Abstract
We show how the difficulties faced by quantum field theory in advancing beyond QED led to various models, one of which was Regge theory, with the addition of the dual resonance idea. This model achieved significant empirical successes, had several powerful theoretical virtues, and was therefore pursued with some excitement. We trace the story from Regge’s introduction of complex angular momentum into quantum mechanics, to its extension into the relativistic domain. This combined with ‘bootstrap’ physics according to which the properties of elementary particles, such as coupling constants, could be predicted from a few basic principles coupled with just a small amount of empirical input. This journey culminated in the finite energy sum rules of Dolen, Horn, and Schmid, which were elevated to the status of a duality principle. The primary researcher network guiding research in this period was fairly narrowly confined, and can be charted quite precisely, with Geoffrey Chew featuring as a key hub leading an anti-QFT school.
As you can see, the new mistress is full of mystery but correspondingly full of promise. The old mistress is clawing and scratching to maintain her status, but her day is past.
Geoffrey Chew, Rouse Ball Lecture. Cambridge 1963
An errata to this chapter is available at DOI 10.1007/978-3-642-45128-7_11
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-642-45128-7_11
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Notes
- 1.
In fact, the beginnings of an erosion of confidence in quantum field theory (the orthodox framework for describing elementary particles) can be traced back to at least the 1950s, when the likes of Heisenberg, Landau, Pauli, and Klein were debating whether field theoretic infinities could be dealt with by invoking some natural (possibly gravitational) cutoff—of course, at this time non-Abelian gauge theories (and asymptotic freedom) were not known. (There was also a positivistic distaste with the notion of unobservable ‘bare’ masses and coupling constants.) One might also note that a new spirit flowed through the rest of physics at this time; not simply because it was a time of great social upheaval (being post-WWII = “the physicist’s war”), but also because many of the ‘old guard’ of physics had passed away. In the immediate aftermath of WWII, there was extreme confidence in the available theoretical frameworks, and little concern with foundational issues. By the late-1950s and into the 1960s, this confidence was beginning to wane, as Chew’s remarks in the above quotation make clear—the “new mistress” is S-matrix theory, while the “old mistress” is quantum field theory (amusingly, Marvin Goldberger had used the terminology of “old, but rather friendly, mistress” to describe quantum field theory in his Solvay talk from 1961—clearly Chew’s remarks are a reference to this).
- 2.
The name ‘hadron’ was introduced by Lev Okun in a plenary talk on “The Theory of Weak Interaction” at CERN in 1962, invoking the Greek word for large or massive (in contrast to lepton: small, light).
- 3.
Recall that the combination of relativity and quantum mechanics implies that particles (quanta of the field) can be created and destroyed at a rate depending on the energy of the system. Therefore, any such combination of relativity and quantum will involve many-body physics. This is compounded as the energy is increased. If the coupling constant is less than 1 then one can treat the increasing number of particles as negligible ‘corrections’ to the lowest order terms—note that the simpler, non-relativistic field theoretic case (the ‘potential-scattering’ problem) does not involve varying particle number. If the coupling constant is greater than 1, then going to higher order in the perturbation series (and adding more and more particles) means that the corrections will not be negligible so that the first few terms will not give a good approximation to the whole series.
- 4.
Not everyone was enchanted by this new S-matrix philosophy. As Leonard Susskind remembers it, the “general opinion among leaders of the field was that hadronic length and time scales were so small that in principle it made no sense to probe into the guts of a hadronic process—the particles and the reactions were unopenable black boxes. Quantum field theory was out; Unitarity and Analyticity were in. Personally, I so disliked that idea that when I got my first academic job I spent most of my time with my close friend, Yakir Aharonov, on the foundations of quantum mechanics and relativity.” [51, p. 262]. As mentioned in the preceding chapter, Susskind would go on to make important contributions to the earliest phase of string theory research, including the discovery that if you break open the black box that is the Veneziano amplitude, you find within it vibrating strings. In his popular book The Cosmic Landscape Susskind compares this black box ideology to the behaviourst psychology of B. F. Skinner [50, pp. 202–203].
- 5.
Indeed, Stanley Deser remarked that the reason he got into general relativity and quantum gravity, after a background in particle physics, was precisely because “quantum field theory appeared to be degenerating while gravitational physics looked like a new frontier” (interview with the author, 2011—available via the AIP oral history archives [Call number OH 34507]). This suggests that there was something like a ‘crisis’ in Kuhn’s sense. It was, of course, resolved to the satisfaction of many physicists (in quantum chromodynamics [QCD]) by a complex series of discoveries, culminating in a solid understanding of scaling and renormalization, dimensional regularization, non-Abelian gauge theories, and asymptotic freedom—recall that QCD is based on quark theory, where the ‘chromo’ refers to the extra degree of freedom postulated by Oscar Greenberg (in addition to space, spin, and flavour), labeled ‘colour’ by Gell-Mann. I don’t discuss these discoveries in any detail in this book. For a good recent historical discussion, see [2] (see also: [29, 44]). However, QCD, while an excellent description of the high-energy behaviour of hadrons, still cannot explain certain low energy features that the earliest dual models (leading to string theory) had at least some limited success with.
- 6.
Naturally, many important concepts (from the point of understanding the development of string theory) have fallen out of fashion as the theories and models to which they belonged have been superseded.
- 7.
Of course, in QCD the strong interaction is governed by the exchange of gluons (massless, spin-1 bosons) which are coupled to any objects with strong charge or ‘colour’ (i.e. quarks). This has many similarities to QED, albeit with a coupling \(\alpha _{strong} = g_{s}^{2}/4\pi \approx 1\), instead of the much weaker \(\alpha _{\textit{EM}} = e^{2}/4\pi \approx 1/137\). However, in the early days of hadron physics quarks were seen as convenient fictions used as a mere book-keeping tool for the various properties of hadrons—the nomenclature had some resistance: Victor Weisskopf, for example, wanted to call them ‘trions,’ while George Zweig wanted to call them ‘aces’! The gluons are themselves coloured which implies that they self-interact. This results in a characteristic property of quarks, namely that they are confined within hadrons, unable to be observed in their singular form. The gluons attract the field lines of the colour field together, forming a ‘tube’. Accounting for this tube-like behaviour was considered to be an empirical success of the early string models of hadrons, as we see below.
- 8.
Primarily the Proton Synchrotron [PS], turned on in 1959, becoming the highest energy accelerator at that time, attaining a beam energy of 28 GeV. By comparison, the Cosmotron at Brookhaven reached energies of just 3 GeV, though at the time of its first operation it was six times more powerful than other accelerators. For a good, technical review of these experiments see [31].
- 9.
The tracks of these particles are bent using strong magnetic fields. The quantum numbers of the particles can then be computed from the curvature of paths, thus enabling (under the assumption of energy-momentum conservation) the identification of various particle types.
- 10.
Such resonance particles are too short-lived and localised to leave a directly observable trace. Resonances possess lifetimes of the order of \(10^{-24}\) s. They would simply not travel far enough to leave a track before decaying. Given that particles travel at the speed of light \(c\), solving for the distance traveled gives just \(\approx 10^{-15}\) m. They are simply not stable enough to warrant the title ‘particle,’ which implies some degree of robust and continued existence. Of course, bubble chambers cannot allow one to see such particles, but one can infer their existence by observing decay products via various channels (see Figs. 2.1 and 2.2). (However, Chew [8, pp. 81–82] argued that, since both were to be represented by S-matrix poles, particles and resonances should not be distinguished in any significant way.)
- 11.
As we will see below, it was consideration of hard scattering processes that led to quantum field theory once again providing the framework in which to couch fundamental interactions. What such processes revealed was a hard, point-like interior structure of hadrons, much as the classic gold foil experiments of Rutherford had revealed a point-like atomic nucleus.
- 12.
Of course, the quark model postulated a deeper layer of elements of which the new particles were really bound states. Although I won’t discuss it, mention should be made here of the ‘current algebra’ approach to strong interactions, of Murray Gell-Mann (see, e.g., [24]). In this programme, although the underlying theory of quarks and their interactions wasn’t determined, certain high-level algebraic aspects of the free theory were, and these were believed to be stable under the transition to the interacting theory. The current algebra is an \({\textit{SU}}(n) \otimes {\textit{SU}}(n)\) algebra (with \(n\) the number of what would now be called ‘flavours’), generated by the equal-time commutation relations between the vector current \(V_{\mu }^{a}(x)\) and the axial vector current \(A^{a}_{\mu }(x)\). One of the crucial approximation methods employed in the construction of dual models (that of infinitely many narrow hadronic resonances) was developed in the context of current algebra. (See [2] for a conceptually-oriented discussion of current algebra, including an extended argument to the effect that this amounts to a ‘structural realist’ position in which the structural (broad algebraic) aspects constituted a pivotal element of the development of the theory.)
- 13.
James Cushing’s Theory Construction and Selection in Modern Physics [12] is a masterly account of the historical development of this new way of doing particle physics. In it he argues that the S-matrix methodology, of employing general mathematical principles to constrain the physics (at least, of the strong interaction), was perfectly viable and bore much fruit, despite the confirmation of QCD that knocked S-matrix theory off its pedestal. I agree with this general sentiment, and string theory can be found amongst such fruit.
- 14.
A concept Gell-Mann had labeled ‘nuclear democracy’—surely a term coloured by the political and social climate of Berkeley in the 1960s. For a discussion of the context surrounding Chew’s ‘democratic’ physics, see [32]. To this idea was appended the notion of ‘bootstrapping’ strongly interacting particle physics, in the sense that hadrons are bound states of other hadrons, that are themselves held together by hadron exchange forces—a purely endogenous mechanism.
- 15.
Holger Nielsen notes that he gave a talk on string theory while Heisenberg was visiting the Niels Bohr Institute at a conference given in his honour, but, as he puts it, “I do not think though that I managed to make Heisenberg extremely enthusiastic about strings” [40, p. 272]. Interestingly, David Olive also spoke on multi-Veneziano theory (that is, the generalised Veneziano model) and its relationship to quarks and duality diagrams, on the occasion of Heisenberg’s 70th birthday, in Munich, June 1971. He notes that Heisenberg’s reaction was a protest denying that the quark model was physics [37, p. 348].
- 16.
This connection was at the core of Freeman Dyson’s equivalence proof of Feynman’s and the Schwinger-Tomonaga formulations of QED [16], which employed the S-matrix to knit them together—the method of proof was to derive from both approaches the same set of rules by which the matrix element of the S-matrix operator between two given states could be written down.
- 17.
In this sense, Heisenberg’s way with the S-matrix corresponds to a repetition of the ideas that led to his matrix mechanics in the context of high-energy particle physics. Once scattering matrix elements have been fixed, then all cross-sections and observables have thereby been determined. Heisenberg’s view was that one needn’t ask for more (e.g., equations of motion are not required—on which, see Dirac [14]). The rough chronology that follows is that renormalisation techniques are developed, leading to quantum electrodynamics (with its phenomenal precision), leading to the demise of S-matrix theory. It was the subsequent fall from grace of quantum field theory at the hands of mesons that led to the resurrection of S-matrix theory, as we will see (see Fig. 2.5 for a visual impression of this “resurrection”). The trouble was that the finite, short range nature of the forces behind mesons seemed to imply that the particles were massive (in the context of Yukawa’s exchange theory). Chen Ning Yang and Robert Mills had argued otherwise, of course, in order to preserve gauge invariance (now generalised to non-Abelian cases), but this view (famously discredited by Pauli) had to wait for an understanding of confinement and the concept of asymptotic freedom to emerge. Fortunately, by that time S-matrix theory had enough time to spawn string theory—’t Hooft gives a good description of this progression (including the impact of dual models and hadronic string theory) in [54] (see also [27, 28]).
- 18.
In the case of quantum mechanics this will be with respect to a Lebesgue measure, \(d\mu (p) = \varPi d^{3}p_{i}\). In the context of a relativistic quantum theory the measure must be Lorentz-invariant, so one has a mass term: \(d\mu (p) = \varPi ( m^{2} + p^{2}_{i})^{-\frac{1}{2}} d^{3}p_{i}\) (with \(m\) the particle mass).
- 19.
The term ‘dispersion’ harks back to Kramers and Kronig’s work in optics and the theory of material dispersion involving the absorption and transmission (in the form of a spectrum of different colours, or rainbow) of white light through a prism (or, more generally, some dispersive medium). In this case, a dispersion relation connects the frequency \(\nu \), wavelength, \(\lambda \), and velocity of the light, \(v\): \(\nu = v(\lambda )\). The spatial dispersion of light into different colours occurs because the different wavelengths possess different (effective) velocities when traveling through the prism. A good guide to dispersion relations is [41]. It was Murray Gell-Mann (at the 1956 Rochester conference [23]) who had initially suggested that dispersion relations might be useful in computing observables for the case of strong interaction physics. In simple terms, the idea is to utilise S-matrix dispersion relations to tie up experimental facts about hadron scattering with information about the behaviour of the resonances (independently of any underlying field theory). More technically, this would be achieved by expressing an analytic S-matrix in terms of its singularities, using Cauchy-Riemann equations. Chew developed this (initially in collaboration with Goldgerber, Low, and Nambu: [4]) into the general idea that strong forces correspond to singularities of an analytic S-matrix.
- 20.
In more orthodox terms, crossed processes are represented by the same amplitude and correspond to continuing energies from positive to negative values (whence the particle-antiparticle switch)—this corresponds, of course, to CPT symmetry. This idea of crossing also harks back to Murray Gell-Mann, this time to a paper coauthored with Marvin Goldberger [22]. Of course, if analyticity is satisfied, then the operation of analytic continuation can amplify knowledge of the function in some region of its domain to other regions—as Cushing puts it, “an analytic function is determined globally once it has been precisely specified in the neighbourhood of any point” [13, p. 38].
- 21.
More generally, it is more appropriate to think about channels of particles. One can think of a channel, loosely, as a providing a possible ‘route’ from which the final state emerges. There might be many such possible routes, in which case one has a multichannel collision process, otherwise one has a single channel process. Such channels are indexed by the kinds of particles they involve and their relative properties. In scattering theory one is interested in inter-channel transitions; i.e. the transition from some process generated through an input channel and decaying through an output channel. Given a set of available channels, unitarity in this case is simply the property that every intermediate state must decay through some channel, so that \(\sum _{out} |S_{\langle in,out\rangle }|^{2} = 1\).
- 22.
The variables \(s\) and \(t\) are known as Mandelstam variables, with a third, \(u = (p_{a} - p_{d})^{2} = (p_{b} - p_{c})^{2}\), completing the set of Lorentz invariant scalars. These variables are not all independent because of the presence of the constraint \(s + t + u = \sum _{i=1}^{i=4} m_{i}^{2}\), so any two variables can be used to construct the scattering amplitudes, therefore we can dispense with \(u\) for convenience.
- 23.
The \(u\)-channel would be obtained from the \(t\)-channel by switching particles \(c\) and \(d\): \(u = (p_{a} - p_{d})^{2} = (p_{b} - p_{c})^{2}\). In the \(u\)-channel is the reaction: \(a + \overline{d} \rightarrow \overline{b}+d\), where the physical region is \(u \ge (m_{a} + m_{d})^{2}\).
- 24.
In other words, a tree graph in the sense of Landau is understood to represent, directly, physical hadrons via the lines. Landau’s singularity conditions are satisfied by a classical process sharing the topological (network) structure of the graph. Coleman and Norton later provided a proof of this graph-process correspondence. As they put it: “a Feynman amplitude has singularities on the physical boundary if and only if the relevant Feynman diagram can be interpreted as a picture of an energy- and momentum-conserving process occurring in space-time, with all internal particles real, on the mass shell, and moving forward in time” [11, p. 438].
- 25.
This expansion into the complex plane has a significant impact on the mathematics employed. For example, integration takes on a different appearance since, whereas given the real numbers one follows a single path to integrate between two points, in complex analysis one can take many different paths in the plane, leading to planar diagrams and contour integration. Note, however, that all were taken with the complex expansion. ’t Hooft mentions that his PhD supervisor, Martinus Veltman, was of the opinion “Angular momentum aren’t complex. They’re real. Why do you have to go to a complex thing? What does it mean?” (interview with the author, 10 February 2010). See [17] for a good general overview of Regge theory, including its place within Veneziano’s dual resonance model.
- 26.
A Riemann surface provides a domain for a many-valued complex function.
- 27.
To put some ‘physical’ flesh on these concepts, it is safe in this context to think of simple poles as particle exchanges at a vertex, while a cut is a singularity corresponding to pair production (of particles). Technically, of course, a branch cut is a kind of formal ‘barrier’ that one imposes on a domain in order to keep a complex function single valued.
- 28.
The singularity is of the form \(\frac{1}{J-\alpha }\) (where \(\alpha \), the Regge slope, is a function of the collision energy of the process in which the particle is involved).
- 29.
The slope \(\alpha ^{\prime }\) of the Regge trajectories was one of the concepts that would enter string theory in a rather direct way. It was suggested later that the slope has the air of a universal constant of nature, and one that might be connected to the extended, non-point-like character of hadrons, leading to a fundamental length scale set by hadron constituents, \(\lambda \approx \sqrt{\alpha '}\) of the order \(10^{-14}\) cm [36]. As Daniel Freedman and Jiunn-Ming Wang showed in 1966, in addition to the ‘leading trajectory,’ one would also have ‘daughter trajectories’ lying parallel (with spins separated by one unit), underneath the leading trajectory, and separated by a spacing of integer multiples of a half.
- 30.
The spin values of the resonances themselves can be inferred from the angular distribution of the decay products in the various reactions.
- 31.
Quantum field theories face severe problems with conservation of probability (i.e. unitarity) for particles of spins greater than 1, in which case the amplitudes diverge at high energies. One of Regge theory’s key successes was the ability to deal with the exchange of particles of very high spins by conceptualizing the process in terms of ‘Reggeon’ and ‘multi-Reggeon’ exchange (where Reggeons are composite objects associated with \(\alpha (t)\)).
- 32.
This story begins in 1958, with Mandelstam’s paper marking the beginning of the so-called ‘double dispersion representation’ (in both energy and momentum transfer): [37]. Such double dispersion relations were later renamed the ‘Mandelstam representation’. Mandelstam was explicitly taking up the suggestion made by Gell-Mann in [23], that one might “actually replace the more usual equations of field theory and ... calculate all observable quantities in terms of a finite number of coupling constants” [37, p. 1344]. Elliot Leader has written that “Tullio Regge’s great imaginative leap, the introduction of complex angular momentum in non-relativistic quantum mechanics, might have ended in oblivion, weighed down by its overpowering mathematical sophistry and rigour, had not S. Mandelstam, seizing upon its crucial element and casting off the mathematical shroud, demonstrated a direct and striking consequence in the behaviour of high-energy elementary particle collision processes [35, p. 213]. Mandelstam’s insight was the realization that unphysical regions of the scattering plane (involving very large values of the cosine of the scattering angle \(\theta \)), for a scattering event like \(A+B \rightarrow A+B\), is mathematically related to the physical reaction \(A+\overline{A} \rightarrow B+\overline{B}\).
- 33.
As Chew describes the origination of the bootstrap idea, it was in discussion with Mandelstam before the 1959 Kiev Conference when they discovered that “a spin 1 \(\pi \pi \) resonance could be generated by a force due to Yukawa-like exchange of this same resonance” [9, p. 605]—a resonance that was later to be named the \(\rho \)-meson. The bootstrap, more generally, refers to the notion that one can build up a pole in some variable via an infinite sum of singularities in some other variable—that is, a pole generates singularities in the crossed-channel, and these singularities generate the original pole. A pole thus generated can then be viewed as a bound state of other particles: “\(\rho \) as a force generates \(\rho \) as a particle” [9, p. 606]. Or, in more general terms, hadrons are to be viewed as bound states of other hadrons (see [5] for the more general bootstrap theory).
- 34.
Gravitation is not incorporated in this scheme, and is modelled only classically. The particle physics approach to quantum gravity was being pursued at around the same time that the standard model was being formed. Indeed, the tools and methods used to construct the standard model were very much bound up with work in quantum gravity. The electroweak, the strong force, and the gravitational force were, after all, described by non-Abelian theories. The properties powerfully represented by the standard model form a target that any future theory that hopes to probe still higher energies (‘beyond the standard model’) will have to hit. This includes string theory.
- 35.
Yukawa had attempted to construct a quantum field theory along the lines suggested by quantum electrodynamics in 1935. His approach proposed a connection between the mass of a particle and its interaction range.
- 36.
The Pomeron was later understood to be the trajectory given by \(2+ \frac{\alpha '}{2}J^{2}\) (the Pomeron sector) corresponding to the massless states of gravitons and dilatons (associated with closed strings). Its defining quality is that it is, in some sense, ‘without qualities,’ carrying no quantum numbers (or equivalently, it has ‘vacuum quantum numbers’: that is, no charge, spin, baryon number, etc.). This latter basic idea of the Pomeron was introduced in Chew and Frautschi’s “Principle of Equivalence for all Strongly Interacting Particles Within the S-Matrix Framework” [5]. They were to be distinguished from Reggeons (later interpreted in terms of open strings). It was subsequently found that the states of the Pomeron sit on a Regge trajectory with twice the intercept and half slope of the Reggeon trajectory. As we see, the vacuum quantum numbers are later explained by the fact that closed string worldtubes have no boundaries on which to ‘attach’ quantum numbers using the then standard ‘Paton-Chan method.
- 37.
This chapter also introduced the representation of Regge trajectories (as in Fig. 2.5, now known as a ‘Chew-Frautschi plot’). The original Chew-Frautschi plot consisted of a line draw between just two points (the only two then known experimentally)—cf. [5, pp. 57–58]. As Frautschi noted in an interview, “Originally, we had just drawn a straight line between two points, because two points were all we had for the data. And then as more data occurred, the straight line continued through the next particle discovered and through the Yukawa exchanges in a different kinematic region. So the straight lines we’d originally drawn for our Regge particles turned out to be a pervasive feature, and eventually that came to be regarded as very strong evidence for strings. ” [21, p. 19].
- 38.
Indeed, James Cushing referred to the combined S-matrix theory \(+\) duality framework as “the ultimate bootstrap” [12, p. 190]. However, duality really is just an implementation of the bootstrap principle of generating a pole (particle) by summing over (infinitely many) singularities in some other amplitude variable. In the case of duality one has a physical (that is, observational) equivalence between a description without forces (but with resonance production: i.e. fermions, though without spin degrees of freedom) and one with forces (mediated by an exchange particle: i.e. bosons).
- 39.
A simple expression of the duality is through the symmetry of the amplitude under the interchange of energy \(s\) and momentum transfer \(t\): \(A(s,t) = A(t,s)\). One can think in terms of \(s-t\) duality or resonance-Regge pole duality—for this reason it is sometimes called ‘\(s-t\) duality’.
- 40.
As Pierre Ramond notes, this was “elevated to a principle to be added to the Chew bootstrap program, regarding resonance and Regge trajectories as aspects of the same entities” [46, p. 505].
- 41.
I borrow the term “epistemic gain” from Ralf Krömer to refer to the fact that there are circumstances in which “dual objects are epistemically more accessible than the original ones” [33, p. 4]. The most significant case of this is seen in the final chapter when we look at S-duality, relating strongly coupled to weakly coupled limits of certain theories.
- 42.
According to Mahiko Suzuki, who shared an office with Horn and Schmid and collaborated with them briefly, it was Horn that coined the name “finite energy sum rule”. Richard Dolen entered the collaboration (as Suzuki departed) because of his computational and data handling skills (private communication).
- 43.
By contrast in the competing interference model scheme, mentioned above, one would have the sum rule: \(f (\mathrm {Resonance}) + f (\mathrm {Regge}) \) (see [1]).
- 44.
At this stage the quarks were, in general, not invested with any physical reality, but were merely viewed as a kind of book-keeping method. George Zweig was entertaining the idea that quarks were real, but Gell-Mann’s view that they were purely formal prevailed. Of course, he would later receive his Nobel prize, in 1969, for the discovery that hadrons are bound states of quarks. In fact, it should be pointed out that this does not appear to have been Gell-Mann’s actual position, and his usage of the term “mathematical” to describe certain quarks was non-standard (cf. [53, p. 634]): he simply meant ‘unliberated’ or “permanently confined” and chose “mathematical” to avoid what he called “the philosopher problem”! He was worried that philosophers would grumble about the possibility of unobservable entities—and, indeed, we saw earlier that Heisenberg objected on just such grounds. David Fairlie goes further, arguing that the positivistic commandment against talking about “unobservable features of particle interactions, but only about properties of asymptotic states...inhibited the invention of the concept of quarks” [19, p. 283].
- 45.
Rosner notes in his paper that he became aware of Harari’s work once the bulk of his own work was completed [20, p. 691]. This feature of multiple near-simultaneous discoveries is especially rife in the history of string theory—it surely points to an underlying common set of heuristics.
- 46.
The resonance width gives us an indication of the uncertainty about the particle’s mass. The terminology of ‘narrow-resonance’ is something of an oxymoron of course, since if a resonance is wide then the particle will be short-lived (a resonance particle!).
- 47.
Chu, Epstein, and Kaus [10] argued that Mandelstam’s scheme for computing the subtraction constants depends too sensitively on both the cutoff used in the FESR and on the specific value of momentum transfer \(s\) at which the FESR are evaluated.
- 48.
Note that Veneziano’s paper, “Construction Of A Crossing-Symmetric Regge-Behaved Amplitude For Linearly Rising Trajectories,” was (by far) the highest cited paper to have been influenced by Mandelstam’s. Note also, that the most highly cited paper to have in turn been influenced by Veneziano’s paper was Neveu and Schwarz’s paper introducing the dual pion model: “Factorizable Dual Model Of Pions”. Continuing, the paper on “Vacuum Configurations for Superstrings” of Candelas, Horowitz, Strominger, and Witten is the highest cited citer of the Neveu-Schwarz paper—again, by a fairly large margin. (Citation analysis performed with Thomson-Reuters, Web of Science.) This gives some indication of the level of continuity between the earliest work on duality and modern superstring theory.
- 49.
Though this is a very selective network, of course, and misses many other important contributors, many of those associated with what have been labelled ‘revolutionary’ developments in string theory are located on this graph. (Note that neither circle size nor overlap has any representational relevance in this diagram.)
- 50.
See p. 10 of Frautschi, Steven C. Interview by Shirley K. Cohen. Pasadena, California, June 17 and 20, 2003. Oral History Project, California Institute of Technology Archives. Retrieved [24th July, 2013]: http://resolver.caltech.edu/CaltechOH:OH_Frautschi_S. Frautschi, also a postdoc under Chew, shared an office with Mandelstam in 1960. Interestingly, Frautschi mentions (pp. 18–19) his later work on the so-called “statistical bootstrap” (employing some of Rolf Hagerdorn’s ideas) reproduced facts of the Regge phenomenology (such as equal-spacing between successive spin states and exponential growth in particle species with mass increases) without invoking string theory, or being aware that what he was doing had any connection to the derivation of equal-spacing from the oscillations of a string system. By this stage, 1971–1972, Frautschi was at Cornell, and that he wasn’t aware of the work that had by then been carried out using string models perhaps indicates that work on dual models and string models did not travel so widely and easily outside of the primary groups.
- 51.
Chew had referred disparagingly to quarks, in 1965, as “strongly interacting aristocrats” [7, p. 95].
- 52.
Schwarz was much aided by Murray Gell-Mann’s advocacy during the quieter years of superstring theory. In his closing talk at the 2nd Nobel Symposium on Particle Physics, Gell-Mann pointed out that Sergio Fubini joked that he (Gell-Mann) had “created at Caltech, during the lean years ... a nature reserve for an endangered species—the superstring theorist” [25, p. 202].
- 53.
At a session on dual models at CERN in 1974, Harry Lipkin put forth the following as a ‘motto’ of the session: “Dual theory should be presented in such a way that it becomes understandable to non-dualists. At least as understandable as East Coast theories are for West Coast physicists and vice versa” (http://www.slac.stanford.edu/econf/C720906/papers/v1p415.pdf). John Polkinghorne speaks of “Californian free-wheeling (bootstrappers)” and “New England Sobriety (field theory)” [45, p. 138]. Peter Woit’s book [57, p. 150] includes a discussion of the East-West divide.
- 54.
- 55.
Kikkawa later joined CUNY (with another dual model/string theorist Bunji Sakita) in 1970.
- 56.
I might add to this brief review of networks the fact that Amati took a sabbatical year in Orsay, while Andrè Neveu and Jöel Scherk were doing their PhDs there, spreading the gospel of dual resonance models to two of its future central proponents. Note that Neveu and Scherk later joined Schwarz in Princeton (in 1969) on NATO fellowships. However, since French higher degrees were not called PhDs, Neveu and Scherk were mistakenly classified as graduate students and assigned to Schwarz as such.
- 57.
David Fairlie himself oversaw a significant dual model group at Durham University in the UK, supervising several PhD theses on the subject in the 1970s.
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Rickles, D. (2014). Particle Physics in the Sixties. In: A Brief History of String Theory. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45128-7_2
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