Inphysics,Goldstone bosons orNambu–Goldstone bosons (NGBs) arebosons that appear necessarily in models exhibitingspontaneous breakdown ofcontinuous symmetries. They were discovered byYoichiro Nambu within the context of theBCS superconductivity mechanism,^[1] and subsequently elucidated byJeffrey Goldstone,^[2] and systematically generalized in the context ofquantum field theory.^[3] Incondensed matter physics such bosons arequasiparticles and are known as Goldstone modes^[4] or Anderson–Bogoliubov modes.^[5][6][7]

Thesespinless bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by thequantum numbers of these.They transform nonlinearly (shift) under the action of these generators, and can thus be excited out of the asymmetric vacuum by these generators. Thus, they can be thought of as the excitations of the field in the broken symmetry directions in group space—and aremassless if the spontaneously broken symmetry is not alsobroken explicitly.^[jargon]

If, instead, the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then thebosons that emerge are not massless, though they typically remain relatively light; they are calledpseudo-Goldstone bosons or pseudo–Nambu–Goldstone bosons.

Goldstone's theorem examines a genericcontinuous symmetry which isspontaneously broken; i.e., its currents are conserved, but theground state is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact)scalar particles appear in the spectrum of possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve theground state. The Nambu–Goldstone mode is a long-wavelength fluctuation of the correspondingorder parameter.

By virtue of their special properties in coupling to the vacuum of the respective symmetry-broken theory, vanishing momentum ("soft") Goldstone bosons involved in field-theoretic amplitudes make such amplitudes vanish ("Adler zeros").

• Influids, thephonon is longitudinal and it is the Goldstone boson of the spontaneously brokenGalilean symmetry. Insolids, 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.
• Inmagnets, 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 themagnons, i.e., spin waves in which the local magnetization direction oscillates.
• Thepions are thepseudo-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 issignificantly smaller than typical hadron masses.
• The longitudinal polarization components of theW 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 theStandard Model through theHiggs mechanism. An analogous phenomenon occurs insuperconductivity, 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. theGinzburg–Landau theory.
• Primordial fluctuations duringinflation can be viewed as Goldstone bosons arising due to the spontaneous symmetry breaking oftime translation symmetry of ade Sitter universe. These fluctuations in theinflatonscalar field subsequently seedcosmic structure formation.^[8]
• 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.^[9] 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).^[10] 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,^[11][12] and by E. Del Giudice, G. Preparata and G. Vitiello.^[13] Mari Jibu andKunio Yasue^[14] and Giuseppe Vitiello,^[15] based on these findings, discussed the implications for consciousness.

Consider acomplexscalar fieldϕ, with the constraint that${\displaystyle {\varphi }^{\ast }\varphi =v^2}$, a constant. One way to impose a constraint of this sort is by including apotential interaction term in itsLagrangian density,

${\displaystyle \lambda ({\varphi }^{\ast }\varphi -v^2{)}^2,}$

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

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

${\displaystyle \varphi =v{e}^{i\theta }}$

whereθ is the Nambu–Goldstone boson (actually${\displaystyle v\theta }$ is) and theU(1) symmetry transformation effects a shift onθ, namely

${\displaystyle \delta \theta =ϵ,}$

but does not preserve the ground state|0〉 (i.e. the above infinitesimal transformationdoes 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 correspondingLagrangian density is given by

${\displaystyle \mathcal{L}=\frac{1}{2}({\mathrm{\partial }}^{\mu }{\varphi }^{\ast }){\mathrm{\partial }}_{\mu }\varphi -m^2{\varphi }^{\ast }\varphi =\frac{1}{2}(-iv{e}^{-i\theta }{\mathrm{\partial }}^{\mu }\theta )(iv{e}^{i\theta }{\mathrm{\partial }}_{\mu }\theta )-m^2{v}^2,}$

and thus

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

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 thekinetic term for a massless scalar.

The symmetry-induced conservedU(1) current is

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

The charge,Q, resulting from this current shiftsθ and the ground state to a new, degenerate, ground state. Thus, a vacuum with〈θ〉 = 0 will shift to adifferent 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 modeϕ_g ismassless, and parameterises the curve of possible (degenerate) vacuum states. Its hallmark under the broken symmetry transformation isnonvanishing vacuum expectation〈δϕ_g〉, anorder 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 theeffective 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.

The principle behind Goldstone's argument is that the ground state is not unique. Normally, by current conservation, the charge operator for any symmetry current is time-independent,

${\displaystyle \frac{d}{dt}Q=\frac{d}{dt}{\int }_x{J}^0(x)=0.}$

Acting with the charge operator on the vacuum eitherannihilates the vacuum, if that is symmetric; else, ifnot, as is the case in spontaneous symmetry breaking, it produces a zero-frequency state out of it, through its shift transformation feature illustrated above. Actually, here, the charge itself is ill-defined, cf. the Fabri–Picasso argument below.

But its better behaved commutators with fields, that is, the nonvanishing transformation shifts〈δϕ_g〉, are, nevertheless,time-invariant,

${\displaystyle \frac{d⟨\delta {\varphi }_g⟩}{dt}=0,}$

thus generating aδ(k⁰) in its Fourier transform.^[16] (This ensures that, inserting a complete set of intermediate states in a nonvanishing current commutator can lead to vanishing time-evolution only when one or more of these states is massless.)

Thus, if the vacuum is not invariant under the symmetry, action of the charge operator produces a state which is different from the vacuum chosen, but which has zero frequency. This is a long-wavelength oscillation of a field which is nearly stationary: there are physical states with zero frequency,k⁰, so that the theory cannot have amass gap.

This argument is further clarified by taking the limit carefully. If an approximate charge operator acting in a huge but finite regionA is applied to the vacuum,

${\displaystyle \frac{d}{dt}{Q}_A=\frac{d}{dt}{\int }_x{e}^{-\frac{x^2}{2A^2}}{J}^0(x)=-{\int }_x{e}^{-\frac{x^2}{2A^2}}\mathrm{\nabla }\cdot J={\int }_x\mathrm{\nabla }\left(e^{-\frac{x^2}{2A^2}}\right)\cdot J,}$

a state with approximately vanishing time derivative is produced,

${\displaystyle \Vert \frac{d}{dt}{Q}_A|0⟩\Vert \approx \frac{1}{A}\Vert Q_A|0⟩\Vert .}$

Assuming a nonvanishing mass gapm₀, the frequency of any state like the above, which is orthogonal to the vacuum, is at leastm₀,

${\displaystyle \Vert \frac{d}{dt}|\theta ⟩\Vert =\Vert H|\theta ⟩\Vert \ge m_0\Vert |\theta ⟩\Vert .}$

LettingA become large leads to a contradiction. Consequentlym₀ = 0. However this argument fails when the symmetry is gauged, because then the symmetry generator is only performing a gauge transformation. A gauge transformed state is the same exact state, so that acting with a symmetry generator does not get one out of the vacuum (seeHiggs mechanism).

Fabri–Picasso Theorem.Q does not properly exist in the Hilbert space, unlessQ|0〉 = 0.

The argument^[17][18] requires both the vacuum and the chargeQ to be translationally invariant,P|0〉 = 0,[P,Q]= 0.

Consider the correlation function of the charge with itself,

so the integrand in the right hand side does not depend on the position.

Thus, its value is proportional to the total space volume,${\displaystyle \Vert Q|0⟩{\Vert }^2=\mathrm{\infty }}$ — unless the symmetry is unbroken,Q|0〉 = 0. Consequently,Q does not properly exist in the Hilbert space.

There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-vacuum states with arbitrarily small energies. Take for example a chiralN = 1super QCD model with a nonzerosquarkVEV which isconformal in theIR. The chiral symmetry is aglobal symmetry which is (partially) spontaneously broken. Some of the "Goldstone bosons" associated with this spontaneous symmetry breaking are charged under the unbroken gauge group and hence, thesecomposite bosons have a continuousmass spectrum with arbitrarily small masses but yet there is no Goldstone boson with exactlyzero mass. In other words, the Goldstone bosons areinfraparticles.

A version of Goldstone's theorem also applies tononrelativistic theories.^[19][20] It essentially states that, for each spontaneously broken symmetry, there corresponds somequasiparticle which is typically a boson and has noenergy 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 isH−μN−α→⋅P→ and notH as it would be in a relativistic case. However, twodifferent spontaneously broken generators may now give rise to thesame 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^[21][22]

As a second example, in asuperfluid, both theU(1) particle number symmetry andGalilean symmetry are spontaneously broken. However, thephonon is the Goldstone boson for both.^[23][24]

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^[25][26]

In contrast to the case of the breaking of internal symmetries, when spacetime symmetries such asLorentz, 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${\displaystyle ⟨\varphi (\mathit{r})⟩}$ that spontaneously breaks a spacetime symmetry, the number of broken generators${\displaystyle T^a}$ minus the number non-trivial independent solutions${\displaystyle c_a(\mathit{r})}$ to

${\displaystyle c_a(\mathit{r})T^a⟨\varphi (\mathit{r})⟩=0}$

is the number of Goldstone modes that arise.^[27] 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${\displaystyle c_a(\mathit{r})}$ 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^[28] symmetry.

Spontaneously broken global fermionic symmetries, which occur in somesupersymmetric models, lead to Nambu–Goldstonefermions, orgoldstinos.^[29][30] These have spin⁠ 1 / 2⁠, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously.

Spontaneous supersymmetry breaking smashes up ("reduces") supermultiplet structures into the characteristicnonlinear realizations of broken supersymmetry, so that goldstinos are superpartners ofall particles in the theory, ofany spin, and the only superpartners, at that. That is, to say, two non-goldstino particles are connected to only goldstinos through supersymmetry transformations, and not to each other, even if they were so connected before the breaking of supersymmetry. As a result, the masses and spin multiplicities of such particles are then arbitrary.

• Pseudo-Goldstone boson
• Majoron
• Higgs mechanism
• Mermin–Wagner theorem
• Vacuum expectation value
• Noether's theorem

• Bosons
• Quantum field theory
• Mathematical physics
• Physics theorems
• Subatomic particles with spin 0

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