Hello all, my teacher assigned a problem related to the yang-mills equation in my general relativity class and I just wanted to ask a couple of questions about this problem. I believe it is a simplified version of the Yang-Mills you encounter in particle physics.(adsbygoogle = window.adsbygoogle || []).push({});

Basicly assuming that [itex] F_{\mu\nu} → F_{\mu\nu} + A_\mu A_\nu - A_\nu A_\mu [/itex] (for some non-commuting potentials) we were instructed to derive the field equations from our lagrangian [itex] S = \frac{1}{4} \int tr ( F_{\mu\nu}F^{\mu \nu}) d^4x [/itex]. Firstly I wondered if we could just ignore the fact that we are taking a trace.. I am really not sure what difference this makes to the equations. He presented the problem with basically only the details I described above so perhaps the trace is used when dealing with a more complex problem.. I'm not entirely sure. Anyways I decided to try varying with respect to [itex] \delta A_\nu[/itex] the math went like:

[itex] \delta S = \frac{1}{4} \int tr (( F_{\mu\nu} \delta F^{\mu \nu} + \delta F^{\mu\nu} F_{\mu \nu} ) d^4x [/itex]

[itex] \delta S = \frac{1}{2} \int tr ((\delta F_{\mu\nu} F^{\mu \nu} )) d^4x [/itex]

[itex] \delta S = \frac{1}{2} \int tr (F^{\mu\nu} (\partial_\mu \delta A_\nu - \partial_\nu \delta A_\mu + \delta(A_\mu A_\nu - A_\nu A_\mu))) d^4x [/itex]

[itex] \delta S = \int tr (F^{\mu\nu} (\partial_\mu \delta A_\nu + (\delta A_\mu A_\nu + A_\mu \delta A_\nu))) d^4x [/itex]

Now using integration by parts and boundary conditions on the first term and similar index swapping from above I made it about as far as I could on my own and came upon

[itex] \delta S = \int tr ((-\partial_\mu F^{\mu\nu} \delta A_\nu +F^{\mu\nu} (-\delta A_\nu A_\mu + A_\mu \delta A_\nu))) d^4x [/itex]

So then I supposed that I could multiply by some sort of inverse matrix or something of the sort. Also I assumed that we could ignore the trace (which again I'm not sure is correct, but I have never varied a trace of a matrix before). My final equation came out to be.

[itex] \partial_\mu F^{\mu\nu} + (\delta A_\nu A_\mu (\delta A_\nu)^{-1} - A_\mu) F^{\mu\nu} = 0 [/itex]

However something has definitely gone wrong, I know I need to get rid of that pesky [itex] \delta A_\nu [/itex].. I also tried using Euler-Lagrange equations but I had even less luck than here.. I would just like to bounce some ideas off you guys and perhaps get pointed in the direction of some good literature.

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# (Simple) Derivation of Yang-Mills Equations

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