Steven Weinberg, Nobel Laureate, 1979

I met Steve Weinberg in 1961, when he was a visiting fellow at Imperial College, and I had just been appointed to an assistant lectureship in Physics. He showed me his work on the proof of Goldstone's theorem in relativistic quantum field theory, and asked me my opinion. I did not know what was going on; I was puzzled why the symmetry was said to be broken, yet had a conserved current. My unsatisfactory answers to his questions about its rigour might have induced Weinberg to ask Salam about it. Salam sent it to Goldstone. I don't know if there were any changes to Steve's text as a result of this scrutiny by Salam and Goldstone, but a new preprint, headed by all three names, was then produced. This is the famous proof referred to in my paper.

It took a couple of years for me to catch the point of this work. First, I met Jan Tarski in 1963, who explained the ideas of new work by Haag and Kastler, which was then only in preprint form. This work proposed that it is better to regard the algebra of observables as a C*algebra rather than a W*algebra (a special case), as had been done up till then by Haag, and Araki. Then Jan explained, there are inequivalent representations, which can be labelled by superselection parameters. (There are also inequivalent representations of W*algebras, but these might not be weakly closed, and might require the axiom of choice for their construction). I then realised that automorphisms of C*algebras need not be spatial in a given representation, so a symmetry, defined as an automorphism commuting with the time-evolution, might not obey Wigner's criterion, that of preserving transition probabilities. My next awakening was caused by Gerald Guralnik, who was visiting Imperial College. His thesis was on spontaneous breakdown of Lorentz symmetry, which was noticed by Salam or Kibble as interesting (I did not yet see the point). To explain to me how it could happen in a simple theory, Gerry pointed out that a similar thing happens in the free boson of zero mass: the Lagrangian is invariant under translations of the field by a real constant c-number (the gauge transformations), but the vacuum is not invariant under this map. This led me to rewrite this well-known model in terms of C*algebras.

Segal could not have discovered this effect in his theory (he was the first, by far, to advocate the use of C*algebras) since he included in his C*algebra some algebraically-defined weak limits which were not local in the sense of Haag. The gauge automorphisms were defined on the local elements, and could not be extended to Segal's algebra. They could, however, be extended to the norm limits of the local algebra, now called the quasi-local algebra. Segal says that he wants to include the weak limits in order to include all operators that can be measured; unfortunately, their measurement involves the whole of space.

My idea, that a spontaneously broken internal symmetry is an automorphism, commuting with the time-evolution, which is not spatial in the given representation, is more convenient than the commonly expressed definition. There, the symmetry is said to be spontaneously broken if the ground state in not invariant under the symmetry. The latter concept does not distinguish whether there is vacuum degeneracy in the representation or not; if so, the various vacua might be related by a unitary operator realising the "broken" symmetry, which would be a good symmetry in Wigner's sense. However it still might be that the generators of the symmetry group are not conserved quantities. The old definition of a broken symmetry as occurring when the ground state is not invariant under the group is not useful in a sector (=representation) without a vacuum state. The sector of charge 1 has no translation-invariant states, but momentum is (usually) conserved in such a representation.

In modern terms, an anomaly occurs if the symmetry is given by a unitary group, but there is a non-trivial multiplier between the implementing unitary group and the dynamical group. See my paper on this subject.