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 Dμ:
G α β = ± 1 i g s [ D α , D β ] , {\displaystyle G_{\alpha \beta }=\pm {\frac {1}{ig_{\text{s}}}}[D_{\alpha },D_{\beta }],}
where
D μ = ∂ μ ± i g s t a A μ a , {\displaystyle D_{\mu }=\partial _{\mu }\pm ig_{\text{s}}t_{a}{\mathcal {A}}_{\mu }^{a},}
in which
i is the imaginary unit; gs is the coupling constant of the strong force; ta = λa/2 are the Gell-Mann matrices λa 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;
A μ = t a A μ a {\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;
A μ {\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 A μ a {\displaystyle {\mathcal {A}}_{\mu }^{a}} are 32 real-valued functions, the four components for each of the eight four-vector fields. Note that different authors choose different signs. Expanding the commutator gives
G α β = ∂ α A β − ∂ β A α ± i g s [ A α , A β ] . {\displaystyle G_{\alpha \beta }=\partial _{\alpha }{\mathcal {A}}_{\beta }-\partial _{\beta }{\mathcal {A}}_{\alpha }\pm ig_{\text{s}}[{\mathcal {A}}_{\alpha },{\mathcal {A}}_{\beta }].}
Substituting t a A α a = A α {\displaystyle t_{a}{\mathcal {A}}_{\alpha }^{a}={\mathcal {A}}_{\alpha }} and using the commutation relation [ t a , t b ] = i f a b
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