# Classical Yang-Mills Force Law ?

1. Mar 16, 2009

### confinement

Even though classical (as opposed to quantized) non-abelian gauge theories do not have any physical applications at this time, it is mathematically valid to say that these classical Yang-Mills fields generalize Maxwell's equations of E&M in some sense i.e. the Yang-Mills equations reduce to the covariant formulation of electromagnetism for the case of the most basic compact lie group i.e. that one having only one parameter, the abelian gauge group U(1) .

But in classical electromagnetism Maxwell's equations are only half of the story, we also need the Lorentz force law which describes the motion of charges. The covariant version of this law is given in terms of the mass m, the four velocity u, the charge q and the field tensor F by:

$$m \frac{d u_{\alpha}}{d \tau} =q F_{\alpha \beta}u^{\beta}$$

Until recently I thought that the corresponding equation would maintain the same form when generalized to the non-abelian case, but with the more complicated Yang-Mills field tensor F taking the place of the Faraday tensor. Recently I read a comment in these informal lecture notes:

http://philsci-archive.pitt.edu/archive/00003476/

to paraphrase, although the simple generalization I described in the above paragraph is the first thing we all think of, this is not correct, and that in principle one could work out the Yang-Mills force law from the Lagrangian.

My question is, does anyone know where this has been worked out in detail ?

2. Mar 17, 2009

### Haelfix

Why not simply try it for yourself? I'm sure its been done before in some textbook somewhere (though probably an unlovely calculation) but you probably will get more from the exercise by actually working through it rather than just seeing the result.

If I had to guess, I suspect the answer is that you will derive some messy linear combination of field strength powers.

On my competency exam, I had to basically do something like that in order to derive the classical YM Maxwell equations for SU(3). Its a PITA, but the calculation was of more value than the result (which is messy and physically irrelevant).

3. Mar 17, 2009

### confinement

It won't do me any good to brute force my way through some index gymnastics only to obtain a messy result that I cannot interpret. The Yang-Mills equations are properly understood in terms of geometry, and if I just power-push through a bunch of component based terms I won't be able to recognize the geometrical meanings of the final answer --- that's why I am looking for an explicit treatment of this topic.

On a related note, I was wondering if anyone knows of a version of the Euler-Lagrange equations that is suitable for directly manipulating differential forms (without using components)?