Abstract
A generalization of Goldstone's theorem is presented which is valid for theories either with or without relativistic invariance. The central suggestion is that, under certain specified assumptions, all Goldstone bosons can be divided into two classes, termed type I and type II, in accordance with the behaviour of their dispersion laws. A Goldstone boson is a member of either the first or the second class according as its energy, in the limit of long wavelengths, is proportional to an odd or an even power of its momentum, respectively. The major result then is that, if each Goldstone boson of type I is counted once and that of type II is counted twice, the total number of “bosons” so obtained is always equal to or greater than the number of symmetry generators that are spontaneously broken. An immediate corollary is the familiar result that for relativistically invariant theories the number of Goldstone bosons can never be less than the number of spontaneously broken generators. Throughout the proof of the above result particular emphasis is placed on theories which are not Lorentz invariant.
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