How to work with a non-Abelian gauge field

[jtceleron]

Homework Statement

Homework Equations

I am learning QFT, and I am confused of such transformations. For example, first, in these equations, especially the one that defines the transformation of A(x), whether should the partial derivative acts on U(or U^-1), or just take U as a constant? Second, the partial derivative acts only on U^-1 or all the things after it?
Another question is should we consider A(x) and F(x) as an operator, so that when calculating, it is convenient to put a function after it.

The Attempt at a Solution

 

[andrien]

I am not sure about your background but if you want to show it then you can take a simple say SU(2) model,and note that the U's are not constant .In most representation it can be modeled as,U=exp(ζ^at_a).you can look up here for this and most noteworthy is chriss quigg book on every kind of interactions.
https://docs.google.com/viewer?a=v&q=cache:rIlRjFXgsUgJ:www.staff.science.uu.nl/~wit00103/ftip/Ch12.pdf+non+abelian+gauge+field+theory&hl=en&gl=in&pid=bl&srcid=ADGEEShqJlXAn76qqji-voYWnDTnSwkJelRaIib5JXx5oLZGhl30sk1OqYHIo2GTAQBaBfT6RQyAyl3itZF7VAYITk8vDdTHbp2BjVoK1NwdzydbX0ZkAEyvea3rkpV9U90W8FYs1XuP&sig=AHIEtbRTdHObuHKb1PeRx73K-wOszayn0Q

 

[jtceleron]

I am not sure about your background but if you want to show it then you can take a simple say SU(2) model,and note that the U's are not constant .In most representation it can be modeled as,U=exp(ζ^at_a).you can look up here for this and most noteworthy is chriss quigg book on every kind of interactions.
https://docs.google.com/viewer?a=v&q=cache:rIlRjFXgsUgJ:www.staff.science.uu.nl/~wit00103/ftip/Ch12.pdf+non+abelian+gauge+field+theory&hl=en&gl=in&pid=bl&srcid=ADGEEShqJlXAn76qqji-voYWnDTnSwkJelRaIib5JXx5oLZGhl30sk1OqYHIo2GTAQBaBfT6RQyAyl3itZF7VAYITk8vDdTHbp2BjVoK1NwdzydbX0ZkAEyvea3rkpV9U90W8FYs1XuP&sig=AHIEtbRTdHObuHKb1PeRx73K-wOszayn0Q

You mean that, for example, in the last equation of transformation for A(x), the partial derivative should act on U^-1, rather than taking U^-1 as a constant?

 

[andrien]

You mean that, for example, in the last equation of transformation for A(x), the partial derivative should act on U^-1, rather than taking U^-1 as a constant?

Yes,of course because U used here as a local gauge transformation.the transformation of gradient takes the form
∂_μψ=U∂_μψ+(∂_μU)ψ
For the derivation which you are looking for,I will refer you to chris quigg book on'gauge theory of strong,weak,electromagnetic' page 55-60.I am not going to derive it here because as always I am out of time.