Goldstone theorem

Goldstone’s theorem states that spontaneously breaking a continuous global symmetry produces massless modes, called Goldstone bosons, associated with the broken symmetry generators.

Spontaneous symmetry breaking: A system chooses a state that lacks a symmetry obeyed by its governing laws. The theorem begins when the vacuum or ground state breaks a continuous symmetry.
Pion: A light meson that appears as a pseudo-Goldstone boson of approximate chiral symmetry breaking in quantum chromodynamics. Its small but nonzero mass illustrates how explicit symmetry breaking modifies the ideal prediction.
Anderson–Higgs mechanism: The absorption of would-be Goldstone modes into massive gauge-field excitations in a gauge theory or superconductor. It contrasts the theorem’s massless modes for global symmetries with the outcome of gauging a symmetry.
Yoichiro Nambu: A Japanese-American theoretical physicist known for work on spontaneous symmetry breaking and superconductivity. His 1960 analysis of superconductivity helped identify the massless-mode problem in particle physics.
Vacuum expectation value: The expectation value of a field operator in the vacuum state. A nonzero value can mark the symmetry-breaking vacuum from which Goldstone modes arise.
Phonon: A quantized collective vibration of a crystal lattice. Long-wavelength acoustic phonons are Goldstone modes associated with broken continuous translation symmetry.
Explicit symmetry breaking: A direct change to a system’s laws or parameters that removes a symmetry. Unlike spontaneous breaking, it generally gives the associated modes nonzero masses.
Jeffrey Goldstone: A British theoretical physicist who proved a massless-particle result for spontaneously broken continuous symmetries. His 1961 paper gave the theorem its standard name.
Broken symmetry generator: A symmetry generator whose transformation changes the chosen vacuum state. Each independent broken generator corresponds to a Goldstone mode under the theorem’s standard conditions.
Spin wave: A propagating disturbance in the ordered spin configuration of a magnetic material. In a magnet with continuous spin-rotation symmetry, spin waves can realize Goldstone modes.

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