Goldstone Theorem & SSB

Goldstone's theorem (1961): When a continuous global symmetry G is spontaneously broken to H, there are dim(G/H) massless Goldstone bosons. For U(1) → {1}: 1 Goldstone (phonon, magnon, pion for approximate). The Mexican hat potential V(φ) = μ²|φ|² + λ|φ|⁴ has a ring of vacua at |φ|² = -μ²/2λ when μ²<0. Radial fluctuations → massive Higgs mode; angular fluctuations → massless Goldstone (flat direction = zero mode). In gauge theories (Higgs mechanism), Goldstone bosons are "eaten" by gauge bosons which become massive. Examples: pions (≈ Goldstone of broken chiral SU(2)×SU(2) → SU(2)), spin waves (magnons), phonons.