Abstract
This opening chapter considers the orthodox story of string theory’s genesis, and indicates how it might be refined. We describe the broad outlines (or ‘periodisation’) of the history as presented in this book. We also include a brief guide to the physics and mathematics of vibrating strings and string theory, showing their similarities and differences.
The superstring theory has perhaps the weirdest history in the annals of science.
Michio Kaku
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Notes
- 1.
The duality (to be discussed more fully in the next chapter) states that what were seen to be distinct diagrams (at least in the context of orthodox Feynman diagram-based quantum field theory) were really two representations (or rather approximations) of one and the same underlying process. The so-called Harari-Rosner duality diagram that represents this equivalence class of diagrams (the dual processes, captured by Veneziano’s formula) played a crucial role in the genesis, early development and understanding of string theory qua theory of one dimensional objects and their two dimensional worldsheets. Indeed, the original introduction of the worldsheet concept used it to provide an explanation of duality (via conformal symmetry and the extreme deformability of worldsheets)—see Sect. 4.1.
- 2.
The “gang of four,” as Veneziano himself describes them [28, p. 185].
- 3.
This, in part, reflects the belief, true for most theoretical physicists, in deep links between mathematics and physics: where there is some piece of physics to be described (involving certain lawlike features), there will usually be a piece of mathematics that can do the job, if one only looks hard enough. Eugene Wigner referred in this context to “the unreasonable effectiveness of mathematics in the natural sciences” [29]. In fact, Dirac had earlier drawn attention to this feature, in 1939, pointing to “some mathematical quality in Nature” that enables one to infer results “about experiments that have not been performed” [4, p. 122]—note: it is from this basis that Dirac develops his principles of simplicity and beauty that characterise his work. See [8] for a superstring-relevant discussion of this subject.
- 4.
See, for example, his talk, “The Beginning of String Theory or: How Nature Deceived us on the Sixties”: http://online.itp.ucsb.edu/online/colloq/veneziano1. In [28] he writes, with somewhat less amusement than in the aforementioned, that “[o]ne of the things that upsets me most these days is to hear that I found the Beta-function ansatz almost by chance, perhaps while browsing through a book on special functions. ...[T]here is nothing further from the truth. Had one not found a simple solution for the (average) imaginary part, and recognised the importance of imposing crossing on the full scattering amplitude, the Beta-function would have stayed idle in maths books for some time...” (p. 185).
- 5.
See Merton and Barber’s study of serendipity for more on this notion of ‘chance’ versus ‘preparedness’ [17, see especially p. 259].
- 6.
Claud Lovelace [15, pp. 198–199] describes a remarkably similar route to his discovery [14] that the Poincaré theta series, which provides a method for constructing Abelian integrals, which can in turn be used to solve hydrodynamics on non-simply connected Riemann surfaces (which Holger Nielsen had already shown to correspond to “momentum flowing across the world-sheet”) provides a way of visualizing the Veneziano formula. This work presaged much later work lying at the intersection of mathematics and string theory (especially that involving automorphic forms and other sectors of number theory).
- 7.
Mahiko Suzuki did not publicly present nor publish his result as Veneziano did. Like Veneziano, Suzuki ceased active research in the field of dual resonance models not so long after discovering the Beta function ansatz. Joseph Polchinski noted that when taking a course on dual theory given by Suzuki in 1980 (according to Suzuki on “S-matrix theoryá la Chew with less emphasis on nuclear democracy”—private communication), he had prefaced the course by stating: “This is the last time this will ever be taught at Berkeley” (Interview with the author; transcript available at: http://www.aip.org/history/ohilist/33729.html). I describe Suzuki’s route to the discovery of the formula in Chap. 3.
- 8.
That is, a formula describing the probabilities for a process with both an input and output of two particles.
- 9.
Though as we shall see, this ‘experimental superiority’ claim is not so simple. QCD, and the theory of quarks, performed exceptionally well when applied to high-energy (deep) scattering experiments, but not so well in low-energy situations, due to its still ill-understood property of colour confinement. In these situations, string theory holds up remarkably well and, in a sense to be explained, was integrated into QCD to deal with such regimes.
- 10.
Namely: “Prediction is hard. Especially about the future”.
- 11.
- 12.
The history of this fruitful interaction deserves a book of its own, and I only discuss those portions of the history that are directly relevant to the development of string theory, and must omit very many interesting examples.
- 13.
I make no claim for uniqueness or canonicity with this periodization. It is somewhat arbitrary, though it tries as much as possible to home in on genuine ‘critical points’ in the history of string theory and is underwritten, as far as is possible, by citation analysis highlighting these critical historical points by their impact on the research literature. The citation analysis reveals an explosion (or implosion) in publication numbers and citations at the outset of each new period. Of course, restricting the analysis to the published literature does not reveal the fine structure that moves a discipline, and for this reason I also utilise a range of ‘external’ sources, including interviews and archives. Note also that I do not always stick to a strict chronological ordering, but often cluster according to thematic links.
- 14.
That is, of the slope parameter in the theory, \(\alpha '\), determining the scale at which ‘stringy’ effects appear.
- 15.
Peter Galison, in one of the few early historical studies of superstring theory, refers to the increased importance of extra-empirical constraints in string theory as “a profound and contested shift in the position of theory in physics” [6, p. 372]. Helge Kragh reiterates this in his recent book on theories of everything, writing that “[p]ublications by many physicists ...show a tendency to unrestrained extrapolation of physics into domains that according to the traditional view [of scientific methodology] are inaccessible to the methods of physics” [13, p. 367]. Here I also point the reader towards Richard Dawid’s recent book [3], which focuses on the issue of how these non-empirical theory assessment criteria can lead to a reasonable level of trust in string theory. See also [20], in which I argue for the legitimacy of (a certain level of) trust in string theory (despite the absence thus far of confirmation via novel experiments predicted by the theory). A more general examination of philosophers accounts of scientific methodology in the light of string theory is [11]. John Schwarz has pointed out that experiment has undergone its own kinds of shifts—e.g. to big collaborative experimental ventures, distributed in a modular fashion across local experts, necessarily involving a different approach to the gathering of evidence—that modify its methodological foundations too, so that the two situations (theory and experiment) are not so different [23, p. 201].
- 16.
A stretched string can of course possess multiple modes of vibration (the harmonics) which is analogous to the various modes of vibration corresponding to different particles (infinitely many of them) in the case of quantized superstrings.
- 17.
See [7] for a nice account of this episode, along with its impact on the development of analysis. The same episode served as a background for the development of the notion of a mathematical function.
- 18.
Here the constant \(c\) is the speed at which waves may propagate across the string. It has a value equal to the square root of the string tension divided by mass per unit length.
- 19.
The fundamental mode of the string was in fact discovered by Brooke Taylor (of Taylor series fame) and laid out, using Newton’s method of fluxions, in his Methodus Incrementorum of 1715. However, Taylor made no attempt to account for the overtones in which a string can vibrate in many ways simultaneously—a notion that required the concept of superposition, not yet discovered.
- 20.
We might mention Dirac’s [5] earlier attempt to come up with a higher-dimensional theory (modelling it as a charged conducting surface in fact) of the electron according to which Rabi’s question ‘Who ordered that?’ (of the muon) could be answered by treating it as the first excited state of the electron (associated with the size and shape oscillations of what is visualizable as a kind of bubble). Crucially, such a system had a finite self-energy. The idea was to have the surface tension counterbalanced by electrostatic repulsion, so as to avoid the problem of instability inherent in three-dimensional rotating objects—I should add, of course, that strings, being one-dimensional objects, do not need a special counterbalancing force since the centrifugal force alone can prevent the collapse. David Fairlie recalls attending a seminar by Dirac on this subject, accompanied by David Olive [18]—both Fairle and Olive are important figures in string theory’s history, and we shall meet them again.
- 21.
Initially, given that string theory was applied to hadron physics, \(\sqrt{T}\) was set to the mass scale of hadrons (which translates to distances of \(10^{-13}\) cm). In the refashioning of string theory as a quantum gravity theory (amongst other things) John Schwarz and Jöel Scherk set \(\sqrt{T} = 10^{19}\,{\mathrm {MeV}} = M_{{\mathrm {Planck}}}\) (in terms of distances, this is \(10^{-33}\) cm: a 20 order of magnitude adjustment!), since the Planck mass sets the scale of quantum gravitational effects.
- 22.
Ultimately, of course, quantum chromodynamics (with its asymptotic freedom) would win out—see ’t Hooft [26] for a nice discussion of the ‘rehabilitation’ of field theory, leading to the construction and acceptance of the quark theory and its basis in QFT (this discussion also includes the role of string theory).
- 23.
More precisely, certain modes (in this case spin-1 and spin-2) survive the limit-taking procedure and become massless particles in the low-energy limiting theory. These particles turn out to have exactly the properties of gravitons (mediating the gravitational force) and gauge bosons (such as the photon). That there exist massless (and therefore infinite range) particles is clearly not good for describing hadrons (which are short-range forces). This was one of the many factors responsible for the demise of string theory as a fundamental theory of hadrons.
- 24.
Or, as Edward Witten succinctly puts it, string theory is “a quantum theory that looks like Einstein’s General Relativity at long distances” [30, p. 1577].
- 25.
The fact that the interactions are ‘smeared’ (or ‘soft,’ as is often said) renders string theory unable to cope with the hard-scattering events leading to the ultimate hadronic crowning of the quark theory, with its point-like particles. However, the same softness also impacts positively on the properties of the amplitudes, making their high-energy (short-distance) behaviour far better than for point-like theories, which must model interactions as occurring at spacetime points. This improved behaviour (which controls the usual problematic divergences) was a major motivating factor in the continued pursuit of string theory beyond hadrons. (I should, however, point out that recent work on the so-called gauge/string duality points to string theory’s ability to cope with hard scattering—see, e.g., [19].)
- 26.
This is in exactly the same sense that a doughnut is ‘the same’ (topologically speaking) as a coffee cup: if all that matters to you is the holes, then they are identical. Though it was initially implicit, the conformal invariance of the theory was understood early on and heavily constrained its development.
- 27.
This worldsheet has a metric \(h_{\alpha \beta }\) defined on it in the version of string theory due to Alexander Polyakov. In the original Nambu-Gotō version the worldsheet was metric-free, with distance measurements made possible by the metric it inherits from the embedding into spacetime.
- 28.
In order that only physical quantities are included in the sum, it is performed over ‘moduli space’: i.e. the space of inequivalent 2D Riemann surfaces. The orbit space of metrics modulo conformal and diffeomorphisms symmetries is known as Teichmüller space. Moduli space is more tightly circumscribed, involving also ‘large’ diffeomorphisms (those not connected to the identity). When one further quotients Teichmüller space by the modular group of transformations, one has moduli space (of a Riemann surface), over which the path-integral in Eq. 1.7 is performed. As we will see, mirror symmetry (and other string dualities) have the effect of producing unexpected identifications of points in the moduli space of a string theory, further reducing it. If interactions are included, then the sum will include holed surfaces.
- 29.
Though Polyakov studied ‘non-critical’ string theories which departed from these constraints, there are other constraints that must be enforced to retain the conformal symmetry (which involves a so-called ‘Liouville mode’ on the worldsheet)—see Sect. 8.2.
- 30.
This is defined to be a compact \(n\)-dimensional complex Ricci flat manifold with Kähler metric, with trivial first Chern class. Ricci flatness means that the metric is a solution of the vacuum Einstein equations for general relativity. The first Chern class \(c_{1}({\fancyscript{X}})\) of a metric-manifold is represented by the two-form \((1/2\pi )\rho \) (with \(\rho \) the Ricci tensor \(R_{i\overline{j}} dz^{i} \wedge d\overline{z}^{\overline{j}}\)). Calabi proved Yau’s conjecture that when \(c_{1}({\fancyscript{X}}) = 0\), there exists a unique Ricci-flat Kähler metric for any choice of compact Kähler space. If one has a Ricci flat metric then one also gets the desired single supersymmetry since Ricci flatness is a sufficient condition for an \(SU(3)\) holonomy group. See [9] for a good, friendly introduction to all things Calabi-Yau.
- 31.
That requires some qualification: the Glashow-Weinberg-Salam theory of weak and electromagnetic unification had been developed in 1967, but stagnated for some time before the conditions were ripe enough to recognize the importance of what they had achieved.
- 32.
There is a certain irony in how things have developed from S-matrix theory since its primary virtue was that it meant one was dealing entirely in observable quantities (namely, scattering amplitudes). Yet string theory grew out of S-matrix theory. Of course, most of the complaints with string theory, since its earliest days, have been levelled at its detachment from measurable quantities—let’s call it ‘the tyranny of (experimental) distance’ (involving what Nambu is said to have called “postmodern physics”: physics without experiments!). However, S-matrix theory, though crucial for the emergence of string theory as we know it today, was a single rung on a ladder of many, and though certain philosophical residues (such as the distaste for arbitrariness in physics) from the S-matrix programme stuck to string theory, it soon became a very different structure.
- 33.
Initially, a host of different terms were employed to describe the one-dimensional structures: from rubber bands to threads to sticks.
- 34.
Of course, though the frameworks match initially (apart from the change of scale), the fact that the new theory is a theory of gravitation and other interactions suggests a host of new possibilities for developing the framework that would simply not arise in the older theory.
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Rickles, D. (2014). History and Mythology. In: A Brief History of String Theory. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45128-7_1
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