Gluon Field Strength Tensor
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In theoretical
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) and ...
, the gluon field strength tensor is a second order
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analys ...
characterizing the
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bi ...
interaction between
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All common ...
s. The
strong interaction The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called th ...
is one of the
fundamental interaction In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electr ...
s of nature, and the
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 a ...
(QFT) to describe it is called ''
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a ty ...
'' (QCD).
Quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All common ...
s 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
with values in the adjoint bundle of the chromodynamical SU(3) gauge group (see
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
for necessary definitions).


Convention

Throughout this article, Latin indices (typically ) take values 1, 2, ..., 8 for the eight gluon color charges, while Greek indices (typically ) take values 0 for timelike components and 1, 2, 3 for spacelike components of
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as ...
s 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 , (or , , or some variant), and has components defined
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...
to the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of the quark
covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differ ...
: : G_ = \pm \frac _\alpha, D_\beta,, where: :D_\mu =\partial_\mu \pm ig_\text t_a \mathcal^a_\mu\,, in which * is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
; * is the
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
of the strong force; * are the Gell-Mann matrices divided by 2; * 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; * is a spacetime index, 0 for timelike components and 1, 2, 3 for spacelike components; *\mathcal_\mu = t_a \mathcal^a_\mu expresses the gluon field, a spin-1 gauge field or, in differential-geometric parlance, a connection in the SU(3) principal bundle; * \mathcal_\mu are its four (coordinate-system dependent) components, that in a fixed gauge are traceless Hermitian matrix-valued functions, while \mathcal^a_\mu are 32
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...
s, the four components for each of the eight four-vector fields. Different authors choose different signs. Expanding the commutator gives; :G_ =\partial_\mathcal_\beta-\partial_\beta\mathcal_\alpha \pm ig_\text mathcal_, \mathcal_/math> Substituting t_a \mathcal^a_\alpha = \mathcal_ and using the commutation relation _a, t_b = i f_^ t_c for the Gell-Mann matrices (with a relabeling of indices), in which 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: :\begin G_ & = \partial_\alpha t_a \mathcal^a_ - \partial_\beta t_a \mathcal^a_\alpha \pm i g_\text \left _b ,t_c \right \mathcal^b_\alpha \mathcal^c_\beta \\ & = t_a \left( \partial_\alpha \mathcal^a_ - \partial_\beta \mathcal^a_\alpha \pm i^2 f_^ag_\text \mathcal^b_\alpha \mathcal^c_\beta \right) \\ & = t_a G^a_ \\ \end\,, so that: :G^a_ = \partial_\alpha \mathcal^a_ - \partial_\beta \mathcal^a_\alpha \mp g_\text f^_ \mathcal^b_\alpha \mathcal^c_\beta \,, where again are color indices. As with the gluon field, in a specific coordinate system and fixed gauge are traceless Hermitian matrix-valued functions, while 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 form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s, specifically as an adjoint bundle-valued
curvature 2-form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algeb ...
(note that fibers of the adjoint bundle are the su(3)
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
); :\mathbf =\mathrm\boldsymbol \mp g_\text\,\boldsymbol\wedge \boldsymbol\,, where \boldsymbol is the gluon field, a vector potential 1-form corresponding to and is the (antisymmetric)
wedge product A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
of this algebra, producing the structure constants . 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 \boldsymbol 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 ) in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
, given by the electromagnetic four-potential describing a spin-1
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
; :F_=\partial_A_-\partial_A_\,, or in the language of differential forms: :\mathbf = \mathrm\mathbf\,. 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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
, making the corresponding Lie algebra non-trivial.


QCD Lagrangian density

Characteristic of field theories, the dynamics of the field strength are summarized by a suitable
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 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: :\mathcal=-\frac\mathrm\left(G_G^\right)+ \bar\left(iD_\mu \right)\gamma^\mu\psi where "tr" denotes trace of the matrix , and are the gamma matrices. In the fermionic term i\bar\left(iD_\mu\right)\gamma^\psi, both color and spinor indices are suppressed. With indices explicit, \psi_ where i=1,\ldots ,3 are color indices and \alpha=1,\ldots,4 are Dirac spinor indices.


Gauge transformations

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: :( i\hbar \gamma^\mu D_\mu - mc ) \psi = 0 which is like the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac pa ...
, and the equations of motion for the gluon (gauge) fields are: :\left _\mu , G^ \right= g_\text 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 Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associa ...
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 :j^\nu = t^b j_b^\nu \,, \quad j_b^\nu = \bar\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 A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
: :D_\nu j^\nu = 0 \,.


See also

* Quark confinement * Gell-Mann matrices *
Field (physics) In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each poin ...
* Yang–Mills field * Eightfold Way (physics) * Einstein tensor * Wilson loop * Wess–Zumino gauge * Quantum chromodynamics binding energy *
Ricci calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to b ...
* Special unitary group


References


Notes


Further reading


Books

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Selected papers

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External links

* * {{tensors Gauge theories Quantum chromodynamics Gluons