Nambu Goldstone TheoremEdit
The Nambu–Goldstone theorem is a central principle that links the symmetries of a system to its low-energy excitations. In broad terms, when a system with a continuous global symmetry develops a ground state that does not share that symmetry (spontaneous symmetry breaking), it necessarily hosts low-energy excitations known as Goldstone bosons. These massless modes reflect the directions in which the ground state can be transformed by the broken symmetry, and they dominate the long-wavelength physics of the broken phase. The theorem has wide reach, from the world of high-energy particle physics to the collective behavior of quantum condensed matter systems. For foundational discussions, see spontaneous symmetry breaking and the precise statement of the Nambu–Goldstone theorem.
In relativistic quantum field theories with exact Lorentz invariance, the correspondence is particularly clean: the number of Goldstone bosons equals the number of broken continuous symmetry generators. However, once one steps beyond strict Lorentz invariance—into condensed matter systems, finite-density physics, or curved spacetime—the counting becomes more subtle. The field has developed a refined understanding, with generalized counting rules and distinctions between different kinds of Goldstone modes. See, for example, the discussions surrounding the Nielsen–Chadha bound and the later refinement by Watanabe–Murayama theorem for nonrelativistic systems.
This article surveys the theorem’s core ideas, its standard examples, and the key extensions and debates. It also situates the theorem in the broader tradition of exploiting symmetry to organize physical phenomena, from the behavior of elementary particles to the collective modes of quantum fluids and magnets. For context on the mechanisms by which symmetries shape interactions, readers may consult the pages on global symmetry, continuous symmetry, and coset space.
Formal statement and foundational ideas
Symmetry and order parameters: A continuous global symmetry G is said to be spontaneously broken to a subgroup H when the ground state (or vacuum) is invariant under only the subgroup H, and not under the full group G. The space of degenerate ground states forms the vacuum manifold G/H, and motion on this manifold corresponds to the action of the broken generators.
Goldstone modes in the relativistic case: For each broken generator of G that does not annihilate the vacuum, a massless scalar excitation—the Goldstone boson—emerges. These modes typically have linear dispersion relations at low momentum, reflecting their origin as long-wavelength distortions of the order parameter.
Gauge theories and the Higgs mechanism: If the broken symmetry is gauged rather than global, the would-be Goldstone modes are not physical massless particles. Instead, they are absorbed as additional polarization degrees of freedom of gauge bosons, giving rise to massive vector bosons via the Higgs mechanism. This is a central mechanism in the explanation of electroweak symmetry breaking.
Nonrelativistic and finite-density extensions: In systems without Lorentz invariance, the relation between the number of broken generators and the number of Goldstone modes can be altered. The modern understanding distinguishes between different types of Goldstone modes (for example, type-A and type-B in some classifications) and provides refined counting rules. See Nielsen–Chadha bound and Watanabe–Murayama theorem for more details.
Examples across physics
Quantum chromodynamics and chiral symmetry breaking: In the theory of the strong interaction, the approximate global chiral symmetry SU(N)L × SU(N)R is spontaneously broken to the vector subgroup SU(N)V in the limit of small quark masses. This yields N^2 − 1 Goldstone bosons, identified with the light mesons (pions, kaons, eta in the SU(3) case). The quark masses explicitly break the symmetry and endow these would-be Goldstone bosons with small masses, making them pseudo-Goldstone bosons. See chiral symmetry and pions.
Magnons in magnets: A ferromagnet spontaneously breaks the rotational symmetry of spin space, leaving a residual U(1) symmetry intact. Although two generators are broken, there is typically a single gapless mode—the magnon—whose existence reflects the broken symmetry directions. In an antiferromagnet, two independent Goldstone modes emerge, corresponding to the two independent broken directions. These modes are observable as low-energy spin waves and illustrate how the same symmetry principles yield different spectra depending on the ground state structure. See ferromagnet and antiferromagnet.
Superfluids and condensed-matter fluids: A neutral superfluid or superconductor (in cases where the global U(1) is broken) hosts a Goldstone mode in the form of a phonon-like excitation at long wavelengths. The concrete realization depends on the system, but the common thread is that the phase of the order parameter embodies the broken U(1) degree of freedom. See superfluid and phonon.
Cosmology and early-universe phase transitions: Spontaneous breaking of symmetries in the early universe can produce Goldstone modes or related phenomena, influencing the dynamics of phase transitions and, in certain scenarios, the formation of topological defects. See cosmic phase transition and topological defect for related ideas.
The role of the topological and geometric structure of G/H: The manifold of broken-symmetry states, G/H, determines the spectrum and interactions of Goldstone modes. In many cases, the geometry and topology of this coset space imprint distinctive low-energy dynamics, a perspective developed in the context of effective field theories. See effective field theory and coset space.
Extensions, refinements, and subtle cases
Type-A and Type-B Goldstone modes: In systems lacking Lorentz invariance, broken generators may lead to Goldstone modes with different dispersion relations and counting rules. The refined frameworks distinguish modes that disperse linearly (type-A) from those with quadratic dispersion (type-B), and they provide a more accurate account of the spectrum in many condensed-matter contexts. See Nielsen–Chadha bound and Watanabe–Murayama theorem for details.
Pseudo-Goldstone bosons: When the symmetry is only approximate (due to small explicit breaking terms), the would-be massless modes acquire small masses. This is a familiar situation for the pions in QCD, where quark masses are not zero but small compared to typical hadronic scales. See pseudo-Goldstone boson.
Spontaneous breaking of spacetime symmetries: The idea that continuous spacetime symmetries can be spontaneously broken is subtle and subject to strong constraints. In many physical situations, the realizations involve collective modes tied to the broken spacetime symmetries, but the analysis differs from internal symmetry breaking and requires careful treatment of the underlying dynamics.
Finite temperature and explicit breakings: At nonzero temperature, long-range order can be disrupted, altering the presence and character of Goldstone modes. Explicit symmetry-breaking perturbations (external fields, lattice potentials, impurities) can gap or damp the would-be massless modes, modifying the spectrum in experimentally accessible ways.
Controversies and debates
Scope and universality: Some discussions stress that the Goldstone paradigm provides a universal organizing principle for low-energy physics across disparate domains. Others emphasize caution: the precise counting and the nature of the modes depend on the symmetries present, the dynamics, and whether the symmetry is global or gauged. This region of discussion is enriched by general theorems (like the Nielsen–Chadha bound and the Watanabe–Murayama counting) that go beyond the textbook statement.
Condensed-matter complications: In solids and fluids, the lattice, disorder, and finite-size effects can complicate a clean Goldstone interpretation. For instance, in crystals the notion of spontaneously broken continuous translation is subtle because the lattice explicitly breaks translation symmetry; the long-wavelength acoustic phonons behave like Goldstone modes of an approximate continuous translation symmetry. See phonon and crystal structure for related ideas.
Gauge symmetries and the hierarchy of explanations: The distinction between global and gauge symmetries remains crucial. The Nambu–Goldstone theorem applies to global symmetries; when a symmetry is gauged, the apparent Goldstone modes are absorbed, changing the spectrum in a way that can be counterintuitive if one conflates global and gauge degrees of freedom. See Higgs mechanism for the mechanism by which gauge bosons acquire mass.
Relation to naturalness and theory choice: In the broader practice of theoretical physics, symmetry considerations are a powerful heuristic. Critics sometimes argue that overreliance on symmetry aesthetics can lead to premature or overly constrained models, while proponents maintain that symmetry principles have historically yielded highly predictive and testable structures (e.g., in the development of the Standard Model). The debate is part of a larger conversation about methodology in fundamental physics, not unique to the Nambu–Goldstone framework.