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) .(adsbygoogle = window.adsbygoogle || []).push({});

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:

[tex]m \frac{d u_{\alpha}}{d \tau} =q F_{\alpha \beta}u^{\beta} [/tex]

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 ?

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# Classical Yang-Mills Force Law ?

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