QCD vacuum

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/QCD_vacuum

The QCD vacuum is the quantum vacuum state of quantum chromodynamics (QCD). It is an example of a non-perturbative vacuum state, characterized by non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter.

Unsolved problem in physics:

QCD in the non-perturbative regime: confinement. The equations of QCD remain unsolved at energy scales relevant for describing atomic nuclei. How does QCD give rise to the physics of nuclei and nuclear constituents?

Symmetries and symmetry breaking

Symmetries of the QCD Lagrangian

Like any relativistic quantum field theory, QCD enjoys Poincaré symmetry including the discrete symmetries CPT (each of which is realized). Apart from these space-time symmetries, it also has internal symmetries. Since QCD is an SU(3) gauge theory, it has local SU(3) gauge symmetry.

Since it has many flavours of quarks, it has approximate flavour and chiral symmetry. This approximation is said to involve the chiral limit of QCD. Of these chiral symmetries, the baryon number symmetry is exact. Some of the broken symmetries include the axial U(1) symmetry of the flavour group. This is broken by the chiral anomaly. The presence of instantons implied by this anomaly also breaks CP symmetry.

In summary, the QCD Lagrangian has the following symmetries:

• Poincaré symmetry and CPT invariance
• SU(3) local gauge symmetry
• approximate global SU(N_f) × SU(N_f) flavour chiral symmetry and the U(1) baryon number symmetry

The following classical symmetries are broken in the QCD Lagrangian:

• scale, i.e., conformal symmetry (through the scale anomaly), giving rise to asymptotic freedom
• the axial part of the U(1) flavour chiral symmetry (through the chiral anomaly), giving rise to the strong CP problem.

Spontaneous symmetry breaking

Main articles: Spontaneous symmetry breaking and Symmetry breaking

When the Hamiltonian of a system (or the Lagrangian) has a certain symmetry, but the vacuum does not, then one says that spontaneous symmetry breaking (SSB) has taken place.

A familiar example of SSB is in ferromagnetic materials. Microscopically, the material consists of atoms with a non-vanishing spin, each of which acts like a tiny bar magnet, i.e., a magnetic dipole. The Hamiltonian of the material, describing the interaction of neighbouring dipoles, is invariant under rotations. At high temperature, there is no magnetization of a large sample of the material. Then one says that the symmetry of the Hamiltonian is realized by the system. However, at low temperature, there could be an overall magnetization. This magnetization has a preferred direction, since one can tell the north magnetic pole of the sample from the south magnetic pole. In this case, there is spontaneous symmetry breaking of the rotational symmetry of the Hamiltonian.

When a continuous symmetry is spontaneously broken, massless bosons appear, corresponding to the remaining symmetry. This is called the Goldstone phenomenon and the bosons are called Goldstone bosons.

Symmetries of the QCD vacuum

The SU(N_f) × SU(N_f) chiral flavour symmetry of the QCD Lagrangian is broken in the vacuum state of the theory. The symmetry of the vacuum state is the diagonal SU(N_f) part of the chiral group. The diagnostic for this is the formation of a non-vanishing chiral condensate ⟨ψ_iψ_i⟩, where ψ_i is the quark field operator, and the flavour index i is summed. The Goldstone bosons of the symmetry breaking are the pseudoscalar mesons.

When N_f = 2, i.e., only the up and down quarks are treated as massless, the three pions are the Goldstone bosons. When the strange quark is also treated as massless, i.e., N_f = 3, all eight pseudoscalar mesons of the quark model become Goldstone bosons. The actual masses of these mesons are obtained in chiral perturbation theory through an expansion in the (small) actual masses of the quarks.

In other phases of quark matter the full chiral flavour symmetry may be recovered, or broken in completely different ways.

Experimental evidence

The evidence for QCD condensates comes from two eras, the pre-QCD era 1950–1973 and the post-QCD era, after 1974. The pre-QCD results established that the strong interactions vacuum contains a quark chiral condensate, while the post-QCD results established that the vacuum also contains a gluon condensate.

Motivating results

Gradient coupling

In the 1950s, there were many attempts to produce a field theory to describe the interactions of pions (${\displaystyle \pi }$) and nucleons (${\displaystyle N}$). The obvious renormalizable interaction between the two objects is the Yukawa coupling to a pseudoscalar:

${\displaystyle L_I=\overline{N}{\gamma }_5\pi N}$

And this is theoretically correct, since it is leading order and it takes all the symmetries into account. But it doesn't match experiment in isolation. When the nonrelativistic limit of this coupling is taken, the gradient-coupling model is obtained. To lowest order, the nonrelativistic pion field interacts by derivatives.^[1] This is not obvious in the relativistic form.^[2] A gradient interaction has a very different dependence on the energy of the pion—it vanishes at zero momentum.

This type of coupling means that a coherent state of low momentum pions barely interacts at all. This is a manifestation of an approximate symmetry, a shift symmetry of the pion field. The replacement

${\displaystyle \pi \to \pi +C}$

leaves the gradient coupling alone, but not the pseudoscalar coupling, at least not by itself. The way nature fixes this in the pseudoscalar model is by simultaneous rotation of the proton-neutron and shift of the pion field. This, when the proper axial SU(2) symmetry is included, is the Gell-Mann Levy σ-model, discussed below.

The modern explanation for the shift symmetry is now understood to be the Nambu-Goldstone non-linear symmetry realization mode, due to Yoichiro Nambu and Jeffrey Goldstone. The pion field is a Goldstone boson, while the shift symmetry is a manifestation of a degenerate vacuum.

Goldberger–Treiman relation

There is a surprising relationship between the strong interaction coupling of the pions to the nucleons, the coefficient in the nucleon-pion-gradient coupling model, and the axial vector current coefficient of the nucleon, which determines the weak decay rate of the neutron. The relation is

${\displaystyle g_{\pi NN}{F}_{\pi }=G_A{M}_N}$

and it is obeyed to 2.5% accuracy.

The constant G_A is the coefficient that determines the neutron decay rate: It gives the normalization of the weak interaction matrix elements for the nucleon. On the other hand, the pion-nucleon coupling is a phenomenological constant describing the (strong) scattering of bound states of quarks and gluons. The weak interactions are current-current interactions ultimately because they come from a non-Abelian gauge theory. The Goldberger–Treiman relation suggests that the pions, by dint of chiral symmetry breaking, interact as surrogates of sorts of the axial weak currents.

Partially conserved axial current

The structure which gives rise to the Goldberger–Treiman relation was called the partially conserved axial current (PCAC) hypothesis, spelled out in the pioneering σ-model paper. Partially conserved describes the modification of a spontaneously-broken symmetry current by an explicit breaking correction preventing its conservation. The axial current in question is also often called the chiral symmetry current.

The basic idea of SSB is that the symmetry current which performs axial rotations on the fundamental fields does not preserve the vacuum: This means that the current J applied to the vacuum produces particles. The particles must be spinless, otherwise the vacuum wouldn't be Lorentz invariant. By index matching, the matrix element must be

${\displaystyle J_{\mu }|0⟩=k_{\mu }|\pi ⟩,}$

where k_μ is the momentum carried by the created pion.

When the divergence of the axial current operator is zero, we must have

${\displaystyle {\mathrm{\partial }}_{\mu }{J}^{\mu }|0⟩=k^{\mu }{k}_{\mu }|\pi ⟩=m_{\pi }^2|\pi ⟩=0.}$

Hence these pions are massless, m²
_π = 0, in accordance with Goldstone's theorem.

If the scattering matrix element is considered, we have

${\displaystyle k_{\mu }⟨N(p)|\pi (k)|N(p^{\prime })⟩=⟨N(p)|J_{\mu }|N(p^{\prime })⟩.}$

Up to a momentum factor, which is the gradient in the coupling, it takes the same form as the axial current turning a neutron into a proton in the current-current form of the weak interaction.

${\displaystyle ⟨N|J^{\mu }|N⟩⟨e|J_{\mu }|\nu ⟩.}$

But if a small explicit breaking of the chiral symmetry (due to quark masses) is introduced, as in real life, the above divergence does not vanish, and the right hand side involves the mass of the pion, now a Pseudo-Goldstone boson.

Soft pion emission

Extensions of the PCAC ideas allowed Steven Weinberg to calculate the amplitudes for collisions which emit low energy pions from the amplitude for the same process with no pions. The amplitudes are those given by acting with symmetry currents on the external particles of the collision.

These successes established the basic properties of the strong interaction vacuum well before QCD.

Pseudo-Goldstone bosons

Experimentally it is seen that the masses of the octet of pseudoscalar mesons is very much lighter than the next lightest states; i.e., the octet of vector mesons (such as the rho meson). The most convincing evidence for SSB of the chiral flavour symmetry of QCD is the appearance of these pseudo-Goldstone bosons. These would have been strictly massless in the chiral limit. There is convincing demonstration that the observed masses are compatible with chiral perturbation theory. The internal consistency of this argument is further checked by lattice QCD computations which allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark mass is as required by chiral perturbation theory.

Eta prime meson

This pattern of SSB solves one of the earlier "mysteries" of the quark model, where all the pseudoscalar mesons should have been of nearly the same mass. Since N_f = 3, there should have been nine of these. However, one (the SU(3) singlet η′ meson) has quite a larger mass than the SU(3) octet. In the quark model, this has no natural explanation – a mystery named the η−η′ mass splitting (the η is one member of the octet, which should have been degenerate in mass with the η′).

In QCD, one realizes that the η′ is associated with the axial U_A(1) which is explicitly broken through the chiral anomaly, and thus its mass is not "protected" to be small, like that of the η. The η–η′ mass splitting can be explained through the 't Hooft instanton mechanism, whose 1/N realization is also known as Witten–Veneziano mechanism.

Current algebra and QCD sum rules

PCAC and current algebra also provide evidence for this pattern of SSB. Direct estimates of the chiral condensate also come from such analysis.

Another method of analysis of correlation functions in QCD is through an operator product expansion (OPE). This writes the vacuum expectation value of a non-local operator as a sum over VEVs of local operators, i.e., condensates. The value of the correlation function then dictates the values of the condensates. Analysis of many separate correlation functions gives consistent results for several condensates, including the gluon condensate, the quark condensate, and many mixed and higher order condensates. In particular one obtains

${\displaystyle \begin{array}{rl}⟨(gG{)}^2⟩\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}⟨g^2{G}_{\mu \nu }{G}^{\mu \nu }⟩& \approx 0.5{\text{GeV}}^4\\ ⟨\overline{\psi }\psi ⟩& \approx (-0.23{)}^3{\text{GeV}}^3\\ ⟨(gG{)}^4⟩& \approx 5:10{⟨(gG{)}^2⟩}^2\end{array}}$

Here G refers to the gluon field tensor, ψ to the quark field, and g to the QCD coupling.

These analyses are being refined further through improved sum rule estimates and direct estimates in lattice QCD. They provide the raw data which must be explained by models of the QCD vacuum.

Models

A full solution of QCD should give a full description of the vacuum, confinement and the hadron spectrum. Lattice QCD is making rapid progress towards providing the solution as a systematically improvable numerical computation. However, approximate models of the QCD vacuum remain useful in more restricted domains. The purpose of these models is to make quantitative sense of some set of condensates and hadron properties such as masses and form factors.

This section is devoted to models. Opposed to these are systematically improvable computational procedures such as large N QCD and lattice QCD, which are described in their own articles.

The Savvidy vacuum, instabilities and structure

The Savvidy vacuum is a model of the QCD vacuum which at a basic level is a statement that it cannot be the conventional Fock vacuum empty of particles and fields. In 1977, George Savvidy showed that the QCD vacuum with zero field strength is unstable, and decays into a state with a calculable non vanishing value of the field. Since condensates are scalar, it seems like a good first approximation that the vacuum contains some non-zero but homogeneous field which gives rise to these condensates. However, Stanley Mandelstam showed that a homogeneous vacuum field is also unstable. The instability of a homogeneous gluon field was argued by Niels Kjær Nielsen and Poul Olesen in their 1978 paper. These arguments suggest that the scalar condensates are an effective long-distance description of the vacuum, and at short distances, below the QCD scale, the vacuum may have structure.

The dual superconducting model

In a type II superconductor, electric charges condense into Cooper pairs. As a result, magnetic flux is squeezed into tubes. In the dual superconductor picture of the QCD vacuum, chromomagnetic monopoles condense into dual Cooper pairs, causing chromoelectric flux to be squeezed into tubes. As a result, confinement and the string picture of hadrons follows. This dual superconductor picture is due to Gerard 't Hooft and Stanley Mandelstam. 't Hooft showed further that an Abelian projection of a non-Abelian gauge theory contains magnetic monopoles.

While the vortices in a type II superconductor are neatly arranged into a hexagonal or occasionally square lattice, as is reviewed in Olesen's 1980 seminar one may expect a much more complicated and possibly dynamical structure in QCD. For example, nonabelian Abrikosov-Nielsen-Olesen vortices may vibrate wildly or be knotted.

String models

String models of confinement and hadrons have a long history. They were first invented to explain certain aspects of crossing symmetry in the scattering of two mesons. They were also found to be useful in the description of certain properties of the Regge trajectory of the hadrons. These early developments took on a life of their own called the dual resonance model (later renamed string theory). However, even after the development of QCD string models continued to play a role in the physics of strong interactions. These models are called non-fundamental strings or QCD strings, since they should be derived from QCD, as they are, in certain approximations such as the strong coupling limit of lattice QCD.

The model states that the colour electric flux between a quark and an antiquark collapses into a string, rather than spreading out into a Coulomb field as the normal electric flux does. This string also obeys a different force law. It behaves as if the string had constant tension, so that separating out the ends (quarks) would give a potential energy increasing linearly with the separation. When the energy is higher than that of a meson, the string breaks and the two new ends become a quark-antiquark pair, thus describing the creation of a meson. Thus confinement is incorporated naturally into the model.

In the form of the Lund model Monte Carlo program, this picture has had remarkable success in explaining experimental data collected in electron-electron and hadron-hadron collisions.

Bag models

Strictly, these models are not models of the QCD vacuum, but of physical single particle quantum states — the hadrons. The model proposed originally in 1974 by A. Chodos et al. consists of inserting a quark model in a perturbative vacuum inside a volume of space called a bag. Outside this bag is the real QCD vacuum, whose effect is taken into account through the difference between energy density of the true QCD vacuum and the perturbative vacuum (bag constant B) and boundary conditions imposed on the quark wave functions and the gluon field. The hadron spectrum is obtained by solving the Dirac equation for quarks and the Yang–Mills equations for gluons. The wave functions of the quarks satisfy the boundary conditions of a fermion in an infinitely deep potential well of scalar type with respect to the Lorentz group. The boundary conditions for the gluon field are those of the dual color superconductor. The role of such a superconductor is attributed to the physical vacuum of QCD. Bag models strictly prohibit the existence of open color (free quarks, free gluons, etc.) and lead in particular to string models of hadrons.

The chiral bag model couples the axial vector current ψγ₅γ_μψ of the quarks at the bag boundary to a pionic field outside of the bag. In the most common formulation, the chiral bag model basically replaces the interior of the skyrmion with the bag of quarks. Very curiously, most physical properties of the nucleon become mostly insensitive to the bag radius. Prototypically, the baryon number of the chiral bag remains an integer, independent of bag radius: the exterior baryon number is identified with the topological winding number density of the Skyrme soliton, while the interior baryon number consists of the valence quarks (totaling to one) plus the spectral asymmetry of the quark eigenstates in the bag. The spectral asymmetry is just the vacuum expectation value ⟨ψγ₀ψ⟩ summed over all of the quark eigenstates in the bag. Other values, such as the total mass and the axial coupling constant g_A, are not precisely invariant like the baryon number, but are mostly insensitive to the bag radius, as long as the bag radius is kept below the nucleon diameter. Because the quarks are treated as free quarks inside the bag, the radius-independence in a sense validates the idea of asymptotic freedom.

Instanton ensemble

Main article: instanton fluid

Another view states that BPST-like instantons play an important role in the vacuum structure of QCD. These instantons were discovered in 1975 by Alexander Belavin, Alexander Markovich Polyakov, Albert S. Schwarz and Yu. S. Tyupkin as topologically stable solutions to the Yang-Mills field equations. They represent tunneling transitions from one vacuum state to another. These instantons are indeed found in lattice calculations. The first computations performed with instantons used the dilute gas approximation. The results obtained did not solve the infrared problem of QCD, making many physicists turn away from instanton physics. Later, though, an instanton liquid model was proposed, turning out to be more promising an approach.

The dilute instanton gas model departs from the supposition that the QCD vacuum consists of a gas of BPST-like instantons. Although only the solutions with one or few instantons (or anti-instantons) are known exactly, a dilute gas of instantons and anti-instantons can be approximated by considering a superposition of one-instanton solutions at great distances from one another. Gerard 't Hooft calculated the effective action for such an ensemble, and he found an infrared divergence for big instantons, meaning that an infinite amount of infinitely big instantons would populate the vacuum.

Later, an instanton liquid model was studied. This model starts from the assumption that an ensemble of instantons cannot be described by a mere sum of separate instantons. Various models have been proposed, introducing interactions between instantons or using variational methods (like the "valley approximation") endeavoring to approximate the exact multi-instanton solution as closely as possible. Many phenomenological successes have been reached. Whether an instanton liquid can explain confinement in 3+1 dimensional QCD is not known, but many physicists think that it is unlikely.

Center vortex picture

A more recent picture of the QCD vacuum is one in which center vortices play an important role. These vortices are topological defects carrying a center element as charge. These vortices are usually studied using lattice simulations, and it has been found that the behavior of the vortices is closely linked with the confinement–deconfinement phase transition: in the confinement phase vortices percolate and fill the spacetime volume, in the deconfinement phase they are much suppressed. Also it has been shown that the string tension vanished upon removal of center vortices from the simulations, hinting at an important role for center vortices.

Gluon field strength tensor

From Wikipedia, the free encyclopedia

 https://en.wikipedia.org/wiki/Gluon_field_strength_tensor

 

Further information: Ricci calculus, Special unitary group, and Quantum chromodynamics

 

Quantum field theory
Feynman diagram

In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.

The strong interaction is one of the fundamental interactions of nature, and the quantum field theory (QFT) to describe it is called quantum chromodynamics (QCD). Quarks interact with each other by the strong force due to their color charge, mediated by gluons. Gluons themselves possess color charge and can mutually interact.

The gluon field strength tensor is a rank 2 tensor field on the spacetime with values in the adjoint bundle of the chromodynamical SU(3) gauge group (see vector bundle for necessary definitions).

Convention

Throughout this article, Latin indices (typically a, b, c, n) take values 1, 2, ..., 8 for the eight gluon color charges, while Greek indices (typically α, β, μ, ν) take values 0 for timelike components and 1, 2, 3 for spacelike components of four-vectors and four-dimensional spacetime tensors. In all equations, the summation convention is used on all color and tensor indices, unless the text explicitly states that there is no sum to be taken (e.g. “no sum”).

Definition

Below the definitions (and most of the notation) follow K. Yagi, T. Hatsuda, Y. Miake and Greiner, Schäfer.

Tensor components

The tensor is denoted G, (or F, F, or some variant), and has components defined proportional to the commutator of the quark covariant derivative D_μ:

${\displaystyle G_{\alpha \beta }=\pm \frac{1}{i{g}_{\text{s}}}[D_{\alpha },D_{\beta }],}$

where:

${\displaystyle D_{\mu }={\mathrm{\partial }}_{\mu }\pm i{g}_{\text{s}}{t}_a{\mathcal{A}}_{\mu }^a,}$

in which

• i is the imaginary unit;
• g_s is the coupling constant of the strong force;
• t_a = λ_a/2 are the Gell-Mann matrices λ_a divided by 2;
• a is a color index in the adjoint representation of SU(3) which take values 1, 2, ..., 8 for the eight generators of the group, namely the Gell-Mann matrices;
• μ is a spacetime index, 0 for timelike components and 1, 2, 3 for spacelike components;
• ${\displaystyle {\mathcal{A}}_{\mu }=t_a{\mathcal{A}}_{\mu }^a}$ expresses the gluon field, a spin-1 gauge field or, in differential-geometric parlance, a connection in the SU(3) principal bundle;
• ${\displaystyle {\mathcal{A}}_{\mu }}$ are its four (coordinate-system dependent) components, that in a fixed gauge are 3×3 traceless Hermitian matrix-valued functions, while ${\displaystyle {\mathcal{A}}_{\mu }^a}$ are 32 real-valued functions, the four components for each of the eight four-vector fields.

Different authors choose different signs.

Expanding the commutator gives;

${\displaystyle G_{\alpha \beta }={\mathrm{\partial }}_{\alpha }{\mathcal{A}}_{\beta }-{\mathrm{\partial }}_{\beta }{\mathcal{A}}_{\alpha }\pm i{g}_{\text{s}}[{\mathcal{A}}_{\alpha },{\mathcal{A}}_{\beta }]}$

Substituting ${\displaystyle t_a{\mathcal{A}}_{\alpha }^a={\mathcal{A}}_{\alpha }}$ and using the commutation relation ${\displaystyle [t_a,t_b]=i{f}_{ab}{}^c{t}_c}$ for the Gell-Mann matrices (with a relabeling of indices), in which f ^abc are the structure constants of SU(3), each of the gluon field strength components can be expressed as a linear combination of the Gell-Mann matrices as follows:

${\displaystyle \begin{array}{rl}{G}_{\alpha \beta }& ={\mathrm{\partial }}_{\alpha }{t}_a{\mathcal{A}}_{\beta }^a-{\mathrm{\partial }}_{\beta }{t}_a{\mathcal{A}}_{\alpha }^a\pm i{g}_{\text{s}}\left[t_b,t_c\right]{\mathcal{A}}_{\alpha }^b{\mathcal{A}}_{\beta }^c\\ & =t_a\left({\mathrm{\partial }}_{\alpha }{\mathcal{A}}_{\beta }^a-{\mathrm{\partial }}_{\beta }{\mathcal{A}}_{\alpha }^a\pm i^2{f}_{bc}{}^a{g}_{\text{s}}{\mathcal{A}}_{\alpha }^b{\mathcal{A}}_{\beta }^c\right)\\ & =t_a{G}_{\alpha \beta }^a\end{array},}$

so that:

${\displaystyle G_{\alpha \beta }^a={\mathrm{\partial }}_{\alpha }{\mathcal{A}}_{\beta }^a-{\mathrm{\partial }}_{\beta }{\mathcal{A}}_{\alpha }^a\mp g_{\text{s}}{f}^a{}_{bc}{\mathcal{A}}_{\alpha }^b{\mathcal{A}}_{\beta }^c,}$

where again a, b, c = 1, 2, ..., 8 are color indices. As with the gluon field, in a specific coordinate system and fixed gauge G_αβ are 3×3 traceless Hermitian matrix-valued functions, while G^a_αβ are real-valued functions, the components of eight four-dimensional second order tensor fields.

Differential forms

The gluon color field can be described using the language of differential forms, specifically as an adjoint bundle-valued curvature 2-form (note that fibers of the adjoint bundle are the su(3) Lie algebra);

${\displaystyle \mathbf{G}=\mathrm{d}\mathcal{A}\mp g_{\text{s}}\mathcal{A}\wedge \mathcal{A},}$

where ${\displaystyle \mathcal{A}}$ is the gluon field, a vector potential 1-form corresponding to G and ∧ is the (antisymmetric) wedge product of this algebra, producing the structure constants f ^abc. The Cartan-derivative of the field form (i.e. essentially the divergence of the field) would be zero in the absence of the "gluon terms", i.e. those ${\displaystyle \mathcal{A}}$ which represent the non-abelian character of the SU(3).

A more mathematically formal derivation of these same ideas (but a slightly altered setting) can be found in the article on metric connections.

Comparison with the electromagnetic tensor

This almost parallels the electromagnetic field tensor (also denoted F ) in quantum electrodynamics, given by the electromagnetic four-potential A describing a spin-1 photon;

${\displaystyle F_{\alpha \beta }={\mathrm{\partial }}_{\alpha }{A}_{\beta }-{\mathrm{\partial }}_{\beta }{A}_{\alpha },}$

or in the language of differential forms:

${\displaystyle \mathbf{F}=\mathrm{d}\mathbf{A}.}$

The key difference between quantum electrodynamics and quantum chromodynamics is that the gluon field strength has extra terms which lead to self-interactions between the gluons and asymptotic freedom. This is a complication of the strong force making it inherently non-linear, contrary to the linear theory of the electromagnetic force. QCD is a non-abelian gauge theory. The word non-abelian in group-theoretical language means that the group operation is not commutative, making the corresponding Lie algebra non-trivial.

QCD Lagrangian density

See also: Classical field theory

Characteristic of field theories, the dynamics of the field strength are summarized by a suitable Lagrangian density and substitution into the Euler–Lagrange equation (for fields) obtains the equation of motion for the field. The Lagrangian density for massless quarks, bound by gluons, is:

${\displaystyle \mathcal{L}=-\frac{1}{2}\mathrm{t}\mathrm{r}\left(G_{\alpha \beta }{G}^{\alpha \beta }\right)+\overline{\psi }\left(i{D}_{\mu }\right){\gamma }^{\mu }\psi }$

where "tr" denotes trace of the 3×3 matrix G_αβG^αβ, and γ^μ are the 4×4 gamma matrices. In the fermionic term ${\displaystyle i\overline{\psi }\left(i{D}_{\mu }\right){\gamma }^{\mu }\psi }$, both color and spinor indices are suppressed. With indices explicit, ${\displaystyle {\psi }_{i,\alpha }}$ where ${\displaystyle i=1,\dots ,3}$ are color indices and ${\displaystyle \alpha =1,\dots ,4}$ are Dirac spinor indices.

Gauge transformations

Main article: Gauge theory

In contrast to QED, the gluon field strength tensor is not gauge invariant by itself. Only the product of two contracted over all indices is gauge invariant.

Equations of motion

Treated as a classical field theory, the equations of motion for the quark fields are:

${\displaystyle (i\hslash {\gamma }^{\mu }{D}_{\mu }-mc)\psi =0}$

which is like the Dirac equation, and the equations of motion for the gluon (gauge) fields are:

${\displaystyle \left[D_{\mu },G^{\mu \nu }\right]=g_{\text{s}}{j}^{\nu }}$

which are similar to the Maxwell equations (when written in tensor notation). More specifically, these are the Yang–Mills equations for quark and gluon fields. The color charge four-current is the source of the gluon field strength tensor, analogous to the electromagnetic four-current as the source of the electromagnetic tensor. It is given by

${\displaystyle j^{\nu }=t^b{j}_b^{\nu },j_b^{\nu }=\overline{\psi }{\gamma }^{\nu }{t}^b\psi ,}$

which is a conserved current since color charge is conserved. In other words, the color four-current must satisfy the continuity equation:

${\displaystyle D_{\nu }{j}^{\nu }=0.}$

Global language system

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Global_language_system

The global language system is the "ingenious pattern of connections between language groups". Dutch sociologist Abram de Swaan developed this theory in 2001 in his book Words of the World: The Global Language System and according to him, "the multilingual connections between language groups do not occur haphazardly, but, on the contrary, they constitute a surprisingly strong and efficient network that ties together – directly or indirectly – the six billion inhabitants of the earth." The global language system draws upon the world system theory to account for the relationships between the world's languages and divides them into a hierarchy consisting of four levels, namely the peripheral, central, supercentral and hypercentral languages.

Theory

Background

According to de Swaan, the global language system has been constantly evolving since the time period of the early 'military-agrarian' regimes. Under these regimes, the rulers imposed their own language and so the first 'central' languages emerged, linking the peripheral languages of the agrarian communities via bilingual speakers to the language of the conquerors. Then was the formation of empires, which resulted in the next stage of integration of the world language system.

Firstly, Latin emerged from Rome. Under the rule of the Roman Empire, which ruled an extensive group of states, the usage of Latin stretched along the Mediterranean coast, the southern half of Europe, and more sparsely to the North and then into the Germanic and Celtic lands. Thus, Latin evolved to become a central language in Europe from 27 BC to 476 AD.

Secondly, there was the widespread usage of the pre-classical version of Han Chinese in contemporary China due to the unification of China in 221 BC by Qin Shi Huang.

Thirdly, Sanskrit started to become widely spoken in South Asia from the widespread teaching of Hinduism and Buddhism in South Asian countries.

Fourthly, the expansion of the Arabic empire also led to the increased usage of Arabic as a language in the Afro-Eurasian land mass.

Military conquests of preceding centuries generally determine the distribution of languages today. Supercentral languages spread by land and sea. Land-bound languages spread via marching empires: German, Russian, Arabic, Hindi, Chinese and Japanese. Languages like Bengali, Tamil, Italian and Turkish too are less considered as land-bound languages. However, when the conquerors were defeated and were forced to move out of the territory, the spread of the languages receded. As a result, some of these languages are currently barely supercentral languages and are instead confined to their remaining state territories, as is evident from German, Russian and Japanese.

On the other hand, sea-bound languages spread by conquests overseas: English, French, Portuguese, Spanish. Consequently, these languages became widespread in areas settled by European colonisers and relegated the indigenous people and their languages to peripheral positions.

Besides, the world-systems theory also allowed the global language system to expand further. It focuses on the existence of the core, semi-peripheral and peripheral nations. The core countries are the most economically powerful and the wealthiest countries. Besides, they also have a strong governmental system in the country, which oversees the bureaucracies in the governmental departments. There is also the prevalent existence of the bourgeois, and core nations have significant influence over the non-core, smaller nations. Historically, the core countries were found in northwestern Europe and include countries such as England, France and the Netherlands. They were the dominant countries that had colonized many other nations from the early 15th century to the early 19th century.

Then is the existence of the periphery countries, the countries with the slowest economic growth. They also have relatively weak governments and a poor social structure and often depend on primary industries as the main source of economic activity for the country.

The extracting and exporting of raw materials from the peripheral nations to core nations is the activity bringing about the most economic benefits to the country. Much of the population that is poor and uneducated, and the countries are also extensively influenced by core nations and the multinational corporations found there. Historically, peripheral nations were found outside Europe, the continent of colonial masters. Many countries in Latin America were peripheral nations during the period of colonization, and today peripheral countries are in sub-Saharan Africa.

Lastly, the presence of the semiperiphery countries, those in between the core and the periphery. They tend to be those which started out as peripheral nations and are currently moving towards industrialization and the development of more diversified labour markets and economies. They can as well come about from declining core countries. They are not dominant players in the international trade market. As compared to the peripheral nations, semi-peripheries are not as susceptible to manipulation by the core countries. However, most of these nations have economic or political relations with the core. Semi-peripheries also tend to exert influence and control over peripheries and can serve to be a buffer between the core and peripheral nations and ease political tensions. Historically, Spain and Portugal were semi-peripheral nations after they fell from their dominant core positions. As they still maintained a certain level of influence and dominance in Latin America over their colonies, they could still maintain their semi-peripheral position.

According to Immanuel Wallerstein, one of the most well-known theorists who developed the world-systems approach, a core nation is dominant over the non-core nations from its economic and trade dominance. The abundance of cheap and unskilled labour in the peripheral nations makes many large multinational corporations (MNCs), from core countries, often outsource their production to the peripheral countries to cut costs, by employing cheap labour. Hence, the languages from the core countries could penetrate into the peripheries from the setting up of the foreign MNCs in the peripheries. A significant percentage of the population living in the core countries had also migrated to the core countries in search of jobs with higher wages.

The gradual expansion of the population of migrants makes the language used in their home countries be brought into the core countries, thus allowing for further integration and expansion of the world language system. The semi-peripheries also maintain economic and financial trade with the peripheries and core countries. That allows for the penetration of languages used in the semi-peripheries into the core and peripheral nations, with the flow of migrants moving out of the semi-peripheral nations to the core and periphery for trade purposes.

Thus, the global language system examines rivalries and accommodations using a global perspective and establishes that the linguistic dimension of the world system goes hand in hand with the political, economic, cultural and ecological aspects. Specifically, the present global constellation of languages is the product of prior conquest and domination and of ongoing relations of power and exchange.^[1]

Q-value

${\displaystyle Q_i}$ is the communicative value of a language i, its potential to connect a speaker with other speakers of a constellation or subconstellation, "S". It is defined as follows:

${\displaystyle Q_i=p_i\times c_i=\left(\frac{P_i}{N^S}\right)\times \left(\frac{C_i}{M^S}\right)}$

The prevalence ${\displaystyle p_i}$ of language i, means the number of competent speakers in i, ${\displaystyle P_i}$, divided by all the speakers, ${\displaystyle N^S}$ of constellation S. Centrality, ${\displaystyle c_i}$ is the number of multilingual speakers ${\displaystyle C_i}$ who speak language i divided by all the multilingual speakers in constellation S, ${\displaystyle M^S}$.

Thus, the Q-value or communication value is the product of the prevalence ${\displaystyle p_i}$ and the centrality ${\displaystyle c_i}$ of language i in constellation S.

Consequently, a peripheral language has a low Q-value and the Q-values increase along the sociology classification of languages, with the Q-value of the hypercentral language being the highest.

De Swaan has been calculating the Q-values of the official European Union (EU) languages since 1957 to explain the acquisition of languages by EU citizens in different phases.

In 1970, when there were only four language constellations, Q-value decreased in the order of French, German, Italian, Dutch. In 1975, the European Commission enlarged to include Britain, Denmark and Ireland. English had the highest Q-value followed by French and German. In the following years, the European Commission grew, with the addition of countries like Austria, Finland and Sweden. Q-value of English still remained the highest, but French and German swapped places.

In EU23, which refers to the 23 official languages spoken in the European Union, the Q-values for English, German and French were 0.194, 0.045 and 0.036 respectively.

Theoretical framework

De Swaan likens the global language system to contemporary political macrosociology and states that language constellations are a social phenomenon, which can be understood by using social science theories. In his theory, de Swaan uses the Political Sociology of Language and Political Economy of Language to explain the rivalry and accommodation between language groups.

Political sociology

This theoretical perspective centres on the interconnections among the state, nation and citizenship. Accordingly, bilingual elite groups try to take control of the opportunities for mediation between the monolingual group and the state. Subsequently, they use the official language to dominate the sectors of government and administration and the higher levels of employment. It assumes that both the established and outsider groups are able to communicate in a shared vernacular, but the latter groups lack the literacy skills that could allow them to learn the written form of the central or supercentral language, which would, in turn allow, them to move up the social ladder.

Political economy

This perspective centres on the inclinations that people have towards learning one language over the other. The presumption is that if given a chance, people will learn the language that gives them more communication advantage. In other words, a higher Q-Value. Certain languages such as English or Chinese have high Q-values since they are spoken in many countries across the globe and would thus be more economically useful than to less spoken languages, such as Romanian or Hungarian.

From an economic perspective, languages are ‘hypercollective’ goods since they exhibit properties of collective goods and produce external network effects. Thus, the more speakers a language has, the higher its communication value for each speaker. The hypercollective nature and Q-Value of languages thus help to explain the dilemma that a speaker of a peripheral language faces when deciding whether to learn the central or hypercentral language. The hypercollective nature and Q-value also help to explain the accelerating spread and abandonment of various languages. In that sense, when people feel that a language is gaining new speakers, they would assign a greater Q-value to this language and abandon their own native language in place of a more central language. The hypercollective nature and Q-value also explain, in an economic sense, the ethnic and cultural movements for language conservation.

Specifically, a minimal Q-value of a language is guaranteed when there is a critical mass of speakers committed to protecting it, thus preventing the language from being forsaken.

Characteristics

The global language system theorises that language groups are engaged in unequal competition on different levels globally. Using the notions of a periphery, semi-periphery and a core, which are concepts of the world system theory, de Swaan relates them to the four levels present in the hierarchy of the global language system: peripheral, central, supercentral and hypercentral.

De Swaan also argues that the greater the range of potential uses and users of a language, the higher the tendency of an individual to move up the hierarchy in the global language system and learn a more "central" language. Thus, de Swaan views the learning of second languages as proceeding up rather than down the hierarchy, in the sense that they learn a language that is on the next level up. For instance, speakers of Catalan, a peripheral language, have to learn Spanish, a central language to function in their own society, Spain. Meanwhile, speakers of Persian, a central language, have to learn Arabic, a supercentral language, to function in their region. On the other hand, speakers of a supercentral language have to learn the hypercentral language to function globally, as is evident from the huge number of non-native English speakers.

According to de Swaan, languages exist in "constellations" and the global language system comprises a sociological classification of languages based on their social role for their speakers. The world's languages and multilinguals are connected in a strongly ordered, hierarchical pattern. There are thousands of peripheral or minority languages in the world, each of which are connected to one of a hundred central languages. The connections and patterns between each language is what makes up the global language system. The four levels of language are the peripheral, central, supercentral and hypercentral languages.

This flowchart depicts the hierarchy of the languages in de Swaan's (2001) global language system theory.

Peripheral languages

At the lowest level, peripheral languages, or minority languages, form the majority of languages spoken in the world; 98% of the world's languages are peripheral languages and spoken by less than 10% of the world’s population. Unlike central languages, these are "languages of conversation and narration rather than reading and writing, of memory and remembrance rather than record". They are used by native speakers within a particular area and are in danger of becoming extinct with increasing globalisation, which sees more and more speakers of peripheral languages acquiring more central languages in order to communicate with others.

Central languages

The next level constitutes about 100 central languages, spoken by 95% of the world's population and generally used in education, media and administration. Typically, they are the 'national' and official languages of the ruling state. These are the languages of record, and much of what has been said and written in those languages is saved in newspaper reports, minutes and proceedings, stored in archives, included in history books, collections of the 'classics', of folk talks and folk ways, increasingly recorded on electronic media and thus conserved for posterity.

Many speakers of central languages are multilingual because they are either native speakers of a peripheral language and have acquired the central language, or they are native speakers of the central language and have learned a supercentral language.

Supercentral languages

At the second highest level, 12 supercentral languages are very widely spoken languages that serve as connectors between speakers of central languages: Arabic, Chinese, English, French, German, Hindi, Japanese, Malay, Portuguese, Russian, Spanish and Swahili.

These languages often have colonial traces and "were once imposed by a colonial power and after independence continued to be used in politics, administration, law, big business, technology and higher education".

Hypercentral languages

At the highest level is the language that connects speakers of the supercentral languages. Today, English is the only example of a hypercentral language as the standard for science, literature, business, and law, as well as being the most widely spoken second language.

Applications

Pyramid of languages of the world

This pyramid illustrates the hierarchy of the world's languages as proposed by Graddol (1997) in his book, 'The future of English? A guide to forecasting the popularity of the English language in the 21st century', published by the British Council.

According to David Graddol (1997), in his book titled The Future of English, the languages of the world comprise a "hierarchical pyramid", as follows:

• The big languages: English, French.
• Regional languages (languages of the United Nations are marked with asterisk): Arabic*, Mandarin*, German, Russian*, Spanish* and Portuguese.
• National languages: around 80 languages serving over 180 nation states (e.g. Nepali).
• Official languages within nation states (and other "safe" languages): around 600 languages worldwide (e.g. Marathi).
• Local vernacular languages: the remainder of the world's 6,000+ languages.

Translation systems

The global language system is also seen in the international translation process as explained by Johan Heilbron, a historical sociologist: "translations and the manifold activities these imply are embedded in and dependent on a world system of translation, including both the source and the target cultures".

The hierarchical relationship between global languages is reflected in the global system for translations. The more "central" a language, the greater is its capability to function as a bridge or vehicular language to facilitate communication between peripheral and semi-central languages.

Heilbron's version of the global system of language in translations has four levels:

Level 1: Hypercentral position — English currently holds the largest market share of the global market for translations; 55–60% of all book translations are from English. It strongly dominates the hierarchical nature of book translation system.

Level 2: Central position — German and French each hold 10% of the global translation market.

Level 3: Semi-central position — There are 7 or 8 languages "neither very central on a global level nor very peripheral", each making up 1 to 3% of the world market (like Spanish, Italian and Russian).

Level 4: Peripheral position — Languages from which "less than 1% of the book translations worldwide are made", including Chinese, Hindi, Japanese, Malay, Swahili, Turkish and Arabic. Despite having large populations of speakers, "their role in the translation economy is peripheral as compared to more central languages".

Acceptance

According to the Google Scholar website, de Swaan's book, Words of the world: The global language system, has been cited by 2990 other papers, as of 25 August 2021.

However, there have also been several concerns regarding the global language system:

Importance of Q-value

Van Parijs (2004) claimed that 'frequency' or likelihood of contact is adequate as an indicator of language learning and language spread. However, de Swaan (2007) argued that it alone is not sufficient. Rather, the Q-value, which comprises both frequency (better known as prevalence) and 'centrality', helps to explain the spread of (super)central languages, especially former colonial languages in newly independent countries where in which only the elite minority spoke the language initially. Frequency alone would not be able to explain the spread of such languages, but Q-value, which includes centrality, would be able to.

In another paper, Cook and Li (2009) examined the ways to categorise language users into various groups. They suggested two theories: one by Siegel (2006) who used 'sociolinguistic settings', which is based on the notion of dominant language, and another one by de Swaan (2001) that used the concept of hierarchy in the global language system. According to them, de Swaan's hierarchy is more appropriate, as it does not imply dominance in power terms. Rather, de Swaan's applies the concepts of geography and function to group languages and hence language users according to the global language system. De Swaan (2001) views the acquisition of second languages (L2) as typically going up the hierarchy.

However, Cook and Li argues that this analysis is not adequate in accounting for the many groups of L2 users to whom the two areas of territory and function hardly apply. The two areas of territory and function can be associated respectively with the prevalence and centrality of the Q-value. This group of L2 users typically does not acquire an L2 going up the hierarchy, such as users in an intercultural marriage or users who come from a particular cultural or ethnic group and wish to learn its language for identity purposes. Thus, Cook and Li argue that de Swaan's theory, though highly relevant, still has its drawbacks in that the concept behind Q-value is insufficient in accounting for some L2 users.

Choice of supercentral languages

There is disagreement as to which languages should be considered more central. The theory states that a language is central if it connects speakers of "a series of central languages". Robert Phillipson questioned why Japanese is included as one of the supercentral languages but Bengali, which has more speakers, is not on the list.

Inadequate evidence for a system

Michael Morris argued that while it is clear that there is language hierarchy from the "ongoing interstate competition and power politics", there is little evidence provided that shows that the "global language interaction is so intense and systematic that it constitutes a global language system, and that the entire system is held together by one global language, English". He claimed that de Swaan's case studies demonstrated that hierarchy in different regions of the world but did not show the existence of a system within a region or across regions. The global language system is supposed to be part of the international system but is "notoriously vague and lacking in operational importance" and therefore cannot be shown to exist. However, Morris believes that this lack of evidence could be from the lack of global language data and not negligence on de Swaan's part. Morris also believes that any theory on a global system, if later proved, would be much more complex than what is proposed by de Swaan. Questions on how the hypercentral language English holds together the system must also be answered by such a global language system.

Theory built on inadequate foundations

Robert Phillipson states that the theory is based on selective theoretical foundations. He claimed that there is a lack of consideration about the effects of globalization, which is especially important when the theory is about a global system: "De Swaan nods occasionally in the direction of linguistic and cultural capital, but does not link this to class or linguistically defined social stratification (linguicism) or linguistic inequality" and that "key concepts in the sociology of language, language maintenance and shift, and language spread are scarcely mentioned".

On the other hand, de Swaan's work in the field of sociolinguistics has been noted by other scholars to be focused on "issues of economic and political sociology" and "politic and economic patterns", which may explain why he makes only "cautious references to socio-linguistic parameters".