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Supersymmetric Strings and Field Theoretic Limits

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A Brief History of String Theory

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Abstract

The first dual models were a strictly bosonic affair. A major challenge was therefore to make them more physically realistic by including fermions in the spectrum. We describe these advances and the events leading up to them. We also discuss how new links to field theories were discovered using the ‘zero-slope’ method.

You claimed to have solved a problem that many people including my colleagues at Berkeley have been trying to solve. I do not know who you are, and from what you have told me, I cannot tell if you have succeeded, but I will study it and let you know.

Stanley Mandelstam (to Pierre Ramond)

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Notes

  1. 1.

    The quotation above is how Ramond recalls Mandelstam’s skeptical response to his claim to have generalised the Dirac equation to the Veneziano model [27, p. 6].

  2. 2.

    In fact, there were five theorists in all, “The NAL Fives” [26, p. 362], hired by Robert Wilson, the then director of NAL—the others were Jim Swank and Don Weingarten, all housed at 27 Sauk Boulevard. Hoddeson, Kolb, and Westfall refer to this period as Fermilab’s “experiment in theory” [16, p. 139], motivated by Wilson and Edwin Goldwasser’s desire to generate dialogue between theory and experiment.

  3. 3.

    Note that the full interpretation of supersymmetry on the worldsheet was to come later than Clavelli indicates.

  4. 4.

    See Lou Clavelli’s article “On the Early Tension between String Theory and Phenomenology”: http://bama.ua.edu/~lclavell/papers/tension1.pdf. (Note that Clavelli was a student of Nambu’s in Chicago.)

  5. 5.

    This bears some resemblance to the so-called Frenkel-Kac mechanism for compactifying degrees of freedom on a lattice—a construction that was crucial in the development of the heterotic string theory (see Sect. 9.2).

  6. 6.

    This is the same Gordon responsible for the influential oscillator formalism paper with Fubini and Veneziano.

  7. 7.

    Neither this nor Ramond’s subsequent paper, introducing the dual Dirac equation, were couched in terms of the string picture. Note also that the Ramond model was a free theory.

  8. 8.

    It’s worth pointing out that Ramond had originally intended to study general relativity in Peter Bergmann’s group, after receiving a four-year graduate fellowship to study at the University of Syracuse. He had been advised to join Sudarshan’s particle physics group instead, later studying under him, though later switching to Balachandran (cf, [26, p. 361]).

  9. 9.

    In his recollections [25, pp. 8–9] André Neveu notes how he first encountered Pierre Ramond quite by chance when they were crossing the Atlantic on the same ship, bound for Princeton and Fermilab respectively. Neveu was studying the Fubini-Veneziano paper on the factorization of dual models in the ship’s library, and left momentarily, at which point Ramond entered, finding the paper: the same paper he was working through! As a result of this fortunate interaction, Ramond thought to send Neveu (and Schwarz, with whom Neveu was collaborating) a copy of his dual (free) fermion paper which inspired them to work out the details for the interacting theory. As Schwarz recalls, he and Neveu had been working on a new bosonic theory, but then noticed that their’s and Ramond’s “were different facets of a single theory containing our bosons and Ramond’s fermions” [31, p. 11]. Moreover, the Ramond model suffered from a lack of manifest duality.

  10. 10.

    This work also had mathematical implications. In fact, they had discovered in this work affine Lie algebras (independently of knowledge of the mathematical work then available) and constructed a concrete, fermionic representation of \(\widehat{\mathfrak {sl}}(3)\). Clavelli produced a broadly similar construction during his time at NAL in his paper “New Dual \(N\)-Point Functions” (NAL Preprint: http://lss.fnal.gov/archive/1971/pub/Pub-71-009-T.pdf).

  11. 11.

    Here I am indebted to Schwarz’s own account presented in [32]. A more recent account can be found in [33].

  12. 12.

    The Veneziano model is sometimes called the “orbital model” on account of the absence of spin degrees of freedom.

  13. 13.

    Though, in a paper Corrigan co-authored with David Olive, they also note that while it is “tantalizing to think of the Ramond fermion as a quark or a baryon ... in fact it is probably neither, but just an important clue on the way to more physically realistic theories” [5, p. 750].

  14. 14.

    Thorne also considered \(N\) pions and two fermions, recovering the spectrum of Neveu and Schwarz’s model (from fermion-meson channels), in addition to the spectrum of Ramond’s propagator (from the fermion-meson channels). Corrigan and Olive generalised this work by constructing a general dual vertex giving the transition of a Ramond fermion into a Neveu-Schwarz meson by the emission of a general excited fermion state [5].

  15. 15.

    It was still, like the Veneziano model, a dual-resonance model. This initial disconnection from the (known) string interpretation (instead, employing operator methods) seems to have been essential for building solid results and moving the field forward in the earliest phase of dual model research—such an approach constitutes one of many such examples of pushing point-particle analogies as far as they will go (e.g. before specific string-specific issues arise).

  16. 16.

    Iwasaki and Kikkawa had, however, given an earlier model of a free spinning string: [17].

  17. 17.

    The full understanding that \(d=10\) in these embryonic superstring theories came in several stages, as with the \(d=26\) result for bosonic strings, with evidence that the theory not only became simpler but also that it was required for the theory to be unitary and ghost-free (see [2, 30]).

  18. 18.

    There was a now well-known incident involving the dismissal of Golfand during Russia’s staff reduction campaign (a thinly disguised anti-semitic campaign). Golfand sent an appeal to Harald Fritzsch, which resulted in a letter being signed by many physicists, including several string theorists [see Gell-Mann Papers: Box 8, Folder 21].

  19. 19.

    More precisely, following Golfand and Likhtman [12, p. 3], a linear space \(L\) is said to be \({\mathbb {Z}}_{2}\)-graded if it possesses a subspace of vectors \(^0{L}\) which are even, with another subspace of odd vectors, \(^1{L}\), and for which the whole of \(L\) is a direct product of these subspaces: \(L = \, ^0{L} \oplus ^1{L} \).

  20. 20.

    This rotation is often described by saying that the rotation happens within a spinorial extension of space, or superspace. One can view this in terms of operators \(Q\) (spinorial charges) acting on bosonic and fermion states as: \(Q|{\textit{fermion}}\rangle = |{\textit{boson}}\rangle \) and \(Q|{\textit{boson}}\rangle = |{\textit{fermion}}\rangle \). The \(Q\)s satisfy the commutation relations \(\{Q_{\alpha }^{i} , Q_{j}^{\dot{\alpha }}\} = -2 (\sigma ^{\mu })_{\alpha }^{\dot{\alpha }} \delta ^{i}_{j} P_{\mu } \) (where \(P_{\mu }\) is the energy-momentum operator—as Haag et al. [14, p. 258] pointed out, the appearance of such operators amongst the elements of the superalgebra implies that a “fusion” occurs between geometric (spacetime) and internal symmetries). The number \(N\) of supersymmetries leads to a classification of theories as follows: \(N=1\) is known as simple symmetry; \(N>1\) cases are known as extended supersymmetry; \(N\le 4\) is demanded by renormalizable gauge theories; and \(N\le 8\) is required for helicity-2 theories like supergravity. (Note that simple supersymmetry is required if one wishes to construct parity-violating theories.)

  21. 21.

    The term ‘supersymmetry’ was introduced by Salam and Strathdee in 1974, in Trieste—it first appears in print, in hyphenated form as ‘super-symmetry,’ in [28].

  22. 22.

    For an excellent survey of the history of supersymmetry, including its intersections with supergravity and superstrings, see: [7].

  23. 23.

    Some did see the potential, of course. For example, Frampton and Wali write that the results are of “considerable interest because they provide a linkage between the hadronic models on the one hand, and Lagrangian field theories on the other” and that since “the latter is considerably more fully explored and understood than the former, we may hope to learn a great deal from the connection” [8, pp. 1879–1880]. Frampton and Wali discuss an interesting non-locality, resulting from the high-spins in hadron scattering experiments (as they say, expected from the string interpretation). They also suggest the possibility of utilising renormalization methods from dual models on the Lagrangians they study.

  24. 24.

    In a later paper on dual models, Scherk and John Schwarz, describe how the string picture makes the zero slope recovery of field theories intuitive: “the length of the strings is characterized by \(\sqrt{\alpha '}\), where \(\alpha '\) is the universal Regge slope parameter” so that “given this situation, it is not surprising that in the limit \(\alpha '\rightarrow 0\) dual models reduce to field theory models of point particles” [29, p. 347].

  25. 25.

    Of course, in the case in which dual models are taken to be models of gravitational (from closed strings) and gauge bosons (from open strings), then the massless states are necessary.

  26. 26.

    Scherk credits Ray Sawyer with the discovery of this zero-slope limit method, and notes that it was also implicit in [18]. Yoneya [36] later argued that Scherk’s result in fact clarified the relationship between the duality diagrams and Feynman diagrams, as discussed in [18].

  27. 27.

    Neveu and Scherk, in 1971, write that the “dual-resonance models seem to be an approach to strong interactions which stands between field theories and the S-matrix” [24, p. 155]. Their reasoning is that on the one hand “one writes down phenomenological amplitudes with desirable physical properties without using a Lagrangian, but on the other hand, the factorizability of those amplitudes allows [one] to compute unitary corrections as in a field theory” (ibid.).

  28. 28.

    Such a method amounts to a low-energy expansion.

  29. 29.

    Here I have in mind examples like ’t Hooft’s large \(N\) expansion in which he proves that the topological structure of the perturbation series in \(1/N\) (for a gauge theory with ‘colour group’ \(U(N)\)) matches that of dual models [34].

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    Dean Rickles

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Rickles, D. (2014). Supersymmetric Strings and Field Theoretic Limits. In: A Brief History of String Theory. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45128-7_5

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