Abelian theories with more than one charge

I have a question about the non-abelian character of QCD. In order to write a gauge-invariant Lagrangian, there must be a term with the strength tensor $$X^{\mu\nu}_{a}X_{\mu\nu}^{a}$$ where $$X^a_{\mu\nu}=\partial_\mu A_\nu^a-\partial_\nu A_\mu+g f^{abc}A_\mu^bA_\nu^c,$$ and the structure constants $$f^{abc}$$ are determined by the Lie group of the theory. Specifically, the group generators $$T^a$$ define the structure with $$[T^a,T^b]=if^{abc}T^c.$$ If $$f^{abc}=0$$, the theory is abelian. Now, my question is: what the group generators are depends on what the theory wants to describe, right? QED has $$U(1)$$ symmetry because there is only one type of electric charge. Similarly, QCD has $$SU(3)$$ symmetry to describe the three color charges. Is this thinking correct? If not, could there be an abelian gauge theory describing three types of charge?

• You could use U(1)xU(1)xU(1) to describe 3 different "electric" charges.
– JGBM
Feb 10, 2021 at 14:47
• I believe the thing is that in the $SU(3)$ case a multiplet in the fundamental representation has three particles which are exchanged among themselves continuously under the symmetry, whereas in a direct product $U(1)\times U(1)\times U(1)$ the three particles would not mix under the symmetry. You would have three singlets which transform independently by phases.
– Gold
Feb 10, 2021 at 15:15
• In the modern way of thinking, the charges are defined as whatever you get by making the particle field gauge invariant, and this, as you note, is pretty locked in by the process and ultimately comes from experiments. At least this is the case so far in nature for all the forces (except gravity) which is pretty convincing and suggests there is some underlying common way this works (local gauge invariance). Otherwise, I assume you can add special couplings to non-gauge theory-force carriers in the Lagrangian but I'm also sure there are heavy constraints on the forms to make it work overall.. Feb 10, 2021 at 16:25
• Also, like @user1620696 wrote, SU(3) is not really "3 types of charge", it's one type of charge ("color charge") that has to be non-abelian if its used as the connection in a local gauge invariance type of theory. I've been struggling myself finding a good analogy of where the non-abelianness comes from, it arises from the non-commutation of rotations in >1 dimension. 3 types of charge would be like, electromagnetic + color charge + weak, for example, and they don't mix. Feb 10, 2021 at 16:31
• @RenanNobuyukiHirayama, if all the fields are charged under that, then they would interact. As mentioned on the other commentaries the number of charges is not the relevant aspect. Indeed SU(3) has 8 generators.
– JGBM
Feb 10, 2021 at 22:35