Abstract
In this chapter we look at the Veneziano model and the rather the wider project of generalising it to multi-particle, multi-loop situations. This was considered to be a genuinely possible route to a full theory of strong interaction physics. Accordingly, many physicist-hours were spent labouring on it, despite the fact that the framework was in many ways utterly detached from most areas of particle physics. There emerged clear early problems with the Veneziano model (the lack of unitarity and the restriction to a 4-particle process) that were resolved with remarkable speed and skill, well within two years, as was the problem of formulating an appropriate mathematical framework. The result was a general, elegant operator formalism for dual models that clearly pointed towards some underlying system responsible for generating the excitation spectrum.
All through 1969 people were adding legs to the Veneziano amplitude, or chopping it in half.
Claud Lovelace
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Notes
- 1.
As Christoph Schmid points out, achieving this was not an obvious possibility at the time: “[l]et me remind you that many people published ‘proofs’ that duality was impossible ... until Veneziano (1968) published his beautiful model. Since one example is stronger than a thousand ‘proofs’ to the contrary, people had to accept the fact that duality was possible” [42, p. 125]. The discovery also seems to have opened the floodgates, for some, as regards the possibility of saying something profound about hadronic scattering amplitudes (behind the various approximations). As David Fairlie writes, “to everyone’s complete surprise Gabriele Veneziano came up with his famous compact form for a dual scattering amplitude, which encompassed contributions from many towers of resonances, and I felt that this was for me!” [20, p. 283]. It is, of course, often the case that the impact of some result is all the more impressive when its prior probability is very low. We see the same ‘high impact’ phenomenon following Michael Green and John Schwarz’s anomaly cancellation result (for specific string theories) which had also been assigned a vanishingly small prior probability by the community of physicists working on it before their discovery.
- 2.
As David Olive recalls, Veneziano presented his discovery “in the ballroom of the Hofburg ... during the Vienna Conference on High Energy Physics (28 August–5 September 1968)” [38, p. 346].
- 3.
The Euler beta function is related to the Gamma function a follows: \(B(a,b)=\frac{\varGamma (a)\varGamma (b)}{\varGamma (a+b)}\). Hence, we can also write Eq. 3.1 simply as \(A(s,t,u) =A(s,t) + A(s,u) + A(t,u)\). This expression encodes the three possible permutations of the four scattered particles’ labels (that are neither cyclic nor anti-cyclic): \((1234)\), \((1243)\), and \((1324)\)—cf. [18, p. 61]. Or, in plain words, these characterise the three perspectives from which one can view the scattering of the particles in the various channels.
- 4.
At this stage there was no known restriction on the value of the intercept, so it seemed it could be fixed to physically reasonable values. However, Virasoro would later show that consistency (specifically, being ghost-free) demanded a unit intercept, \(\alpha (0) = 1\). Other models would require slightly different, but still fixed, intercept values.
- 5.
Rather surprisingly perhaps, Chew was not happy with the Veneziano model because of this approximation. He viewed it as conceding too much to the fundamentalist (read arbitrary) approach. According to Chew, the general S-matrix constraints ought to “fix particle widths as well as particle masses” [12, p. 26].
- 6.
The standard interpretation was to view the Veneziano amplitude as the first term in a Born approximation to a more complete version of the amplitude which would be ‘generated’ by adding loop corrections, hopefully thereby fixing the problem of unitarity.
- 7.
Note that ’t Hooft’s comments have some overlap with my reasons for being cautious about marking the birth of string theory with the construction of the Veneziano’s model.
- 8.
Of course, it is the dissemination that (quite rightly) holds the weight in matters of scientific discovery, so I don’t mean to reduce Veneziano’s place in the history of dual theory and string theory with this discussion. My aim is to flesh out the background to the discovery and to present a piece of the history that has hitherto remained under wraps—my sincere thanks to Professor Suzuki for sharing his story with me.
- 9.
Meaning “mathematical formulas”. These volumes traveled with Suzuki when he left Japan for life in Pasadena. They still grace his shelves in Berkeley. Following an expression of interest in them from David Horn, Suzuki had a set mailed over as a present to Horn in 1966.
- 10.
Though it seems he initially stuck to writing it as the ratio of gamma functions, as expressed in his book of formulas. It was in fact Ling-Lie Wang (now Chau, currently at UC Davis) that mentioned that this ratio was simply the beta function, while chatting about their current research topics in the IAS library reading room.
- 11.
- 12.
This approximation basically means, in modern terms, that it is only carried out at the tree level, with ‘external’ particles, not to all orders.
- 13.
However, an early study of the problem of incorporating spin, in some detail, was that of Yasunori Miyata in Tokyo [36]. Later studies, as we see in a later chapter, correspond to what are now viewed as the first spinning string models of Pierre Ramond and Andrè Neveu and John Schwarz.
- 14.
Fairlie compares the problem of the dual model tachyon to that weighing on Yang-Mills theory in the early days of its existence because of the zero mass particles described by the theory, which seemed clearly inadequate in accounting for spin-1 strongly interacting particles [20, p. 284]. As he notes, though his colleagues were sceptical of resolving the problem, the ground state tachyon was indeed eliminated thanks to a clever projecting out of the physically irrelevant sector of states by Gliozzi, Scherk, and Olive, in 1977 (see Sect. 7.3).
- 15.
The Japanese physicist Ziro Koba died on 28th September 1973. He had been a student of Tomonaga’s. Apparently, he had once shaved his head as a self-punishment for making an error in a self-energy calculation he was carrying out for Tomonaga (see Madhusree Mukerjee’s article on Nambu in Scientific American, February 1995, p. 38). He had neglected to include certain processes involving virtual pairs created via the Coulomb self-interaction of a vacuum electron—see Progress in Theoretical Physics 2(2), pp. 216–217 (for the original calculation) and p. 217 for the retraction. Curiously, Koba had once shared an office with Yoichiro Nambu (who would later become the first to give the full string interpretation of the Veneziano formula) in Tokyo, just after the second world war.
- 16.
See http://theory.fi.infn.it/colomo/string-book/nielsen_note.txt. It seems that Hector Rubinstein was very effective in spreading the news of the Veneziano model. Leonard Susskind also credits Rubinstein with bringing the Veneziano amplitude to his attention while Rubinstein (then based in Israel) was visiting him in New York [48, p. 204].
- 17.
Rather interestingly, these factors would undergo successive transformations (“theoretical exapation” in the terminology of Chap. 7) as the understanding of dual models underwent its own transformation, first into string theory (amounting to an index known as the “Chan-Paton factor”) characterising the endpoints of open strings) and then later as a result of developments leading to D-branes (amounting to an index characterising the surfaces the endpoints of open strings terminate on). (I should also point out, as a matter of historical accuracy, that Chan-Paton factors ought really to be called Paton-Chan factors, given that Paton was the lead author on the original paper.)
- 18.
Note that there exists a two-to-one correspondence between the operators in the original Veneziano model and in the Virasoro-Shapiro model (an important implication of this is a doubling of masses in the latter, as compared to the former). However, both the original Veneziano model and the Virasoro-Shapiro model are consistent only in \(d=26\) (a discovery that would be made in the year following these generalizations). The Virasoro-Shapiro model was later interpreted to be a closed-string analogue of the original Veneziano model.
- 19.
David Kaiser refers to superstring theory as a “sign of the S-matrix program’s afterlife” which came about through “the transmogrification of Gabriele Veneziano’s 1968 ‘duality’ model” [29, p. 385]. He argues that Feynman diagrams (“paper tools”) were at the heart of this transmogrification, claiming that “[t]oday’s superstring theories owe their existence” to such tools. I would, however, say that the duality programme (initiated by DHS duality) initially marked a rather dramatic failure of Feynman diagrams, pointing to a need for diagrams (‘duality diagrams’) with very different representational characteristics. These map in an even more indirect way onto their target processes, as the Harari-Rosner diagrams make clear—functioning as equivalence classes of Feynman diagrams and thus superseding them. It took a little longer to interpret this equivalence class as emerging from the invariance properties of string worldsheets, and strictly speaking there was a discrete jump from pre- to post-duality programme Feynman diagrams. The Kikkawa, Sakita, Virasoro paper [31] was pivotal in the restablishment of Feynman diagram (or ‘Feynman-like’ diagrams, as they make clear) techniques, as were Holger Nielsen and David Fairle’s ‘fishnet diagrams’ (discussed in Chap. 4). Subsequent usage of Feynman-like diagrams in superstring theory truly superseded their original role, since one eventually finds (thanks to modular invariance) that only one diagram is needed at each order of perturbation theory, which defeats their original purpose.
- 20.
Here, the loop corrections (known as ‘\(M\)-loops’) were conceptualised as integrals over a holed surface, with the number of handles (the genus) corresponding to the order in the perturbation series, much as in the modern string theoretic sense.
- 21.
Of course, this corresponds to the “chopping it in half” part of the Claud Lovelace quotation opening this chapter.
- 22.
Note that in their model of rising Regge trajectories in 1968 (still pre-Veneziano’s model), Chu et al. [13] had guessed at the existence of a possible harmonic-oscillator potential as the ‘force’ causing the trajectories to rise.
- 23.
Pierre Ramond describes the creation/annihilation operator formalism as “clearly the window into the structures behind the dual models” ([41, p. 362]).
- 24.
The oscillators were Bose fields in the first dual models. In the next chapter we look at the attempt to generalise this to include a fermionic sector, and also a combination of a bosonic and fermionic sector of states.
- 25.
As John Schwarz writes, “it suggested that these formulas could be viewed as more than just an approximate phenomenological description of hadronic scattering. Rather, they could be regarded as the tree approximation to a full-fledged quantum theory.” [45, p. 55]. A fact that came as a surprise to Schwarz, and many others.
- 26.
Elena Castellani quite rightly puts great stress on the “continuous influence exercised by quantum field theory” in the development of string theory [9, p. 71]. Quantum field theory had at its disposal very many powerful tools for dealing with problems faced by dual models, not least the elimination of the ghosts from the spectrum of states, which were eliminated using a gauge-symmetry device known from QED—though, as we will see, an infinite-dimensional symmetry is required in the case of dual models (a result that would be understood in the string picture as arising from the infinitely many ghosts corresponding to string’s infinite tower of vibrational modes).
- 27.
This is analogous to the situation that arises with timelike photons in QED, in which the spurious states also decouple.
- 28.
Virasoro was, of course, well aware of the overly restrictive nature of the unit intercept case, but expected that his method could be generalised to more physically realistic cases. Fubini and Veneziano [22] later did this using a projective operator language, providing ghost-cancellation up to the third excited level. Brower and Thorn [6] still later extended this to the ninth excited level.
- 29.
Here he was building on earlier work of Gliozzi [25], who had constructed a similar set of operators: \(L_{n} = - \frac{1}{2} \sum _{m=-\infty }^{\infty } : a_{-m} \cdot a_{m+n},\;\; n = -1 , 0 , 1 \) (note that this is Mandelstam’s condensed version of the Gliozzi operators: [35, p. 282]. However, Virasoro extended this to all values of \(n\).
- 30.
Note that the Virasoro algebra is the central extension of the ‘Witt algebra’ (over \(\mathbb {C}\)) defined by \([L_{m}, L_{m}] = (m-n) L_{m+n}\) (generators are \(\{L_{n} : n \in \mathbb {Z}\}\))—see [53]. It acts as \([L_{m}, L_{m}]f = \{ -z^{n+1} \frac{d}{dz} , -z^{m+1} \frac{d}{dz} \}f \). This is the Lie algebra associated with diffeomorphisms of the circle.
- 31.
The central charge\(c\) in the algebra is credited to Joseph H. Weis. Weis died, aged just 35, in a climbing accident in the French Alps (on the Grandes Jorasses) in August, 1978—he was killed with his climbing partner, and CERN physicist, Frank Sacherer. He had taken his PhD under Mandelstam at Berkeley, before taking up a postdoctoral position at MIT. He discovered the central charge during his study of 2D QCD. Though he never published it, he seems to have communicated his discovery to several people, and one can find him credited in, e.g., [6, p. 167], [22].
- 32.
These physical states were also shown to be eigenstates of the twist operator mentioned above [15, p. 587].
- 33.
Goddard and Thorn [27], and also Brower [7], later proved in 1972, that this space is complete when the number of transverse components of the oscillators is 24 (that is to say, the physical Hilbert space of states has dimension \(\fancyscript{T}^{24}\)). There are no ghosts present when this condition, in addition to the unit intercept condition, is met. For \(D>26\) (where \(D\) is the spacetime dimension) ghosts appear, for \(D<26\) there are no ghosts.
- 34.
- 35.
A complete understanding of the elimination of tachyons would take the best part of a decade to achieve, once supersymmetry was better understood.
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Rickles, D. (2014). The Veneziano Model. In: A Brief History of String Theory. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45128-7_3
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