What Does the [A., A.] Term Signify in Non-Abelian Yang-Mills Theory?

[member 11137]

In the yang Mils formulation of the EM field (Faraday Maxwell) tensor, what exactly is (means) the complementary term usually written: [A., A.] (see for example the article of Jackiw: 50 years of the Yang Mils theory and my contribution to it) and related to the fact that this theory can be non Abelian? Thanks for your help

 

[nrqed]

In the yang Mils formulation of the EM field (Faraday Maxwell) tensor, what exactly is (means) the complementary term usually written: [A., A.] (see for example the article of Jackiw: 50 years of the Yang Mils theory and my contribution to it) and related to the fact that this theory can be non Abelian? Thanks for your help

I am not sure what your question is exactly (do you want to know what the *definition* of this expression is? Or what is the physical interpretation?).

The gauge fields are in the adjoint representation (they are of the form $A_\mu^a \lambda_a$ where the lambda's are the generators of the group. Since the group is non abelian, the commutator does not vanish.

The physical interpretation is that there is self-coupling between the gauge bosons. There is no direct coupling between two photons (because QED is abelian) but there *is* self coupling between gluons or between W's and Z bosons.

 

[member 11137]

I am not sure what your question is exactly (do you want to know what the *definition* of this expression is? Or what is the physical interpretation?).

The gauge fields are in the adjoint representation (they are of the form $A_\mu^a \lambda_a$ where the lambda's are the generators of the group. Since the group is non abelian, the commutator does not vanish.

The physical interpretation is that there is self-coupling between the gauge bosons. There is no direct coupling between two photons (because QED is abelian) but there *is* self coupling between gluons or between W's and Z bosons.

Thanks for your rapid answer. My initial question was about the definition of the brackets. In another article concerning the anomalies and the renormalization of the BF YM theory, a new formulation of this term appears. The proposition for it is a quadratic form of the components of the EM field potential vector (the A 4-vector). I am working myself about a reformulation of this tensor and that was the reason of my question (see independent research forum and my home page). I recently did success to get a new expression of the tensor assuming the existence of a "new" mathematical operation (the extended vector product) and assuming a parallel transport of this A vector by respect for the trajectory. I don't know if it owns a physical signification but the formalism looks interesting.

 

[member 11137]

To be clear: my question is not to make publicity for my own and insignifiant research but only motivated by the fact that it seems to exist several propositions and interpretations in the litterature for this complementary term. I was lost. I just try to understand better. The physical interpretation that you give (self coupling between gluons) takes a particular relief when it is confronted with my own calculations which are absolutely not complicated but which are implying a special lecture of the usual formulation of the tensor. Once more time thanks for your help and that was.