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In 1968, Gabriele Veneziano, is going to construct a scattering amplitude (the Veneziano amplitude) with the property of Dolen-Horn-Schmid duality and that will also follow the Regge trajectories.
This is going to be the most successful S-matrix approach. The idea was that there was a consistent expansion that began with stable particles on straight line Regge trajectories. This property of Dolen-Horn-Schmid duality will prove to be what leads Veneziano, to construct his self-consistent scattering amplitude.
Veneziano, was working in Israel at the time. He was interested in figuring out the mathematics of the S-matrix theory. Nobody knew the mathematical expression that satisfied the S-matrix's rules. This search is what led to his proposal of the Veneziano amplitude. The Veneziano amplitude was not an elegant mathematical expression. It is a table of probabilities. It was a formula that tells us what happens when two hadrons collide. When Hector Rubinstein, showed this formula to Leonard Susskind, he made the connection. He believed that the formula had something to do with quantum mechanical harmonic oscillators. A harmonic oscillator is anything in physics that can vibrate, or swing back and forth repeatedly.
Veneziano and Suzuki
Indeed, it was two theorists: Gabriele Veneziano and Mahiko Suzuki, who were working independently from each other at CERN. These theorists will ask a daring question: if the S-matrix is suppose to obey such restrictive properties, than why not just try to guess the answer? These theorists thumbed through many volumes of mathematical formulas, some from as far back as the 18th century.
These theorists found that the Beta function of Swiss mathematician Leonhard Euler from the 1800s could satisfy most of the properties from Geoffrey Chew's S-matrix postulate.
It should be noted that despite this being a kind of celebrated victory of the S-matrix theory over quantum field theory, and there being a flurry of papers being published that attempted to apply the Beta function to the data that was coming out of atom smashers all over the world, there was still a problem with Chew's S-matrix: unitarity. Unitarity, in this sense, is the conservation of probability.
Veneziano’s idea was that the Euler beta function, could be interpreted as a scattering amplitude and that, it could describe many of the properties of strongly interacting particles. The amplitude itself, was for 4 open string tachyons. What the amplitude needed were: poles, where the particles would appear, on straight line trajectories. Veneziano, is going to use the Gamma function (which was used widely in Regge theory), since it has poles that are evenly spaced, and, was able to be manipulated to construct a consistent scattering amplitude, with poles, on straight line trajectories.
Leonhard Euler
Veneziano was attempting to make sense of some experimentally observered properties of the strong nuclear force. Veneziano, a researcher at CERN in Geneva, Switzerland had been working on this project for many years. Veneziano, stumbled upon a formula that was concocted for purely mathematical purposes by the prolific mathematician Leonhard Euler, some 200 years earlier. This was the Euler beta function, and, it appeared to Veneziano to be able to describe some properties of strongly interacting particles. There was then a subsequent intense period of research that revolved around using the Euler beta function to describe data that was being collected from atom smashers. However, nobody knew why the beta function worked for this purpose. It was a mystery. This is going to change in 1970...
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