Goldstone boson
In physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu within the context of the BCS superconductivity mechanism, and subsequently elucidated by Jeffrey Goldstone, and systematically generalized in the context of quantum field theory. In condensed matter physics such bosons are quasiparticles and are known as Goldstone modes or Anderson–Bogoliubov modes.
These spinless bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the quantum numbers of these.
They transform nonlinearly 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 are massless if the spontaneously broken symmetry is not also broken explicitly.
If, instead, the symmetry is not exact, i.e. if it is explicitly broken as well as spontaneously broken, then the bosons that emerge are not massless, though they typically remain relatively light; they are called pseudo-Goldstone bosons or pseudo–Nambu–Goldstone bosons.
Goldstone's theorem
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 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. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding order parameter.By virtue of their special properties in coupling to the vacuum of the respective symmetry-broken theory, vanishing momentum Goldstone bosons involved in field-theoretic amplitudes make such amplitudes vanish.
Examples
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 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⊗U, which, however, are not observable. 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, 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.
- Ricciardi and Umezawa proposed in 1967 a general theory about the possible brain mechanism of memory storage and retrieval in terms of Nambu–Goldstone bosons. This theory was subsequently extended in 1995 by Giuseppe Vitiello taking into account that the brain is an "open" system. 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, and by E. Del Giudice, G. Preparata and G. Vitiello. Mari Jibu and Kunio Yasue and Giuseppe Vitiello, based on these findings, discussed the implications for consciousness.
Theory
and taking the limit as . This is called the "Abelian nonlinear σ-model".
The constraint, and the action, below, are invariant under a U phase transformation, . The field can be redefined to give a real scalar field without any constraint by
where is the Nambu–Goldstone boson and the U symmetry transformation effects a shift on , namely
but does not preserve the ground state , 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
and thus
Note that the constant term 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 current is
The charge, Q, resulting from this current shifts and the ground state to a new, degenerate, ground state. Thus, a vacuum with will shift to a different vacuum with. The current connects the original vacuum with the Nambu–Goldstone boson state,.
In general, in a theory with several scalar fields,, the Nambu–Goldstone mode is massless, and parameterises the curve of possible vacuum states. Its hallmark under the broken symmetry transformation is nonvanishing vacuum expectation, an order parameter, for vanishing, at some ground state |0〉 chosen at the minimum of the potential, . 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,, specify the relevant null eigenvectors of the mass matrix,
and hence the corresponding zero-mass eigenvalues.
Goldstone's argument
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,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. 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, are, nevertheless, time-invariant,
thus generating a in its Fourier transform.
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,, so that the theory cannot have a mass gap.
This argument is further clarified by taking the limit carefully. If an approximate charge operator acting in a huge but finite region is applied to the vacuum,
a state with approximately vanishing time derivative is produced,
Assuming a nonvanishing mass gap , the frequency of any state like the above, which is orthogonal to the vacuum, is at least ,
Letting become large leads to a contradiction. Consequently 0 = 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.
The argument requires both the vacuum and the charge to be translationally invariant,, .
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, — unless the symmetry is unbroken,. Consequently, does not properly exist in the Hilbert space.