Broken Symmetries and Massless Particles

Abstract

The following generalization of a theorem conjectured by Goldstone is proven: In a theory admitting a continuous group of transformations, suppose a set of operators ${\ensuremath{\varphi}}_{i}(x)$, transforming under an irreducible representation of the group, has the property that in the vacuum some expectation values $〈{\ensuremath{\varphi}}_{i}(x)〉\ensuremath{\ne}0$ for $i={i}^{\ensuremath{'}}$. The theorem then asserts that ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{D}}_{\mathrm{ij}}(p)$, the Fourier transform of the propagator of ${\ensuremath{\varphi}}_{i}(x)$, is singular at ${p}^{2}=0$ for some $i\ensuremath{\ne}{i}^{\ensuremath{'}}$. (The maximum number of $〈{\ensuremath{\varphi}}_{i}〉\ensuremath{\ne}0$ is a property of the group representation. The further identification of the singularities as poles and their interpretation as massless particles depends on the usual apparatus of quantum field theory.)The appropriate choice to be made for the field ${\ensuremath{\varphi}}_{i}$ when it describes a boson excitation and when the Lagrangian contains only direct fermion-fermion coupling is discussed. It is suggested that such Fermi interaction theories may be renormalizable when expanded in terms of the coupling between fermions and the collective boson field.The theorem is illustrated by the following models: (A) ${\ensuremath{\gamma}}_{5}$ gauge group (Nambu and Jona-Lasinio), where a massless pseudoscalar meson is predicted; (B) isospin group (Nambu and Jona-Lasinio) where massless charged mesons are predicted; (C) $\mathrm{SU}(3)$ octet model (Baker and Glashow) where six or four massless mesons are predicted; (D) Lorentz group (Bjorken) where the massless photon is predicted. The limitations of the theorem are also discussed.

Broken Symmetries and Massless Particles

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Abstract

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