Does Spontaneous Breakdown of Symmetry Imply Zero - Mass Particles ?

There is relatively intense interest at present in exploring more deeply and widely the suggestion' that the mathematical methods essential to the understanding of material media which exhibit long-range order (ferromagnets, superconductors, etc. ) may also be basic or useful for the theory of elementary particles. The original work had two connected aspects: the generation of fermion masses (by extension of the underlying group, mass differences) by the "spontaneous breakdown of symmetry, " and the consequent occurrence of collective boson excitations. Though the two kinds of consequences appear inextricably linked in the above work, further studies have tended to emphasize one or the other aspect. Illustrative of one kind of study has been the effort to "derive" the mass formula for the octet of fermions in the SU(3) symmetry model of strong interactions, '&' ignoring the possible occurrence —in the particular models employedof boson excitations. By analogy with the results of reference 1 the models in question would, however, appear to offer the embarrassment of nonexistent strongly interacting scalar bosons with zero mass. A second line of investigation dealing largely with this latter problem traces to Goldstone's conjecture that all broken-symmetry models thus far considered are indeed plagued with this difficulty. Goldstone, Salam, and Weinberg' gave two proofs of this conjecture whereas Bludman and Klein' have refined and generalized one of these proofs. In the latter work, it was emphasized, though perhaps insufficiently, that the proof given depended both on the class of models and on the prescribed method of calculation. The second proof of Goldstone, Salam, and Weinberg, however, appears to soar above such detailed considerations and may be summarized as follows: Lorentz invar iance + continuous internal symmetry group (represented by commutation relations between generators and operator representations as well as by conserved currents)+ spontaneous breakdown of symmetry (conditions of long-range order) implies massless bosons. If this proof is correct, it would seem to spell finis to many of the interesting possibilities for the application of the ideas of spontaneous breakdown of symmetry. y, (x) = 4,j'(x)4,t (x) + 4,(x)4,(x), cp, (x) = —i[4,t(x)4, (x) —4, (x)4,(x)],