Quarks interact with each other by the strong force due to their color charge, mediated by gluons.
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).
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.
Below the definitions (and most of the notation) follow K. Yagi, T. Hatsuda, Y. Miake[1] and Greiner, Schäfer.
[2] The tensor is denoted G, (or F, F, or some variant), and has components defined proportional to the commutator of the quark covariant derivative Dμ:[2][3] where: in which Note that different authors choose different signs.
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: so that:[4][5] where again a, b, c = 1, 2, ..., 8 are color indices.
As with the gluon field, in a specific coordinate system and fixed gauge Gαβ are 3×3 traceless Hermitian matrix-valued functions, while Gaαβ are real-valued functions, the components of eight four-dimensional second order tensor fields.
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); where
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.
A more mathematically formal derivation of these same ideas (but a slightly altered setting) can be found in the article on metric connections.
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; or in the language of differential forms: 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.
The word non-abelian in group-theoretical language means that the group operation is not commutative, making the corresponding Lie algebra non-trivial.
The Lagrangian density for massless quarks, bound by gluons, is:[2] where "tr" denotes trace of the 3×3 matrix GαβGαβ, and γμ are the 4×4 gamma matrices.
In contrast to QED, the gluon field strength tensor is not gauge invariant by itself.
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.