# Assorted questions about Srednicki ch78

1. Nov 12, 2011

### LAHLH

Hi,

I have a few questions about ch78 of Srednicki's QFT: background field gauge. I'll just post a couple to see if anyone can help and so this post isn't huge, then maybe some more later.

1) If the background field transforms as $\delta_G \bar{A^a_{\mu}}(x)=0$ and $\delta_{BG} \bar{A^a_{\mu}}(x)=-\bar{D^{ac}_{\mu}\theta^c(x)}$, am I correct in thinking that under the combined transformation we have just $\delta_{G+BG} \bar{A^a_{\mu}}(x)=-\bar{D^{ac}_{\mu}}\theta^c(x)$? If so then I can see that $D_{\mu},\bar{D_{\mu}}$ will have the same transformation under $\delta_{G+BG}$. I guess then for his argument to show the invariance of the ghost Lagrangian under this combined transformation it doesnt matter that the D's are different (D, D bar) so long as they transform the same way under the transformation you might aswell have two D's? Don't suppose anyone could make this more explicit? so I can this this ghost L really is inv under the combined transform.

2) I have read ch21 regarding the quantum action, but I can't see what he means in ch78 about it being the sum of 1PI diagrams with external propagators replaced with fields, where is this shown in ch21? Why exactly is the quantum action not guage invariant? I know we had to fix a guage to do the path integral, but what does this have to do with quantum action not being guage inv?

3) Isn't (78.15) true all the time? by my argument above, and the quantum action is always invariant under $\delta_{G+BG}$...why did we have to set $A=\bar{A}$ in the quantum action?

4) In (78.22) I don't understand again why we are replacing ext props with external fields, but assuming it for now: and I presume he wants to just set $A=\bar{A}$ because of reasons expressed on previous page, how does setting $A=\bar{A}+\mathcal{A}$ achieve this?

OK I will leave it at those questions for now, and see how it goes...thanks for any help