Goldstone Bosons & Zero Modes — Symmetry Breaking
Massless modes emerge when continuous symmetry is spontaneously broken
Physics: The Mexican hat (wine bottle) potential V(φ) = μ²|φ|² + λ|φ|⁴ with μ² < 0 has continuous U(1) symmetry φ → e^{iα}φ. The vacuum manifold is a circle |φ| = v = √(−μ²/2λ). Goldstone's theorem: for each broken continuous symmetry generator, there is a massless boson (zero mode = flat direction along the vacuum circle). The radial excitation (Higgs) has mass² = −2μ² > 0. Adding explicit breaking ε·Re(φ) gaps the Goldstone to a pseudo-Goldstone (pion analogy). Temperature fluctuations can restore symmetry when T > T_c.