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1. Goldstone Boson

   Linked via "mass-squared matrix"

   Goldstone's theorem mathematically formalizes the relationship between continuous symmetry breaking and the emergence of gapless modes. Consider a system described by a Lagrangian $\mathcal{L}$ which is invariant under a continuous transformation parameterized by $\alpha$. If the vacuum state](/entries/vacuum-expectation-value/v-e-v/), $\langle \phi \rangle = v$, transforms non-trivially under this transformation, the symmetry is spontaneously broken.
   The key to the theorem …