Natural

• In fluids, the phonon is longitudinal and it is the Goldstone boson of the spontaneously broken Galilean symmetry. In solids, the situation is more complicated; the Goldstone bosons are the longitudinal and transverse phonons and they happen to be the Goldstone bosons of spontaneously broken Galilean, translational, and rotational symmetry with no simple one-to-one correspondence between the Goldstone modes and the broken symmetries.
• In magnets, the original rotational symmetry (present in the absence of an external magnetic field) is spontaneously broken such that the magnetization points in a specific direction. The Goldstone bosons then are the magnons, i.e., spin waves in which the local magnetization direction oscillates.
• The pions are the pseudo-Goldstone bosons that result from the spontaneous breakdown of the chiral-flavor symmetries of QCD effected by quark condensation due to the strong interaction. These symmetries are further explicitly broken by the masses of the quarks so that the pions are not massless, but their mass is significantly smaller than typical hadron masses.
• The longitudinal polarization components of the W and Z bosons correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2)⊗U(1), which, however, are not observable.^{[nb 1]} Because this symmetry is gauged, the three would-be Goldstone bosons are absorbed by the three gauge bosons corresponding to the three broken generators; this gives these three gauge bosons a mass and the associated necessary third polarization degree of freedom. This is described in the Standard Model through the Higgs mechanism. An analogous phenomenon occurs in superconductivity, which served as the original source of inspiration for Nambu, namely, the photon develops a dynamical mass (expressed as magnetic flux exclusion from a superconductor), cf. the Ginzburg–Landau theory.
• Primordial fluctuations during inflation can be viewed as Goldstone bosons arising due to the spontaneous symmetry breaking of time translation symmetry of a de Sitter universe. These fluctuations in the inflaton scalar field subsequently seed cosmic structure formation.^[7]
• Ricciardi and Umezawa proposed in 1967 a general theory (quantum brain) about the possible brain mechanism of memory storage and retrieval in terms of Nambu–Goldstone bosons.^[8] This theory was subsequently extended in 1995 by Giuseppe Vitiello taking into account that the brain is an "open" system (the dissipative quantum model of the brain).^[9] Applications of spontaneous symmetry breaking and of Goldstone's theorem to biological systems, in general, have been published by E. Del Giudice, S. Doglia, M. Milani, and G. Vitiello,^[10][11] and by E. Del Giudice, G. Preparata and G. Vitiello.^[12] Mari Jibu and Kunio Yasue^[13] and Giuseppe Vitiello,^[14] based on these findings, discussed the implications for consciousness.

Theory

Consider a complex scalar field ϕ, with the constraint that ${\displaystyle \phi ^{*}\phi =v^{2}}$, a constant. One way to impose a constraint of this sort is by including a potential interaction term in its Lagrangian density,

${\displaystyle \lambda (\phi ^{*}\phi -v^{2})^{2}~,}$

and taking the limit as λ → ∞. This is called the "Abelian nonlinear σ-model".^{[nb 2]}

The constraint, and the action, below, are invariant under a U(1) phase transformation, δϕ=iεϕ. The field can be redefined to give a real scalar field (i.e., a spin-zero particle) θ without any constraint by

${\displaystyle \phi =ve^{i\theta }}$

where θ is the Nambu–Goldstone boson (actually ${\displaystyle v\theta }$ is) and the U(1) symmetry transformation effects a shift on θ, namely

${\displaystyle \delta \theta =\epsilon ~,}$

but does not preserve the ground state |0〉 (i.e. the above infinitesimal transformation does not annihilate it—the hallmark of invariance), as evident in the charge of the current below.

Thus, the vacuum is degenerate and noninvariant under the action of the spontaneously broken symmetry.

The corresponding Lagrangian density is given by

${\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial ^{\mu }\phi ^{*})\partial _{\mu }\phi -m^{2}\phi ^{*}\phi ={\frac {1}{2}}(-ive^{-i\theta }\partial ^{\mu }\theta )(ive^{i\theta }\partial _{\mu }\theta )-m^{2}v^{2},}$

and thus

${\displaystyle ={\frac {v^{2}}{2}}(\partial ^{\mu }\theta )(\partial _{\mu }\theta )-m^{2}v^{2}~.}$

Note that the constant term ${\displaystyle m^{2}v^{2}}$ in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.

The symmetry-induced conserved U(1) current is

${\displaystyle J_{\mu }=v^{2}\partial _{\mu }\theta ~.}$

The charge, Q, resulting from this current shifts θ and the ground state to a new, degenerate, ground state. Thus, a vacuum with 〈θ〉 = 0 will shift to a different vacuum with 〈θ〉 = ε. The current connects the original vacuum with the Nambu–Goldstone boson state, 〈0|J₀(0)|θ〉≠ 0.

In general, in a theory with several scalar fields, ϕ_j, the Nambu–Goldstone mode ϕ_g is massless, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is nonvanishing vacuum expectation 〈δϕ_g〉, an order parameter, for vanishing 〈ϕ_g〉 = 0, at some ground state |0〉 chosen at the minimum of the potential, 〈∂V/∂ϕ_i〉 = 0. In principle the vacuum should be the minimum of the effective potential which takes into account quantum effects, however it is equal to the classical potential to first approximation. Symmetry dictates that all variations of the potential with respect to the fields in all symmetry directions vanish. The vacuum value of the first order variation in any direction vanishes as just seen; while the vacuum value of the second order variation must also vanish, as follows. Vanishing vacuum values of field symmetry transformation increments add no new information.

By contrast, however, nonvanishing vacuum expectations of transformation increments, 〈δϕ_g〉, specify the relevant (Goldstone) null eigenvectors of the mass matrix,

and hence the corresponding zero-mass eigenvalues.

Nonrelativistic theories

A version of Goldstone's theorem also applies to nonrelativistic theories.^[18][19] It essentially states that, for each spontaneously broken symmetry, there corresponds some quasiparticle which is typically a boson and has no energy gap. In condensed matter these goldstone bosons are also called gapless modes (i.e. states where the energy dispersion relation is like ${\displaystyle E\propto p^{n}}$ and is zero for ${\displaystyle p=0}$), the nonrelativistic version of the massless particles (i.e. photons where the dispersion relation is also ${\displaystyle E=pc}$ and zero for ${\displaystyle p=0}$). Note that the energy in the non relativistic condensed matter case is H−μN−α→⋅P→ and not H as it would be in a relativistic case. However, two different spontaneously broken generators may now give rise to the same Nambu–Goldstone boson.

As a first example an antiferromagnet has 2 goldstone bosons, a ferromagnet has 1 goldstone bosons, where in both cases we are breaking symmetry from SO(3) to SO(2), for the antiferromagnet the dispersion is ${\displaystyle E\propto p}$ and the expectation value of the ground state is zero, for the ferromagnet instead the dispersion is ${\displaystyle E\propto p^{2}}$ and the expectation value of the ground state is not zero, i.e. there is a spontaneously broken symmetry for the ground state ^[20][21]

As a second example, in a superfluid, both the U(1) particle number symmetry and Galilean symmetry are spontaneously broken. However, the phonon is the Goldstone boson for both.^[22][23]

Still in regards to symmetry breaking there is also a close analogy between gapless modes in condensed matter and the Higgs boson, e.g. in the paramagnet to ferromagnet phase transition^[24][25]

Breaking of spacetime symmetries

In contrast to the case of the breaking of internal symmetries, when spacetime symmetries such as Lorentz, conformal, rotational, or translational symmetries are broken, the order parameter need not be a scalar field, but may be a tensor field, and the number of independent massless modes may be fewer than the number of spontaneously broken generators. For a theory with an order parameter ${\displaystyle \langle \phi ({\boldsymbol {r}})\rangle }$ that spontaneously breaks a spacetime symmetry, the number of broken generators ${\displaystyle T^{a}}$ minus the number non-trivial independent solutions ${\displaystyle c_{a}({\boldsymbol {r}})}$ to

${\displaystyle c_{a}({\boldsymbol {r}})T^{a}\langle \phi ({\boldsymbol {r}})\rangle =0}$

is the number of Goldstone modes that arise.^[26] For internal symmetries, the above equation has no non-trivial solutions, so the usual Goldstone theorem holds. When solutions do exist, this is because the Goldstone modes are linearly dependent among themselves, in that the resulting mode can be expressed as a gradients of another mode. Since the spacetime dependence of the solutions ${\displaystyle c_{a}({\boldsymbol {r}})}$ is in the direction of the unbroken generators, when all translation generators are broken, no non-trivial solutions exist and the number of Goldstone modes is once again exactly the number of broken generators.

In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken translation^[27] symmetry.