Abstract

A personal account is given of the six to seven years (1967–1974) during which the hadronic string rose from down-to-earth phenomenology, through some amazing theoretical discoveries, to its apotheosis in terms of mathematical beauty and physical consistency, only to be doomed suddenly by the inexorable verdict of experiments. The a posteriori reasons forwhy the theorists of the time were led to the premature discovery of what has since become a candidate Theory of Everything, are also discussed.

Introduction and outline

In order to situate historically the developments I will be covering in this Chapter, let me start with a picture (see Figure 2.1) illustrating, with the help of Michelin-guide-style grading, the amazing developments that took place in our understanding of elementary particle physics from the mid-Sixties to the mid-Seventies. Having graduated from the University of Florence in 1965, I had the enormous luck to enter the field just at the beginning of that period which, a posteriori, can rightly be called the ‘golden decade’ of elementary particle physics.

The theoretical status of the four fundamental interactions was very uneven in the mid-Sixties: only the electromagnetic interaction could afford an (almost1) entirely satisfactory description (hence a 3-star status) according to quantum electrodynamics (QED), the quantum-relativistic extension of Maxwell's theory. Gravity too had a successful theoretical description, this time according to Einstein's general relativity, its 2-star rating being related to the failure of all attempts to construct a consistent quantum extension.

Abstract

In this Chapter I describe the ideas and developments that led to the recognition that the structure underlying the N-point scattering amplitude of the Dual Resonance Model was that of a quantum-relativistic string.

Introduction

The history of the origin of string theory is very peculiar and difficult to understand if one looks at it with today's eyes. In particular, the fact that one could understand so many string properties without making reference to any Lagrangian, and without even knowing that one was studying a string, seems almost miraculous.

In fact, the starting point for describing the property of a relativistic string is today the string Lagrangian, which is invariant under reparameterizations of the world-sheet coordinates. From it, using techniques developed for quantizing theories with local gauge invariance, such as the Faddeev–Popov and the BRST quantization, one derives the spectrum of physical states and their scattering amplitudes. However, at the end of the Sixties and at the beginning of the Seventies, these techniques were not yet known and, even though the Nambu–Goto action had already been written, it was not clear how to use it to deduce its physical consequences. Therefore, the historical path that led us to understand the properties of a relativistic string was quite different from the path that one follows today, for instance, when teaching string theory in a university course. But at that time, where did string theory come from?

In from the periphery

At the University of Helsinki, where I took my BSc and MSc degrees, research in and teaching of modern theoretical physics had started in the early Sixties after a long period of stagnation. In particle physics, introduced to Finland by amongst others K. V. Laurikainen, the emphasis was on phenomenology.

Enthusiastic young teachers like Keijo Kajantie taught us the basics of S-matrix and Regge theory. When Veneziano's paper appeared, it was greeted with much interest. Visits to Finland by Chan Hong-Mo, David Fairlie and Holger Bech Nielsen helped to spread the gospel. In the spring of 1970, Eero Byckling organized a study group on the Koba–Nielsen formalism, and in the summer of 1970 I was able to participate in the legendary Summer School in Copenhagen, where Nambu was not present in person, but in the form of his string action; his lecture notes were distributed among the participants.Victor Alessandrini's lectures on the multiloop amplitudes of the Veneziano model [Ale71, AA71] made a deep impression on me, and I long cherished his beautiful handwritten notes on the subject.

I was fortunate enough to receive a three-year scholarship for doctoral studies at ‘an established British university’ from the Osk. Huttunen Foundation, the very first institution in Finland which started awarding grants for long-term postgraduate studies. Being convinced that S-matrix theory was the answer to the riddle of the strong interactions, I naturally applied to Cambridge, the European Mecca of S-matrix theory.

The beginnings

I started research in the year 1957, at the time when aspiring particle physicists were being channelled into the arid field of dispersion relations, and field theory was out of fashion. I did not find the methods of analysis employed in trying to extract information out of dispersion relations to my taste. The only tool available was the analyticity of the S-matrix, as constrained by the requirements of causality, that there should be no output before input. To give an instance of the attitude to mathematics at the time, we graduate students were advised that the only pure mathematical courses worth attending were those on functional analysis, or the theory of several complex variables! The philosophy of logical positivism reigned supreme, in which one was not allowed to talk about the unobservable features of particle interactions, but only about properties of asymptotic states. This was one of the features which inhibited the invention of the concept of quarks. I had been impressed by the tractability of electrodynamics and quantum mechanics as an undergraduate, and what Wigner has called ‘The unreasonable effectiveness of Mathematics in the Natural Sciences’. In the middle of year 1968 I was feeling very pessimistic about the possibility of theorists ever being able to say anything about scattering amplitudes for hadrons, beyond the simple tree and Regge pole approximations, and was contemplating changing fields.

Not by accident

The Sixties in Berkeley were exciting times for a young graduate student, not just because of the free speech movement, but also in elementary particle physics. The strong nuclear interactions appeared to be so different from the beautiful simplicity of QED that a systematic programme was underway to find a new foundation to relativistic quantum interactions. A new set of axioms was sought in the so-called S-matrix programme by abstracting properties of analyticity and unitarity exhibited in Feynman diagrams without recourse to an underlying local field theory. Remarkably this ambitious programme, guided by copious experimental data for hadronic scattering, did result in an alternative solution, now referred to as ‘string theory’. The fact that the nuclear or strong interaction was subsequently formulated by QCD, a local field theory, should not obscure this remarkable discovery. Only recently, with the advent of Maldacena's conjectured AdS/CFT correspondence between string and gauge field theory [Ma198], are we beginning to understand why two alternative solutions to strong interactions (gauge theory and its dual string theory) should co-exist as correct and equivalent theories. Whether string theory ultimately leads to a fundamental string/gauge duality for all forces including gravity, is not clear yet. Nevertheless the discovery of the theory, contrary to a widespread opinion, was not by accident, but was the result of the traditional interaction between theoretical ideas and experimental data.

Here let me give an intentionally anecdotal description of my recollections of early developments in string theory.

The Veneziano model [Ven68], the starting point of string theory, addressed the at that point much studied and phenomenologically successful idea of ‘two-component duality’. Here I would like to recall this idea and give its meaning in modern terms. At the risk of letting the cat out of the bag at too early a stage, let me say right away that the two components in question will turn out to be the open and the closed hadronic strings.

The argument for two-component duality is the following. Unlike quarks, hadrons (mesons and baryons) are obviously not elementary objects. Yet the particles appearing in the initial, final and intermediate states of the hadronic S-matrix, describing all quantum processes involving hadrons, are precisely these composite objects and not their constituent quarks. With elementary particles it is obvious how to calculate the S-matrix, just use the Feynman rules. For instance, when studying lowest order (tree-level) Bhabha scattering (i.e. electron–positron scattering) in QED, we are instructed to add the socalled s- and t-channel photon-pole Feynman diagrams, as in Figure 8.1. But there we have a Lagrangian and the photon is an ‘elementary’ particle whose field appears in this Lagrangian.

When dealing with composite states, these are represented by an infinite sum of Feynman diagrams in the quantum field theory of the elementary fields out of which the composite particles are built. In the simplest case we can think of these diagrams as Bethe–Salpeter ladder-diagrams.

Student in Cambridge

After all this time I do not trust my memory to be accurate in every detail. Besides, I can only provide glimpses from the perspective of a beginning graduate student struggling to catch up, followed by that of a postdoc who found himself quite far from much of the action. I will not attempt to be comprehensive in either the telling or the references, rather I will restrict myself to those aspects I concentrated on at the time, some of the people I knew and who influenced me, and will try not to add any insights that have emerged over subsequent years.

In October 1969, more than forty years ago, I started my three years as a graduate student in the Department of Applied Mathematics and Theoretical Physics, Cambridge. During the previous year, as was customary then, I took Part III of the Mathematics Tripos. This was a thorough grounding in many of the tools then useful in elementary particle theory. However, by today's standards it was lacking in some respects, especially in the area of quantum field theory – the most notable omission being any mention of Yang–Mills gauge theory. This was hardly surprising, of course, because most of the Cambridge group had been actively developing S-matrix theory and ‘Eden, Landshoff, Olive and Polkinghorne’ wasmandatory reading for prospective graduate students. Less surprising, was the omission of any reference to the Veneziano model, it being far too much of a recent innovation to make it into Part III. Instead, I heard about it in a roundabout way.

This is a very personal view of the origin of string theory, and it also makes an interesting prediction.

When I first encountered the Veneziano model in 1968, I was at CERN working on the phenomenology of strong interactions. I saw an application to pion–pion scattering (Lovelace [Lov68] and Shapiro [Sha69]), which occurred independently to Shapiro and Yellin [SY70] at Berkeley. It attracted considerable attention, but now seems misguided. Later I combined dual amplitudes with Gribov's Reggeon calculus [Lov71b]. This application has been revived recently by people exploiting AdS/CFT duality, and should eventually succeed.

All through 1969 people were adding legs to the Veneziano amplitude, or chopping it in half. I will leave this story to other contributors. Amati and coworkers gave a series of seminars at CERN on their SL(2,ℝ) group theoretic formulation which I thought very elegant [AALO71]. This inspired me to drop phenomenology and join them. In modern language, the N-Reggeon vertex [Lov70a] is the open string disc amplitude with arbitrary excited states on all external lines. Several people had written down a complicated formula. I noticed it could be factorized by a single set of oscillators at the centre of the disc, and realized that an easy construction of the complete perturbation expansion to all orders in string loops would follow. Alessandrini and Amati [Ale71, AA71], and I [Lov70b] worked this out.

I am eulexic – my favourite way of developing an idea is to cover a table with piles of possibly relevant books and papers, and rapidly turn pages till I catch a scent like the one in my mind.

I was privileged to be either a participant or an observer in many of the developments involved in string and superstring theory and my attachment to the CERN Theory Division during much of the Seventies gave me a grandstand view of much of it. In what follows I shall try to describe my own experiences, what I did, or tried to do, and what I saw and heard, because those are the things of which I am most certain. So, in particular, this does not aspire to be a comprehensive history.

I was present in the ballroom of the Hofburg when Gabriele Veneziano first presented his dual scattering amplitude to the wider world of theoretical physics during the Vienna Conference on High Energy Physics (28 August–5 September 1968). Despite the bad acoustics of the venue that experience changed my life and makes an appropriate start for my account [CERN68, Ven68], even though there is important prehistory.

The idea that there could be formulae for particle scattering amplitudes that could be, in some sense, almost exact fell on fertile ground. My scientific outlook had been formed in DAMTP, Cambridge, UK where the influence of the then charismatic figure, Geoffrey Chew, of the University of California at Berkeley still held sway. He and his school, influenced by the earlier work of Werner Heisenberg, had argued that scattering amplitudes of hadrons were the appropriate quantities to think about, rather than the quantum fields that create the particles.

Abstract

This Chapter is about my memories of the discovery that the Veneziano model in fact describes interacting strings. Susskind and Nambu also found the string picture by following other, independent approaches. A characteristic feature of my approach was that I used very highorder ‘fishnet’ or planar Feynman diagrams to describe the development of the relativistic string. The arrangement of particles on a chain indeed leads to the dominance of planar diagrams, if only nearest neighbour interactions are relevant. I also mention the work of Ziro Koba and myself on extending the Veneziano model, first to five external particles – as also done by Bardakci and Ruegg, Chan and Tsou, and Goebel and Sakita – and subsequently to an arbitrary number of external mesons.

Introduction

In the unpublished preprint [Nie70], ‘An almost physical interpretation of the integrand of the n-point Veneziano model’, I proposed – independently of Nambu [Nam70] and Susskind [Sus70a, Sus70b] – that the dual model or Veneziano model [Ven68] was describing the scattering of relativistic strings.

In the present Chapter I present my recollections on how I came to understand the scattering of strings in the Veneziano model. The background that led to the string idea can be found in the works with Ziro Koba that extended dual models to an arbitrary number of external particles [KN69a, KN69b, KN69c]. This work is described in Section 22.4. It would, however, be natural in such a reminiscence to describe first what for me was crucial, namely the seminar by Hector Rubinstein given at the Niels Bohr Institute.

String theory describes one-dimensional systems, like thin rubber bands, that move in spacetime in accordance with special relativity. These objects supersede pointlike particles as the elementary entities supporting microscopic phenomena and fundamental forces at high energy.

This simple idea has originated a wealth of other concepts and techniques, concerning symmetries, geometry, spacetimes and matter, that still continue to astonish and puzzle the experts in the field. The question ‘What is string theory?’ is still open today: indeed, the developments in the last fifteen years have shown that the theory also describes higher-dimensional extended objects like membranes, and, in some limits, it is equivalent to quantum field theories of point particles.

Another question which is also much debated outside the circle of experts is: ‘What is string theory good for?’ In its original formulation, the theory could not completely describe strong nuclear interactions; later, it was reproposed as a unified theory of all fundamental interactions including gravity, but it still needs experimental confirmation.

This book will not address these kinds of questions directly: its aim is to document what the theory was in the beginning, about forty years ago, and follow the threads connecting its development from 1968 to 1984. Over this period of time, the theory grew from a set of phenomenological rules into a consistent quantum mechanical theory, while the concepts, physical pictures and goals evolved and changed considerably.

Abstract

We briefly recall the historical environment around our 1971 and 1975 constructions of current algebraic internal symmetry on the open string. These constructions included the introduction of world-sheet fermions, the independent discovery of affine Lie algebra in physics (level one of affine su(3)), the first examples of the affine-Sugawara and coset constructions, and finally – from compactified spatial dimensions on the string – the first vertex-operator constructions of the fermions and level one of affine su(n).

Introduction

In this Chapter we describe the environment around our 1971 paper ‘New dual quark models’ [BH71] and the two companion papers ‘The two faces of a dual pion–quark model’ [Hal71c] in 1971, and ‘Quantum “solitons” which are SU(N) fermions’ [Hal75] in 1975, which laid the foundations of non-Abelian current-algebraic internal symmetry on the string.

The background for our contributions included many helpful discussions with other early workers, including H. M. Chan, C. Lovelace, H. Ruegg and C. Schmid (during KB's 1969 visit to CERN), as well as R. Brower, S. Klein, C. Thorn, M. Virasoro and J. Weis (with MBH at Berkeley). Both of us also acknowledge many discussions with Y. Frishman, G. Segre, J. Shapiro and especially S. Mandelstam. Later discussions with I. Bars and W. Siegel are also acknowledged by MBH. With apologies to many other authors then, we reference here only the work which was most influential in our early thinking: before Veneziano, there had been widespread interest in the quarkmodel of Gell-Mann and Zweig, including the four-dimensional current algebra (Adler and Dashen [AD68]) of quarks.