Goldstone Theorem Preview

Statement

Goldstone theorem says, in its standard relativistic field-theory form, that spontaneously breaking a continuous global symmetry produces gapless excitations. In many-body language, the corresponding low-energy excitations are often called Nambu–Goldstone modes or Goldstone modes.

A compressed symmetry pattern is

G \longrightarrow H,

where the ground state preserves only a subgroup $H$ of the continuous symmetry group $G$ . The broken generators in $G/H$ produce low-energy collective directions in the space of ordered states.

This is a preview card because full Goldstone counting depends on whether the system is relativistic or nonrelativistic, whether charges commute in the ground state, and whether long-range gauge fields are present. The canonical nonrelativistic treatment is Goldstone Modes in Many-Body Systems.

Assumptions

The cleanest textbook statement assumes:

a continuous global symmetry;
spontaneous symmetry breaking in the thermodynamic limit;
locality and a stable ground state;
a conserved current or charge associated with the broken symmetry;
no gauge redundancy masquerading as a physical global symmetry;
no explicit symmetry-breaking term that gives the mode a mass or gap.

Finite systems do not literally choose one broken-symmetry ground state. They may have a symmetry-respecting exact ground state with nearly degenerate low-lying states. Spontaneous symmetry breaking is a thermodynamic-limit concept.

Quantum-Mechanical Meaning

A continuous symmetry gives a flat direction in the energy landscape when the ground state breaks it. Long-wavelength fluctuations along that direction cost little energy, producing low-energy collective modes.

For a physical orientation, see Goldstone Modes Preview. For the commutator-density counting theorem and many-body examples, see Goldstone Modes in Many-Body Systems.

Examples include:

phonons in crystals, associated with broken continuous translation symmetry;
spin waves in magnets with broken spin-rotation symmetry;
the phase mode of a neutral superfluid or Bose condensate;
pion-like modes in idealized chiral symmetry breaking.

The theorem should not be applied blindly to gauge symmetries. In superconductors and Higgs-like systems, gauge fields can absorb the would-be Goldstone mode, producing a different spectrum.

Counting Cautions

In relativistic systems under standard assumptions, the number of Goldstone bosons often equals the number of broken generators. In nonrelativistic many-body systems, this simple equality can fail. Some broken generators pair up, producing fewer modes with quadratic dispersion.

The robust reference-card lesson is:

\text{broken continuous global symmetry} \quad\Rightarrow\quad \text{gapless collective structure}

under the appropriate locality and thermodynamic-limit assumptions. The exact number and dispersion of modes require the more detailed theorem.

What It Does Not Say

Goldstone theorem does not say:

discrete symmetry breaking produces Goldstone modes;
finite systems must have exactly degenerate ground states;
gauge symmetries are broken in the same sense as global physical symmetries;
every gapless excitation is a Goldstone mode;
approximate symmetry produces exactly gapless modes.

If a symmetry is explicitly but weakly broken, the would-be Goldstone modes can become pseudo-Goldstone modes with a small gap.

Canonical Links

Common Mistakes

Applying Goldstone theorem to a broken discrete symmetry.
Forgetting the thermodynamic-limit nature of spontaneous symmetry breaking.
Counting nonrelativistic Goldstone modes by broken generators without checking commutators of charges.
Treating gauge redundancy as an ordinary global symmetry.
Ignoring explicit symmetry-breaking terms.
Calling any gapless excitation a Goldstone mode without identifying the broken continuous symmetry.

Quick Check

Why does breaking a discrete symmetry, such as a spin-flip symmetry, not by itself imply a Goldstone mode?

Solution

Goldstone modes arise from continuous families of symmetry-related ground states. A discrete symmetry has separated alternatives rather than a continuous flat direction. Domain walls or nearly degenerate states may appear, but the Goldstone theorem’s gapless-mode conclusion does not follow from discrete symmetry breaking alone.

References

J. Goldstone, “Field theories with superconductor solutions,” Il Nuovo Cimento 19, 154-164, 1961.
J. Goldstone, A. Salam, and S. Weinberg, “Broken symmetries,” Physical Review 127, 965-970, 1962.
Y. Nambu, “Quasi-particles and gauge invariance in the theory of superconductivity,” Physical Review 117, 648-663, 1960.
H. Watanabe and H. Murayama, “Unified description of Nambu–Goldstone bosons without Lorentz invariance,” Physical Review Letters 108, 251602, 2012.
S. Weinberg, The Quantum Theory of Fields, Volume II, Cambridge University Press, 1996.
P. Coleman, Introduction to Many-Body Physics, Cambridge University Press, 2015.