They were discovered by Yoichiro Nambu in particle physics within the context of the BCS superconductivity mechanism,[1] and subsequently elucidated by Jeffrey Goldstone,[2] and systematically generalized in the context of quantum field theory.
[4][5][6] These spinless bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the quantum numbers of these.
Goldstone's theorem examines a generic continuous symmetry which is spontaneously broken; i.e., its currents are conserved, but the ground 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 the ground state.
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").
One way to impose a constraint of this sort is by including a potential interaction term in its Lagrangian density, and taking the limit as λ → ∞.
is) and the U(1) symmetry transformation effects a shift on θ, namely 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.
In general, in a theory with several scalar fields, ϕj, the Nambu–Goldstone mode ϕg is massless, and parameterises the curve of possible (degenerate) vacuum states.
Its hallmark under the broken symmetry transformation is nonvanishing vacuum expectation 〈δϕg〉, an order parameter, for vanishing 〈ϕg〉 = 0, at some ground state |0〉 chosen at the minimum of the potential, 〈∂V/∂ϕi〉 = 0.
Vanishing vacuum values of field symmetry transformation increments add no new information.
By contrast, however, nonvanishing vacuum expectations of transformation increments, 〈δϕg〉, specify the relevant (Goldstone) null eigenvectors of the mass matrix,
Normally, by current conservation, the charge operator for any symmetry current is time-independent, Acting with the charge operator on the vacuum either annihilates the vacuum, if that is symmetric; else, if not, as is the case in spontaneous symmetry breaking, it produces a zero-frequency state out of it, through its shift transformation feature illustrated above.
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 a mass gap.
If an approximate charge operator acting in a huge but finite region A is applied to the vacuum, a state with approximately vanishing time derivative is produced, Assuming a nonvanishing mass gap m0, the frequency of any state like the above, which is orthogonal to the vacuum, is at least m0, Letting A become large leads to a contradiction.
Consider the correlation function of the charge with itself, so the integrand in the right hand side does not depend on the position.
Take for example a chiral N = 1 super QCD model with a nonzero squark VEV which is conformal in the IR.
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
), the nonrelativistic version of the massless particles (i.e. photons where the dispersion relation is also
However, two different spontaneously broken generators may now give rise to the same Nambu–Goldstone boson.
and the expectation value of the ground state is zero, for the ferromagnet instead the dispersion is
[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] 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.
that spontaneously breaks a spacetime symmetry, the number of broken generators
[26] For internal symmetries, the above equation has no non-trivial solutions, so the usual Goldstone theorem holds.
In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken translation[27] symmetry.
[28][29] 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 characteristic nonlinear realizations of broken supersymmetry, so that goldstinos are superpartners of all particles in the theory, of any spin, and the only superpartners, at that.