Kaon Condensation and Goldstone's Theorem

We consider QCD at a nonzero chemical potential for strangeness. At a critical value of the chemical potential equal to the kaon mass, kaon condensation occurs through a continuous phase transition. We show that, when the isospin symmetry is exact in the Lagrangian, a Goldstone boson with a dispersion relation $E\sim p^2$ appears in the kaon condensed phase. At the same time, the number of the Goldstone bosons is less than the number of broken generators. Both phenomena are familiar in non-relativistic systems. We interpret our results in terms of a Goldstone boson counting rule found previously by Nielsen and Chadha. We also formulate a criterion sufficient for the equality between the number of Goldstone bosons and the number of broken generators.