Yang-Mills Field Strength Tensor

[neevor]

I was wondering why for
[tex] F_{\mu \nu} = [D_{\mu},D_{\nu}] = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\mu},A_{\nu}]<br /> [\tex]<br /> the term<br /> $$A_{\mu}\partial_{\nu} - A_{\nu}\partial_{\mu}<br /> [\tex]<br /> vanishes.$$[/tex]

 

[cristo]

Since no summation is implied, one can simply swap the indices \mu and \nu in the second term of the expression, and hence the result.

Note, to display latex use the [ tex ] and [ / tex ] tags (without spaces inside the square brackets) in place of \begin{displaymath}.

 

[neevor]

Really?
because A_{\mu} is not equal to A_{\nu} in general. So simply swapping the two indecies would just give,
A_{\nu}\partial_{\mu} - A_{\mu}\partial_{\nu}
leaving me with the same problem.

 

[cristo]

Oh yes, sorry. I just glance at it and typed before I thought really! I'm blaming it on the fact that it's late. With regard to the question, I'm not too sure.. is there anything special about A? Sorry I can't be of more help!

 

[George Jones]

The term

$$A_{\mu}\partial_{\nu} - A_{\nu}\partial_{\mu}$$

does not vanish.

There is a similar term, but of opposite sign, that comes from the product rule in terms like $\partial_\mu A_\nu$.

$$\partial_\mu \left(A_\nu \psi \right) = \left( \partial_\mu A_\nu \right) \psi + A_\nu \partial_\mu \psi$$