Generalised Goldstone Theorem

by R. F. Streater; Phys. Rev. Lett., 15, 475-476, 1965.

ABSTRACT

We prove that if a local conserved current in quantum field theory generates a transformation between fields corresponding to particles of different mass, then there must be states in the theory of arbitrarily small mass (in the absence of states of negative norm)

Note. This work was made rigorous, using the axiomatic definition (see my paper ) in my lecture at the Endicott House conference ( q. v.)

This paper received a bad review by C. R. Hagen in Mathematical Reviews; he made the point that the paper was not more general in its result than the results claimed by physicists earlier, and was in fact less general, in that the case of mass-splitting is less general than these results (he might have been thinking of the paper (*) quoted here of Goldstone, Salam and Weinberg). In this, the reviewer does not realise that this famous work (*) (and the original thesis of Goldstone, and that of Nambu, and the work (**) of Higgs) was not done by mathematical physicists, and was not considered rigorous by us. It is not so much that any mathematical mistake is made, but that assumptions are made along the way which are not stated at the beginning, but are "physically" true in any good model. One such is the discarding, in (*) and (**), of time-dependent terms at infinity. Surely, all physical terms must go to zero at infinity? Maybe, maybe not. The proof of Goldstone's theorem was therefore considered to be open by mathematical physicists at this time. One point made in my paper is that the canonical transformation between fields of different mass is not implemented by unitary transformations. To prove this, we need to apply the Hall-Wightman theorem in a version of Haag's theorem. At the time of this paper (1964), it was assumed in all the physical proofs of Goldstone's theorem that canonical transformations were given by unitary operators. If they happened not to commute with the Hamiltonian, there was the the possibility of different masses. This I would have called an "explicit" breaking of isospin symmetry. In a spontaneous breaking, the symmetry is assumed to commute with the time-evolution algebraically. Hagen seems to be saying that the non-implementability of the canonical transformations between the fields of different mass was known to previous authors. In fact, the first general statement of this, apart from the present paper, is in: E. Fabri and L. E. Picasso, Phys. Rev. Lett., 16, 408-410, 1966.

Hagen also makes the point that the zero-mass field predicted by the theorem might be decoupled from all the rest, and so the absence of any scalar particle of zero mass does not rule out a spontaneously broken symmetry, contrary to my claim. This remark was well known to us (Higgs made the same point in 1964), but was not the subject of this PRL. I discuss this point in the Endicott House version of the paper in some detail.