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.

 

[paranormal]

Working with non-Abelian gauge fields can be a complex task, but it is an essential part of studying quantum field theory. To answer your first question, the partial derivative should act on the transformation operator U, as it is a function of spacetime coordinates. The transformation of A(x) should then be calculated by applying this operator to the original A(x) field. As for your second question, the partial derivative should act on all the terms after U-1, as they are all functions of spacetime coordinates.

In terms of considering A(x) and F(x) as operators, it is helpful to think of them as operators that act on states or fields. This allows for a more intuitive understanding of their transformations and how they interact with other operators. However, when calculating, it is important to keep track of the order of operations and apply the transformations correctly. Overall, working with non-Abelian gauge fields requires a strong understanding of the underlying principles and careful attention to detail in calculations. With practice and patience, you will be able to navigate the complexities of non-Abelian gauge fields and further your understanding of quantum field theory.