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:^[2][3]

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

where:

${\displaystyle D_{\mu }=\partial _{\mu }\pm ig_{\text{s}}t_{a}{\mathcal {A}}_{\mu }^{a}\,,}$

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;

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

Substituting ${\displaystyle t_{a}{\mathcal {A}}_{\alpha }^{a}={\mathcal {A}}_{\alpha }}$ and using the commutation relation ${\displaystyle [t_{a},t_{b}]=if_{ab}{}^{c}t_{c}}$ 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:

${\displaystyle {\begin{aligned}G_{\alpha \beta }&=\partial _{\alpha }t_{a}{\mathcal {A}}_{\beta }^{a}-\partial _{\beta }t_{a}{\mathcal {A}}_{\alpha }^{a}\pm ig_{\text{s}}\left[t_{b},t_{c}\right]{\mathcal {A}}_{\alpha }^{b}{\mathcal {A}}_{\beta }^{c}\\&=t_{a}\left(\partial _{\alpha }{\mathcal {A}}_{\beta }^{a}-\partial _{\beta }{\mathcal {A}}_{\alpha }^{a}\pm i^{2}f_{bc}{}^{a}g_{\text{s}}{\mathcal {A}}_{\alpha }^{b}{\mathcal {A}}_{\beta }^{c}\right)\\&=t_{a}G_{\alpha \beta }^{a}\\\end{aligned}}\,,}$

so that:^[4][5]

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

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);

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

where ${\displaystyle {\boldsymbol {\mathcal {A}}}}$ 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 ${\displaystyle {\boldsymbol {\mathcal {A}}}}$ 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;

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

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.

Notes

Further reading