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John Harnad
professor of mathematics and statistics at Concordia University in Montréal

A huge breakthrough came in 1970-71 that changed the nature of theoretical elementary particle physics in its treatment of the ‘weak’ and ‘strong’ interactions from little more than a phenomenological treatment of relations between data, to one where the nuclear weak interactions could be calculated with as much consistency and accuracy as quantum electrodynamics. It began as a speculative model, which embodied the known features of the weak and electromagnetic interactions, but was not yet a viable quantum field theory, like quantum electrodynamic.

The main obstacle to doing ‘higher order’ calculations, was ‘renormalisability’, an essential part of relativistic quantum field theory. On the basis of reasonable notions (gauge fields, similar to the electromagnetic field, providing the mediating forces, together with ‘spontaneous symmetry breaking’ to make these short range), this model was conjectured to work, but the renormalisability was not demonstrated until 1970-71, when Gerard ‘t Hooft, then a graduate student in Utrecht, with his research supervisor Martinus Veltman, proved it, carrying out a number of basic computations that previously had never have been possible in the theory of weak interactions. This was a huge breakthrough, bringing theoretical high-energy physics back onto a consistent track, like what had been developed in earlier decades for quantum electrodynamics.

Almost immediately, people looked for a similar explanation of the ‘strong interactions’, the other type of short-range nuclear forces. But there were two obstacles: the very fact that the forces were so strong implied there was no reasonable way to use the standard successive approximation scheme (‘perturbation theory’) that worked successfully in the weak and electromagnetic case. And the mechanism ‘spontaneous symmetry breaking’ for making the forces short-range did not apply in this case.

The first obstacle was impressively overcome, theoretically, through a proof of the notion of ‘asymptotic freedom’ which said, essentially, that these interactions, within the ‘standard model’, diminished enormously in strength at sufficiently small scales of distance (or, equivalently, at high enough scales of energy). This predication preceded experimental evidence however, and it took several decades of further collection of high-energy data before it could be confirmed.

The second problem, of how to explain the short-range nature of strong interactions remains only understood at a ‘phenomenological’ level, so there is room for future work. Since this development increased the number of fundamental forces of nature that we can treat consistently within the framework of relativistic quantum field theory from one to three (out of a known total of four such forces), it could reasonably be rated as the greatest advance in the field of elementary particle physics of the last half-century.

John Harnad is professor of mathematics and statistics at Concordia University in Montréal, director of the Mathematical Physics Laboratory of the Centre de Recherches Mathematiques at the Université de Montréal, a national research centre in mathematics and its applications.