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 ${\phi }_i\mathrm{}\left(x\right)\mathrm{}$, transforming under an irreducible representation of the group, has the property that in the vacuum some expectation values $〈{\phi }_i\mathrm{}\left(x\right)\mathrm{}〉\ne 0$ for $i=i^{\prime }$. The theorem then asserts that ${\tilde{D}}_{\mathrm{ij}}\mathrm{}\left(p\right)\mathrm{}$, the Fourier transform of the propagator of ${\phi }_i\mathrm{}\left(x\right)\mathrm{}$, is singular at $p^2=0\mathrm{}$ for some $i\ne i^{\prime }$. (The maximum number of $〈{\phi }_i〉\ne 0$ 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.)