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How to work with a non-Abelian gauge field

  1. Nov 4, 2012 #1
    1. The problem statement, all variables and given/known data
    attachment.php?attachmentid=52614&stc=1&d=1352010475.jpg


    2. Relevant 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.

    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Nov 4, 2012 #2
    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(ζata).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
     
  4. Nov 4, 2012 #3
    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?
     
  5. Nov 5, 2012 #4
    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.
     
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