Further Reading

2.9 Further Reading

Trinity College, Cambridge boasts many great scientific achievements. The discovery of Yang-Mills theory is not among the most celebrated.

Nonetheless, in January 1954 a graduate student at Trinity named Ronald Shaw wrote down what we now refer to as the Yang-Mills equations.

Aware that the theory describes massless particles, which appear to have no place in Nature, Shaw was convinced by his supervisor, Abdus Salam, that the result was not worth publishing. It appears only as a chapter of his thesis [181].

Across the Atlantic, in Brookhaven national laboratory, two office mates did not make the same mistake. C. N. Yang and Robert Mills constructed the equations which now bear their name [232].

It seems likely that that they got the result slightly before Shaw, although the paper only appeared afterwards.

Their original motivation now seems somewhat misguided: their paper suggests that global symmetries of quantum field theory – specifically SU (2) isospin – are not consistent with locality.

They write “It seems that this [global symmetry] is not consistent with the localized field concept that underlies the usual physical theories”

From this slightly shaky start, one of the great discoveries of 20th century physics emerged,

In those early days, the role played by Yang-Mills theory was, to say the least, confusing.

Yang gave a famous seminar in Princeton in which Pauli complained so vociferously about the existence of massless particles that Yang refused to go on with the talk and had to be coaxed back to the blackboard by Oppenheimer.

(Pauli had a headstart here: in 1953 he did a Kaluza-Klein reduction on S2 , realising an SU (2) gauge theory but discarding it because of the massless particle [151].

A similar result had been obtained earlier by Klein [122].)

It took a decade to realise that the gauge bosons could get a mass from the Higgs mechanism, and a further decade to realise that the massless particles were never really there anyway: they are an artefact of the classical theory and gain a mass automatically when ~ 6= 0.

Below is a broad brush description of this history. A collection of reminiscences, “50 Years of Yang-Mills” [108], contains articles by a number of the major characters in this story.

Asymptotic Freedom

As the 1970s began, quantum field theory was not in fashion.

Fundamental laws of physics, written in the language of field theory, languished in the literature, unloved and uncited [77, 205]. The cool kids were playing with bootstraps.

The discovery of asymptotic freedom was one of the first results that brought field theory firmly into the mainstream.

The discovery has its origins in the deep inelastic scattering experiments performed in SLAC in the late 1960s. Bjorken [19] and subsequently Feynman [56] realised that the experiments could be interpreted in terms of the momentum distribution of constituents of the proton.

But this interpretation held only if the interactions between these constituents became increasingly weak at high energies.

Feynman referred to the constituents as “partons” rather than “quarks” [57].

It is unclear whether this was because he wanted to allow for the possibility of other constituents, say gluons, or simply because he wanted to antagonise Gell-Mann.

In Princeton, David Gross set out to show that no field theory could exhibit asymp- totic freedom [86]. Having ruled out field theories based on scalars and fermions, all that was left was Yang-Mills. He attacked this problem with his new graduate student – 116 –Frank Wilczek. The minus signs took some getting right, but by April 1973 they re- alised that they had an asymptotically free theory on their hands [83] and were keenly aware of its importance. Meanwhile, in Harvard, Sidney Coleman was interested in the same problem. He asked his graduate student Erick Weinberg to do the calculation but, content that he had enough for his thesis, Erick passed it on to another graduate student, David Politzer.

Politzer finished his calculation at the same time as the Princeton team [156].

In 2004, Gross, Politzer and Wilczek were awarded the Nobel prize. Politzer’s Nobel lecture contains an interesting, and very human, account of the discovery [157].

In fact, both American teams had been scooped. In June 1972, at a conference in Marseilles, a Dutch graduate student named Gerard ’t Hooft sat in a talk by Symanzik on the SLAC experiments and their relation to asymptotic freedom.

After the talk, ’t Hooft announced that Yang-Mills theory is asymptotically free.

Symanzik encouraged him to publish this immediately but, like Shaw 20 years earlier, ’t Hooft decided against it.

His concern was that Yang-Mills theory could not be relevant for the strong force because it had no mechanism for the confinement of quarks [107].

The failure to publish did not hurt ’t Hooft’s career. By that stage he had already shown that Yang-Mills was renormalisable, a fact which played a large role in bringing the theory out of obscurity [93, 94, 95].

This was enough for him to be awarded his PhD [96].

It was also enough for him to be awarded the 1999 Nobel prize, together with his advisor Veltman. We will be seeing much more of the work of ’t Hooft later in these lectures.

The analogy between asymptotic freedom and paramagnetism was made by N. K.

Nielsen [148], although the author gives private credit to ’t Hooft. In these lectures, we computed the one-loop beta function using the background field method.

This method was apparently introduced by (of course) ’t Hooft in lectures which I haven’t managed to get hold of. It first appears in published form in a paper by Larry Abbott [1] (now a prominent theoretical neuroscientist) and is covered in the textbook by Peskin and Schroeder [154].