Talk:Gluon field strength tensor

This is a terrible article and needs to be rewritten; unfortunately I do not have the knowledge nor the fortitude to do so. Thanks in advance for any future efforts. 147.188.248.179 (talk) 09:23, 4 February 2013 (UTC)

Potential inconsistencies/inaccuracies

There could be inconsistencies in signs multiplying g_s and factors of 1/2... Need to check out what a "derivative acting on octet fields" actually means. M∧Ŝc²ħεИ_τlk 07:46, 19 October 2013 (UTC)

Here is the section I'm on about, it's been removed since we don't seem to need it:

Derivative acting on the color octet field
There is similar covariant derivative which acts on the color octet field: ${\mathcal {D}}_{\mu }=\partial _{\mu }\pm ig_{s}T^{a}{\mathcal {A}}_{\mu }^{a}\,,$ where $(T^{a})_{bc}=-if_{abc}\,,$ are 8 × 8 matrices, the adjoint representations of the SU(3) Lie algebra satisfying the commutation relations: $[T^{a},T^{b}]=if_{abc}T^{c}\,.$

just placed here for future reference. M∧Ŝc²ħεИ_τlk 07:59, 19 October 2013 (UTC)

Transplant of curvature 2-form expression from Quantum chromodynamics to this article

See this cut and paste. Quantum chromodynamics gives the tensor expression for the Lagrangian and the gluon field strength:

$G_{\mu \nu }^{a}=\partial _{\mu }{\mathcal {A}}_{\nu }^{a}-\partial _{\nu }{\mathcal {A}}_{\mu }^{a}+gf^{abc}{\mathcal {A}}_{\mu }^{b}{\mathcal {A}}_{\nu }^{c}\,,$

which is enough for that article IMO. M∧Ŝc²ħεИ_τlk 21:18, 20 October 2013 (UTC)

Gibberish

The IP said the article is terrible. Indeed, I am pretty familiar with such stuff as “tensor field”, “gauge group”, “vector bundle”, “differential form”, and “covariant derivative”, but hardly can understand anything here.

First of all, why is it called a tensor? To which bundle, or to the tensor product of which, does it belong? Before stating something about it, the entire section starting with “The tensor is denoted G, … and has components defined proportional” and ending “G_αβ are actually 3 × 3 matrices, while G^a_αβ are scalar components of a four dimensional second order tensor field” reads as a gibberish, especially scalar components. I can compute dimensions myself. What I need is some clue with a definition.

The Gluon field strength tensor #Differential forms section is also poor. If

$\mathbf {G} =\mathrm {d} {\boldsymbol {\mathcal {A}}}\mp g\,{\boldsymbol {\mathcal {A}}}\wedge {\boldsymbol {\mathcal {A}}}$

is a definition, then what is g? Where the article explains the nature of that symbol? What operation is assumed between the enigmatic g and ${\boldsymbol {\mathcal {A}}}$? And how the section may contain a link to Élie Cartan, but miss a link to curvature form? Incnis Mrsi (talk) 20:17, 8 January 2014 (UTC)

I wrote what is in the sources provided.

The link to curvature form is easy to add. Sue me for missing the subscript "s" on the "enigmatic g" for the strong coupling constant. As you can see above from the (now corrected) section, I did not write the differential form equation, but it should be here. As far as I can make out the wedge product reproduces the term with the structure constants, but I don't know exactly, so left it. I haven't yet found a source for this particular expression.

The sources didn't give the relevant bundles. They just gave the definition in terms of the commutator of covariant derivatives.

My question - let's see you provide a source (or more) which gives a crystal clear and technically correct definition of the gluon field strength tensor as used in QCD? M∧Ŝc²ħεИ_τlk 21:57, 8 January 2014 (UTC)

About "why is it a tensor?", because of the commutator of covariant derivative components, surely? Why is it (3 × 3)-matrix-valued? Because the covariant derivatives have contractions over the colour indices with the Gell-Mann matrices. There are 8 gluon colours, 8 GM matrices, and 8 gluon potentials, the covariant derivative contains a linear combination of all of them. M∧Ŝc²ħεИ_τlk 22:08, 8 January 2014 (UTC)

There is a confusion about the word “scalar”. Your “scalar-valued components” is true in the sense “for certain coordinate system and fixed gauge we obtain scalar-valued functions”, but false in the sense “[for a fixed gauge] these functions transform under changes of coordinates as scalars”. It’s better to omit it to avoid ambiguities. Incnis Mrsi (talk) 08:04, 9 January 2014 (UTC)

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