6.1 Historical Outline and Introduction

Fig. 6.1 A schematic view of Veneziano duality in hadron scattering

The process can be thought first as the fusion of A with C into an intermediate state, that successively decays into B and D. The probability of the process is essentially provided by a weighted sum over all possible intermediate states. Alternatively one could interpret the same process as the fusion of A with the antiparticle of B into an intermediate state that decays into D and the antiparticle of C. Also in this interpretation the probability is given by a weighted sum over all possible intermediate states. These two interpretations of the same process are respectively named the s and the t channel of the considered reaction (see Fig. 6.1).

The idea that fascinated the physicists of that time was the following one. Might it exist a scattering amplitude AABCD such the first and the second interpretation are simultaneously valid and the sum over the intermediate states in the s channel is exactly equal to the same sum in the t channel? Such a property was christened duality and preserves such a name to the present day.

The posed question was of a complete mathematical nature. If such a function AABCD existed, the next necessary step was to invent a theory capable of yielding it as scattering amplitude for the considered process.

In a paper sent to the Rivista del Nuovo Cimento in that dense 1968 year, Gabriele Veneziano singled out a function that realizes the desired duality in a mathematical exact way: it is the Euler beta-function introduced two hundred years before by the great swiss mathematician. The same Veneziano contributed a couple of years later, together with Sergio Fubini from Torino University and the MIT, to open the way for the identification of the physical system capable of producing dual scattering amplitudes. Just in a couple of years, by means of the contributions of many scientists throughout the world, Veneziano’s formula for the dual scattering amplitude of four particles was generalized to processes with an arbitrary number N of external legs: in 1970, in another fundamental paper published on Nuovo Cimento, Fubini and Veneziano organized the construction of such amplitudes within a new algorithm defined operatorial formalism which involved the use of an infinite number of harmonic oscillators with frequencies that are integer multiples of a fundamental one.

This infinite spectrum of harmonic oscillators induced an intuition in the brilliant mind of Yoichiro Nambu (see Fig. 6.2), the same Nippon-American physicist who in 1965 had proposed the color charge for the quarks. Nambu observed that anyone who is familiar with string musical instruments perfectly knows a very simple physical system endowed with the spectrum used by Fubini and Veneziano: the vibrating string. A very short and tiny string that besides traveling through space-time can also

Fig. 6.2 The fathers of string theory. From the left Gabriele Veneziano, in the middle Sergio Fubini, on the right Yoichiro Nambu. Gabriele Veneziano was born in Florence, where he studied before transferring to the Weizman Institute in Israel, where he got his Ph.D. Later on, for many years he was permanent staff member of the Theoretical Division of CERN, which he left at retirement age to fill a highly prestigious position at the French Academy in Paris. Sergio Fubini was born in Torino, where he studied and became quite early full professor of Theoretical Physics. Appointed professor of Physics at the Massachusetts Institute of Technology, he lived several years in Boston, until he left it to become permanent staff member of the Theoretical Division of CERN. After retirement he continued to live in Geneva where he died in 2005. Yoichiro Nambu, born in Japan, studied in the United States of America and up to the present day has been full professor of Physics at the University of Chicago. In 2008, professor Nambu was awarded the Nobel prize in Physics for his early contributions to the theory of symmetry breaking

Fig. 6.3 An idealized view of an open string that propagates through space-time, tracing a world-sheet with the topology of a strip

vibrate! This had to be the typical hadron! The infinite spectrum of hadronic states and resonances was thus explained with the infinite number of vibrational modes of the microscopic string. Once started, the string concert rapidly grew and developed. In a series of papers produced by several authors from all countries of the world, the physical system of the quantum-relativistic string was analyzed from all viewpoints. The string can be closed or open, namely its end points can coincide, or not. In the first case the string has the topology of a circle, in the second that of a segment. In both cases propagating through an ambient space-time the strings sweeps a worldsheet that in the closed case has the topology of a cylinder, in the second case that of a strip with boundary (see Figs. 6.3, 6.4). The interpretation of Veneziano ampli-

Fig. 6.4 An idealized view of a closed string that propagates through space-time, tracing a world-sheet with the topology of a tube

Fig. 6.5 The closed string interpretation of a scattering amplitude of seven particles

tudes as the result of the propagation of tiny strings that can join and split became standard and it is schematically illustrated in Fig. 6.5. In the quantization of the system two problems were met, whose solution led the theory very far in the direction of unexpected scenarios of incredible mathematical depth and unparalleled wealth of physical implications. The first problem related with the number of space-time dimensions. The usual four-dimensional space-time was too narrow for the strings to propagate freely without developing deadly anomalies capable of destroying the quantum consistency of the two-dimensional world-sheet theory. In quantum field theory anomalies are obstructions that forbid the extension to the quantum level of global or local symmetries present at the classical level. In the case of local symmetries, anomalies are deadly blows since the quantum theory acquires spurious degrees of freedom and becomes both meaningless and inconsistent. In the case of the strings the anomalous symmetry is the conformal one, namely the invariance