Goldstone boson
Encyclopedia
In particle
and condensed matter physics
, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries
(cf. spontaneously broken symmetry). They were discovered by Yoichiro Nambu
in the context of the BCS superconductivity
mechanism, and subsequently elucidated by Jeffrey Goldstone
, and systematically generalized in the context of quantum field theory
.
These spinless
bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the quantum number
s 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 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 Nambu–Goldstone bosons are not massless, though they typically remain relatively light; they are then called pseudo-Goldstone boson
s or pseudo-Nambu–Goldstone bosons (abbreviated PNGBs).
which is spontaneously broken
; i.e., its currents are conserved, but the ground state (vacuum) 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
. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding order parameter.
There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-vacuum state
s with arbitrarily small energies. Take for example a chiral N = 1 super QCD
model with a nonzero squark
VEV
which is conformal
in the IR
. The chiral symmetry is a global 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, these composite bosons have a continuous mass spectrum
with arbitrarily small masses but yet there is no Goldstone boson with exactly zero mass. In other words, the Goldstone bosons are infraparticle
s.
In theories with gauge symmetry, the Goldstone bosons are "eaten" by the gauge boson
s. The latter become massive and their new, longitudinal polarization is provided by the Goldstone boson.
scalar field
φ, with the constraint that φ*φ = k². One way to get a constraint of this sort is by including a potential
and taking the limit as λ → ∞ (this is called the Abelian nonlinear σ-model).
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
where θ is the Nambu–Goldstone boson (actually kθ 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.
The vacuum is degenerate and noninvariant under the spontaneously broken symmetry.
The corresponding Lagrangian density is given by
and the symmetry-induced conserved U(1) current is
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 state, ⟨0|J0(0)|θ⟩ ≠ 0.
Note that the constant term m²k² in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.
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.
This nonvanishing vacuum expectation of the transformation increment, ⟨δφg⟩, specifies the relevant (Goldstone) null eigenvector of the mass matrix,
and hence the corresponding zero-mass eigenvalue.
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; but its better
behaved commutators with fields, so, then, the transformation shifts, are still time-invariant, d⟨δφg⟩/dt = 0 .)
So, 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 states with zero frequency, so that the theory cannot have a mass gap
.
This argument is clarified by taking the limit carefully. If an approximate charge operator is applied to the vacuum,
a state with approximately vanishing time derivative is produced,
.
Assuming a mass gap
, the frequency of any state like the above, which is orthogonal to the vacuum, is at least 
,
Letting A become large leads to a contradiction.
This argument fails, however, 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.
It essentially states that, for each spontaneously broken symmetry, there corresponds some quasiparticle
with no energy gap—the nonrelativistic version of the mass gap
. (Note that the energy here is really
and not 
.) However, two different spontaneously broken generators may now give rise to the same Nambu–Goldstone boson. For 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.
In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken Galilean
/Lorentz symmetry. However, in contrast to the case of internal symmetry breaking, when spacetime symmetries are broken, the order parameter need not be a scalar field, but may be a tensor field, and the corresponding independent massless modes may now be fewer than the number of spontaneously broken generators, because the
Goldstone modes may now be linearly dependent among themselves: e.g., the Goldstone modes for some generators might be expressed as gradients of Goldstone modes for other broken generators.
models, lead to Nambu–Goldstone fermions, or goldstinos. These have spin ½, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously.
(Vestigial bosonic superpartners of these goldstinos under supersymmetries, called sgoldstinos, might sometimes also appear. Strictly speaking, however, spontaneous symmetry, and supersymmetry, breaking smashes up ("reduces") multiplet and supermultiplet structures, into the characteristic nonlinear realizations of broken supersymmetry, so that, in a technical
sense, goldstinos are superpartners of all particles in the theory, of any spin.)
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
and condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries
Continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...
(cf. spontaneously broken symmetry). They were discovered by Yoichiro Nambu
Yoichiro Nambu
is a Japanese-born American physicist, currently a professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded a one-half share of the Nobel Prize in Physics in 2008 for the discovery of the mechanism of spontaneous broken symmetry in...
in the context of the BCS superconductivity
BCS theory
BCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...
mechanism, and subsequently elucidated by Jeffrey Goldstone
Jeffrey Goldstone
Jeffrey Goldstone is a British-born theoretical physicist and an emeritus physics faculty at MIT Center for Theoretical Physics.He worked at the University of Cambridge until 1977....
, and systematically generalized in the context of quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.
These spinless
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the quantum number
Quantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
s 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 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 Nambu–Goldstone bosons are not massless, though they typically remain relatively light; they are then called pseudo-Goldstone boson
Pseudo-Goldstone boson
Pseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken , and would thus engender massless Goldstone bosons. The additional explicit symmetry breaking...
s or pseudo-Nambu–Goldstone bosons (abbreviated PNGBs).
Goldstone's theorem
Goldstone's theorem examines a generic continuous symmetrySymmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
which is spontaneously broken
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
; i.e., its currents are conserved, but the ground state (vacuum) is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact) scalar
Scalar field theory
In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...
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
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding order parameter.
There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-vacuum state
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
s with arbitrarily small energies. Take for example a chiral N = 1 super QCD
Super QCD
In theoretical physics, super QCD is a supersymmetric gauge theory which resembles quantum chromodynamics but contains additional particles and interactions which render it supersymmetric....
model with a nonzero squark
Sfermion
In particle physics, a sfermion is the spin-0 superpartner particle of its associated fermion. In supersymmetric extensions to the Standard Model each particle has a superpartner with spin that differs by . Fermions in the SM have spin- and therefore sfermions have spin 0.In general the name...
VEV
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
which is conformal
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
in the IR
Infrared
Infrared light is electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 micrometres , and extending conventionally to 300 µm...
. The chiral symmetry is a global symmetry
Global symmetry
A global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.Global symmetries require conservation laws, but not forces, in physics.-See also:...
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, these composite bosons have a continuous mass spectrum
Mass spectrum
A mass spectrum is an intensity vs. m/z plot representing a chemical analysis. Hence, the mass spectrum of a sample is a pattern representing the distribution of ions by mass in a sample. It is a histogram usually acquired using an instrument called a mass spectrometer...
with arbitrarily small masses but yet there is no Goldstone boson with exactly zero mass. In other words, the Goldstone bosons are infraparticle
Infraparticle
An infraparticle is an electrically charged particle and its surrounding cloud of soft photons—of which there are infinite number, by virtue of the infrared divergence of quantum electrodynamics. That is, it is a dressed particle rather than a bare particle...
s.
In theories with gauge symmetry, the Goldstone bosons are "eaten" by the gauge boson
Gauge boson
In particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...
s. The latter become massive and their new, longitudinal polarization is provided by the Goldstone boson.
A simple example
Consider a complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
scalar field
Scalar field theory
In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...
φ, with the constraint that φ*φ = k². One way to get a constraint of this sort is by including a potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...
and taking the limit as λ → ∞ (this is called the Abelian nonlinear σ-model).
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
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
(i.e., a spin-zero particle) θ without any constraint by
where θ is the Nambu–Goldstone boson (actually kθ 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.
The vacuum is degenerate and noninvariant under the spontaneously broken symmetry.
The corresponding Lagrangian density is given by
and the symmetry-induced conserved U(1) current is
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 state, ⟨0|J0(0)|θ⟩ ≠ 0.
Note that the constant term m²k² in the Lagrangian density has no physical significance, and the other term in it is simply the kinetic term for a massless scalar.
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.
This nonvanishing vacuum expectation of the transformation increment, ⟨δφg⟩, specifies the relevant (Goldstone) null eigenvector of the mass matrix,
and hence the corresponding zero-mass eigenvalue.
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 istime-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; but its better
behaved commutators with fields, so, then, the transformation shifts, are still time-invariant, d⟨δφg⟩/dt = 0 .)
So, 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 states with zero frequency, so that the theory cannot have a mass gap
Mass gap
In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest...
.
This argument is clarified by taking the limit carefully. If an approximate charge operator is applied to the vacuum,
a state with approximately vanishing time derivative is produced,
.Assuming a mass gap
, the frequency of any state like the above, which is orthogonal to the vacuum, is at least ,Letting A become large leads to a contradiction.
This argument fails, however, 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.
Nonrelativistic theories
A version of Goldstone's theorem also applies to nonrelativistic theories (and also relativistic theories with spontaneously broken spacetime symmetries, such as Lorentz symmetry or conformal symmetry, rotational, or translational invariance).It essentially states that, for each spontaneously broken symmetry, there corresponds some quasiparticle
Quasiparticle
In physics, quasiparticles are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in free space...
with no energy gap—the nonrelativistic version of the mass gap
Mass gap
In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest...
. (Note that the energy here is really
and not .) However, two different spontaneously broken generators may now give rise to the same Nambu–Goldstone boson. For example, in a superfluidSuperfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...
, both the U(1) particle number symmetry and Galilean symmetry are spontaneously broken. However, the phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
is the Goldstone boson for both.
In general, the phonon is effectively the Nambu–Goldstone boson for spontaneously broken Galilean
Galilean
Generically, a Galilean is an inhabitant of Galilee. Galileans were also the members of a fanatical sect , followers of Judas of Galilee, who fiercely resented the taxation of the Romans, and whose violence contributed to induce the latter to vow the extermination of the whole race...
/Lorentz symmetry. However, in contrast to the case of internal symmetry breaking, when spacetime symmetries are broken, the order parameter need not be a scalar field, but may be a tensor field, and the corresponding independent massless modes may now be fewer than the number of spontaneously broken generators, because the
Goldstone modes may now be linearly dependent among themselves: e.g., the Goldstone modes for some generators might be expressed as gradients of Goldstone modes for other broken generators.
Nambu–Goldstone fermions
Spontaneously broken global fermionic symmetries, which occur in some supersymmetricSupersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
models, lead to Nambu–Goldstone fermions, or goldstinos. These have spin ½, instead of 0, and carry all quantum numbers of the respective supersymmetry generators broken spontaneously.
(Vestigial bosonic superpartners of these goldstinos under supersymmetries, called sgoldstinos, might sometimes also appear. Strictly speaking, however, spontaneous symmetry, and supersymmetry, breaking smashes up ("reduces") multiplet and supermultiplet structures, into the characteristic nonlinear realizations of broken supersymmetry, so that, in a technical
sense, goldstinos are superpartners of all particles in the theory, of any spin.)
Goldstone bosons in nature
- In fluidFluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
s, the phononPhononIn physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
is longitudinal and it is the Goldstone boson of the spontaneously broken Galilean symmetry. In solidSolidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s, 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 magnetMagnetA magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
s, the original rotational symmetry (present in the absence of an external magnetic field) is spontaneously broken such that the magnetization points into a specific direction. The Goldstone bosons then are the magnonMagnonA magnon is a collective excitation of the electrons' spin structure in a crystal lattice. In contrast, a phonon is a collective excitation of the crystal lattice atoms or ions. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. As a...
s, i.e., spin waves in which the local magnetization direction oscillates. - The pionPionIn particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
s are the pseudo-Goldstone bosonPseudo-Goldstone bosonPseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken , and would thus engender massless Goldstone bosons. The additional explicit symmetry breaking...
s 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 bosonsW and Z bosonsThe W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...
correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2)⊗U(1), which, however, are not observable. Because this symmetry is gauged, the three would-be Goldstone bosons are "eaten" 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 ModelStandard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
through the Higgs mechanismHiggs mechanismIn particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
.
See also
- Pseudo-Goldstone bosonPseudo-Goldstone bosonPseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken , and would thus engender massless Goldstone bosons. The additional explicit symmetry breaking...
- MajoronMajoronIn particle physics, majorons are a hypothetical type of Goldstone boson that are theorized to mediate the neutrino mass violation of lepton number or B − L in certain high energy collisions such as...
- Higgs mechanismHiggs mechanismIn particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
- Mermin-Wagner theoremMermin-Wagner theoremIn quantum field theory and statistical mechanics, the Mermin–Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2...