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 ...
:
:
where:
:
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;
*
expresses the
gluon field, a
spin-1 gauge field or, in differential-geometric parlance, a
connection in the SU(3)
principal bundle;
*
are its four (coordinate-system dependent) components, that in a fixed gauge are traceless
Hermitian matrix-valued functions, while
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;
: