In
particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
and
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
that appear necessarily in models exhibiting
spontaneous breakdown of
continuous symmetries. They were discovered by
Yoichiro Nambu
was a Japanese-American physicist and professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded half of the Nobel Prize in Physics in 2008 for the discovery in 1960 of the mechanism ...
in
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
within the context of the
BCS superconductivity mechanism, and subsequently elucidated by
Jeffrey Goldstone
Jeffrey Goldstone (born 3 September 1933) is a British theoretical physicist and an ''emeritus'' physics faculty member at the MIT Center for Theoretical Physics.
He worked at the University of Cambridge until 1977. He is famous for the discove ...
, and systematically generalized in the context of
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
such bosons are
quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exam ...
s and are known as Anderson–Bogoliubov modes.
These
spinless bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the
quantum number
In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
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 bosons or pseudo-Nambu–Goldstone bosons (abbreviated PNGBs).
Goldstone's theorem
Goldstone's theorem examines a generic
continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant u ...
which is
spontaneously broken; i.e., its currents are conserved, but the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; 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. ...
is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact)
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
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 ground state of a quantum-mechanical system is its stationary state of lowest energy; 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
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
.
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").
Examples
Natural
*In
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s, the
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
is longitudinal and it is the Goldstone boson of the spontaneously broken
Galilean symmetry
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotati ...
. In
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
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
magnet
A 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, steel, nickel, ...
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 ''
magnon
A magnon is a quasiparticle, a collective excitation of the electrons' spin structure in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of e ...
s'', i.e., spin waves in which the local magnetization direction oscillates.
*The ''
pion
In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
s'' 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
In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2)⊗U(1), which, however, are not observable.
[In theories with gauge symmetry, the Goldstone bosons are absent. Their degrees of freedom are absorbed ("eaten", gauged out) by ]gauge boson
In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge ...
s, through the Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
. The latter become massive and their new, longitudinal polarization is provided by the would-be Goldstone boson, in an elaborate rearrangement of degrees of freedom . 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
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
through the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
. An analogous phenomenon occurs in
superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
, 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
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenol ...
.
Theory
Consider a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
, with the constraint that
, a constant. One way to impose a constraint of this sort is by including a
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
interaction term in its
Lagrangian density
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
,
:
and taking the limit as . This is called the "Abelian nonlinear σ-model".
[It corresponds to the Goldstone sombrero potential where the tip and the sides shoot to infinity, preserving the location of the minimum at its base.]
The constraint, and the action, below, are invariant under a ''U''(1) phase transformation, . The field can be redefined to give a real
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
(i.e., a spin-zero particle) without any constraint by
:
where is the Nambu–Goldstone boson (actually
is) and the ''U''(1) symmetry transformation effects a shift on , namely
:
but does not preserve the ground state (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
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
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''(1) 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 (degenerate) vacuum states. Its hallmark under the broken symmetry transformation is ''nonvanishing vacuum expectation'' , an
order parameter
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
, 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
The effective potential (also known as effective potential energy) combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the 'opposing' centrifugal potential energy with the potential energy of a ...
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 (Goldstone) ''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. (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, , so that the theory cannot have a
mass gap
In quantum field theory, the mass gap is the difference in energy between the lowest energy state, 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 ...
.
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 (see
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
).
:Fabri–Picasso Theorem. does not properly exist in the Hilbert space, unless .
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.
Infraparticles
There is an arguable loophole in the theorem. If one reads the theorem carefully, it only states that there exist non-
vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
s with arbitrarily small energies. Take for example a chiral
N = 1
super QCD model with a nonzero
squark VEV
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
which is
conformal in the
IR. The chiral symmetry is a
global symmetry
In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.
A family of particular transformations may be ''continuo ...
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
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
bosons have a continuous
mass spectrum
A mass spectrum is a histogram plot of intensity vs. ''mass-to-charge ratio'' (''m/z'') in a chemical sample, usually acquired using an instrument called a ''mass spectrometer''. Not all mass spectra of a given substance are the same; for example ...
with arbitrarily small masses but yet there is no Goldstone boson with exactly
zero mass. In other words, the Goldstone bosons are
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 ...
s.
Extensions
Nonrelativistic theories
A version of Goldstone's theorem also applies to
nonrelativistic
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
theories. It essentially states that, for each spontaneously broken symmetry, there corresponds some
quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exam ...
which is typically a boson and has no
energy gap
In solid-state physics, an energy gap is an energy range in a solid where no electron states exist, i.e. an energy range where the density of states vanishes.
Especially in condensed-matter physics, an energy gap is often known more abstractly as ...
. In condensed matter these goldstone bosons are also called gapless modes (i.e. states where the energy dispersion relation is like
and is zero for
), the nonrelativistic version of the massless particles (i.e. photons where the dispersion relation is also
and zero for
). Note that the energy in the non relativistic condensed matter case is and not 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 goldostone bosons, where in both cases we are breaking symmetry from SO(3) to SO(2), for the antiferromagnet the dispersion is
and the expectation value of the ground state is zero, for the ferromagnet instead the dispersion is
and the expectation value of the ground state is not zero, i.e. there is a spontaneously broken symmetry for the ground state
As a second example, in a
superfluid
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
, both the ''
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
'' particle number symmetry and
Galilean symmetry
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotati ...
are spontaneously broken. However, the
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
is the Goldstone boson for both.
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
Breaking of spacetime symmetries
In contrast to the case of the breaking of internal symmetries, when spacetime symmetries such as
Lorentz
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboar ...
, 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
that spontaneously breaks a spacetime symmetry, the number of broken generators
minus the number non-trivial independent solutions
to
:
is the number of Goldstone modes that arise. 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
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 symmetry.
Nambu–Goldstone fermions
Spontaneously broken global fermionic symmetries, which occur in some
supersymmetric
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
models, lead to Nambu–Goldstone
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
, or ''
goldstinos''.
These have spin , 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 realization In mathematical physics, nonlinear realization of a Lie group ''G'' possessing a Cartan subgroup ''H'' is a particular induced representation of ''G''. In fact, it is a representation of a Lie algebra \mathfrak g of ''G'' in a neighborhood of its ...
s of broken supersymmetry, so that goldstinos are superpartners of ''all'' particles in the theory, of ''any 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.
See also
*
Pseudo-Goldstone boson
*
Majoron
*
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
*
Mermin–Wagner theorem
In quantum field theory and statistical mechanics, the Mermin–Wagner theorem (also known as Mermin–Wagner–Hohenberg theorem, Mermin–Wagner–Berezinskii theorem, or Coleman theorem) states that continuous symmetries cannot be spontaneously ...
*
Vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
*
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
Notes
References
{{particles
Bosons
Quantum field theory
Mathematical physics
Physics theorems
Subatomic particles with spin 0