory. On page 4 of a new edition of their classic book, Aitchinson and Hey write: ‘‘Despite remarkable recent developments of strings (Green et al 1987, Polchinski 1998), it is fair to say that the vision of the unification of all the forces, which possessed Einstein, is still some way from realization.’’ At the end of chapter 2 they comment again on superstring theory and Planck-scale physics.[source] Weinberg’s affection for string theory is well known; for example, in the first of his volumes on QFT he says: ‘‘We have learned in recent years to think of our successful quantum field theories, including quantum electrodynamics, as ‘effective field theories,’ low-energy approximations to a deeper theory that may not even be a field theory, but something different like a string theory,’’ and, in another place, ‘‘the underlying theory might not be a theory of fields or particles, but perhaps of something quite different, like strings.’’[source] In the final section of Peskin and Schroeder’s textbook on QFT, ‘‘Toward an Ultimate Theory of Nature,’’ there is an extensive discussion on superstring theory’s potentials. The character of the conclusion is not very surprising if we consider that Michael Peskin, the main author of this new classic, has lectured many courses on superstring theory.[source] Of course, Michio Kaku’s textbook on quantum field theory also discusses at length, it dedicates a full chapter, the basics of superstring theory.[source]

D. The general theory of relativity:

Einstein himself publicly declared on many occasions what he considered to be the beauty of general relativity. In his popular account Relativity: The Special and General Theory, first published in late 1916, he said: ‘‘The theory of gravitation derived in this way from the general postulate of relativity excels not only in its beauty … .’’[source] The beauty of the theory resided, according to Einstein’s explanation, in its ability to correctly interpret and predict the whole variety of gravitational phenomena from a single principle: the principle of equivalence. Until his death, which occurred four decades later, Einstein maintained this opinion. Actually, it is fair to say that this belief became stronger as he got older, turning into a sort of religious dogma.

The simplicity of the theory, what gave it its beauty, was not quickly acknowledged by Einstein’s colleagues. In fact, for most physicists at the time the theory was a conceptually difficult and convoluted mathematical construct. It was only during the years that followed the theory’s first experimental verification, in 1919, that Einstein’s point of view started to gain wide acceptance. But the experimental confirmation was not the only factor that played a role in the growing positive reception of Einstein’s idea. This was also due to the efforts made by scientists and science amateurs to popularize the basics of the theory. For instance, the work of Arthur Eddington was a significant contribution to English popular literature on the subject. There, Einstein’s theory of the universe was presented as a great achievement of mathematical physics and its search for simpler explanations. Eddington also published an advanced book on general relativity; a book that received a favourable response from the specialist reader and which was highly praised by Einstein. Around the same time the beauty of the theory was conveyed to a larger number of professional physicists by one of the first widely used textbooks on the subject: Relativity, Thermodynamics, and Cosmology, by Richard Tolman. Here, the spirit of Einstein’s legacy was guaranteed:
In addition to these observational verifications, which justify the introduction of the principle of equivalence, we must also assign a high importance to our intuitive appreciation of the rationality of assuming the abolition of gravitational effects for a freely falling observer, and to our intellectual appreciation of the simplicity, clarity, and effectiveness of the postulate that we thus obtain. These qualities of intuitive rationality and of intellectual simplicity, clarity, and effectiveness, which bespeak so unmistakably the insight and genius of Einstein, furnish of themselves of course no evidence of correspondence with experimental and observational fact. They are, nevertheless, necessary qualities for those principles which the human mind is willing to use as the fundamental postulates for science, and their presence must hence be regarded as also furnishing important justification for the acceptance of the principle of equivalence.[source]

In other words: independently of its experimental verification, simplicity of nature and its mental representations must be a basic prerequisite when doing physics. The scientific enquiry, the ‘‘human mind’’ in Tolman’s words, must be guided by the fact that nature is ruled by a few simple fundamental principles. The human mind can bypass the intricate phenomenal world and reach by pure thought these essential principles; once there, the physical world can be reconstructed: this is what Einstein did from 1905 to 1915. To reconstruct the physical world from a few principles, maybe even a single one, should be the ultimate task for every real theoretical physicist: these constituting principles are beautiful by definition.

For many decades, and thanks to Einstein’s influence and the personal effort he put into the propagation of his metaphysical thoughts, this common conception was normally associated with his theory. After the Second World War and the ensuing recession of the atomic projects on both sides of the Iron Curtain, there was a revival of fundamental research in gravitational physics.

In 1973 the voluminous Gravitation, by Charles Misner, Kip Thorne and John Wheeler was published; a textbook extensively employed in general relativity courses and the inspiration for a new approach to the field: the “bible” of general relativity as some still call it. In the Preface the authors talk about their motivations for writing a new book on the topic: ‘‘We have not seen any way to meet our responsibilities to our students at our three institutions except by a new exposition, aimed at establishing a solid competence in the subject, contemporary in its mathematics, oriented to the physical and astrophysical applications of greatest present-day interest, and animated by belief in the beauty and simplicity of nature.’’[source] For them, the use of modern mathematics, such as differential geometry and topology, and the analysis of new gravitational phenomena, such as relativistic stars and gravitational collapse, had left unaltered the essence of the theory: the beauty of the laws governing gravitational phenomena and its incorporation into the theory by a few general principles.

In a lesser known textbook, written by Norbert Straumann, based on personal notes of his course taught at Zurich in 1979, he recalled what Pauli once said on the occasion of the unveiling of a bust of Einstein: ‘‘The general theory of relativity then completed and – in contrast to the special theory – worked out by Einstein alone without simultaneous contributions by other researchers, will forever remain the classical example of a theory of perfect beauty in its mathematical structure.’’[source] It must be remembered that Wolfgang Pauli was not only an extraordinarily talented nuclear physicist; he was also a great admirer of Einstein’s theory. When he was still very young he wrote a short, but very influential, monograph compiling the most important results of the novel theory. His high regard for Einstein’s relativity lasted his entire lifetime.

Another devotee of general relativity was the eminent Indian-American astrophysicist Subrahmanyan Chandrasekhar. He has a lecture entitled ‘‘The General Theory of Relativity: Why ‘‘It is Probable the Most Beautiful of All Existing Theories’’?’’ (A title inspired, according to Chandrasekhar himself, by a comment he found in the weighty Classical Fields of Landau and Lifschitz.) After considering the major achievements of the theory at both theoretical and experimental level, he declares:
The foregoing examples provide evidence that a theory developed by a scientist with an exceptionally well-developed aesthetic sensibility can turn out to be true even if at the time of its formulation, it did not appear relevant to the physical world.
It is, indeed, an incredible fact that what the human mind, at its deepest and most profound, perceives as beautiful finds its realization in external nature.
What is intelligible is also beautiful.[source]

Chandrasekhar wrote this in the mid-eighties. Ten years later, Kip Thorne echoes this tenet in his bestseller Black Holes and Time Warps. He asks himself rhetorically: ‘‘What is the single most important thing that you want your reader to learn?’’ to which he replies, ‘‘My answer: the amazing power of the human mind – by fits and starts, blind eyes, and leaps of insight – to unravel the complexities of our Universe, and reveal the ultimate simplicity, the elegance, and the glorious beauty of the fundamental laws that govern it.’’[source]

The reader must have noted that the previous quotations by Tolman, Chandrasekhar and Thorne, have a common trait: they talk about the ‘‘human mind.’’ The person unacquainted with these sorts of interpretations would think it incongruent to combine physics and philosophy in this way. Yet, discourses of this type are indeed very frequent and constitute the sanctioned metaphysical foundation of the general theory of relativity. For admirers of this theory, nature is not only ontologically beauty but the ‘‘human mind’’ is actually capable of deciphering such beauty. Many things have changed since Einstein originally formulated the general theory of relativity — we now have a different and more powerful mathematics which allows us to understand an ever growing number of physical phenomena — however, the philosophical foundation remains intact: fundamental principles are beautiful and as human beings, we are able to grasp them. What must be retained from this discussion is that beauty, according to the established interpretation of general relativity, means basically reduction to fundamental physical principles. (The relationship between the human and the natural world will be discussed more extensively in the next essay.)

Before bringing this section to a close, let us conclude by referring to a frequent evaluation of the work of two of the most prominent theoretical physicists of the last decades: Roger Penrose and Stephen Hawking. These two physicists have played a leading role in the contemporary infusion of Platonism in general relativity, exalting even more the role of mathematical beauty in theoretical physics. By a way of example, in his influential book The Emperor’s New Mind Penrose, a declared Platonist, asserts that ‘‘Plato’s world consists not of tangible objects, but of ‘mathematical things.’ This world is accessible to us not in the ordinary physical way but, instead, via the intellect. One’s mind makes contact with Plato’s world whenever it contemplates a mathematical truth, perceiving it by the exercise of mathematical reasoning and insight.’’[source] (Italics in the original)

Have string theorists been preoccupied with this issue? If yes, what are the different attempts they have made in order to include in their theory a principle comparable to the equivalence principle of general relativity? Though they have tried, this has been the most elusive task of all. Indeed, so far string theorists have failed to propose a unique principle embracing all the properties of the theory, and, consequently, of the physical world.

As early as 1987, Green, Schwarz, and Witten, wrote: ‘‘In fact, historically, in the case of general relativity, it was the concepts that came first; Einstein first identified the concepts on which a relativistic theory of gravity should be based, and then found the theory. String theory has been the other way around. … At best we have perhaps just begun to scratch the surface of this question.’’[source] Around the same time Michael Green was asking a younger audience: ‘‘How can the logic of superstring theory be discovered? The principles of general relativity must be a special case of the more general principles of superstring theory, and so in a sense general relativity can serve as a guide.’’[source]

The first comprehensive proposal came many years later. This occurred in the mid-nineties. At that time, a fresh, numerous, and robust generation of string theorists were at work. Leonard Susskind proposed to them that this ‘‘simple organizing principle that can be expressed in a phrase or two’’ could be the Holographic Principle: ‘‘My own view is that the lasting idea will be the holographic principle …, the assertion that the number of possible states of a region of space is the same as that of a system of binary degrees of freedom distributed on the boundary of the region.’’[source] The AdS/CFT correspondence has provided some evidence in support of this idea, however, as we saw in the first essay, much remains to be done. Witten’s query: ‘‘What is the core idea [of string theory] analogous to the principle of equivalence in the case of general relativity?’’[source] still awaits an answer.

In this section I have exemplified by means of four theories, electromagnetism, quantum mechanics, particle physics, and general relativity, how the assessment of beauty in twenty-century physics changed over time. I depicted the evolution of these four cases, showing that they are not reducible to a single common pattern of change. I have also described how string theorists have taken advantage of these consensual interpretations of beauty in order to advance their own theory as a beautiful construct. These interpretations, already existing and widely accepted, have provided the basis for string theorists’ rhetoric. In turn, string theorists’ main criterion of beauty, unification, has begun to influence the way these more standard subjects are conceived.

It must be noted that the aesthetic appreciation of a particular physical theory is not indifferent to the way beauty is seen in neighbouring theories. For this reason, the previous demarcation between electromagnetism, quantum mechanics, particle physics, and general relativity, must be considered a schematic one. In fact, physicists from diverse areas share a set of ideas and attitudes, including sometimes aesthetic criteria, and the different subdomains are in constant interaction. An expert in one subfield can inoculate his colleagues in other subfields with the aesthetic criteria he generally uses to assess his theory. The constant reinterpretation and adaptation of aesthetic criteria from other subfields is what I had in mind when I observed that Pauli and Dirac were deep admirers of Einstein’s theory. The same applies when Steven Weinberg, a particle physicist, refers to Hermann Weyl, a mathematical physicist, in his textbook on general relativity: ‘‘Symmetry, as a wide or as a narrow as you may define it, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.’’[source] Many physicists also quote Weyl saying: ‘‘My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.’’ In conclusion, a systematic analysis of the role of aesthetics in physics must consider that the idea of beauty changes over time and travels across subfields.

*******You can read this blog for free! Please, do not copy its content.*******