Assorted questions about Srednicki ch78

[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 [itex] \delta_G \bar{A^a_{\mu}}(x)=0[/itex] and [itex] \delta_{BG} \bar{A^a_{\mu}}(x)=-\bar{D^{ac}_{\mu}\theta^c(x)}[/itex], am I correct in thinking that under the combined transformation we have just [itex] \delta_{G+BG} \bar{A^a_{\mu}}(x)=-\bar{D^{ac}_{\mu}}\theta^c(x)[/itex]? If so then I can see that [itex] D_{\mu},\bar{D_{\mu}} [/itex] will have the same transformation under [itex] \delta_{G+BG} [/itex]. I guess then for his argument to show the invariance of the ghost Lagrangian under this combined transformation it doesn't matter that the D's are different (D, D bar) so long as they transform the same way under the transformation you might as well 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 gauge invariant? I know we had to fix a gauge to do the path integral, but what does this have to do with quantum action not being gauge inv?

3) Isn't (78.15) true all the time? by my argument above, and the quantum action is always invariant under [itex]\delta_{G+BG}[/itex]...why did we have to set [itex]A=\bar{A} [/itex] 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 [itex]A=\bar{A} [/itex] because of reasons expressed on previous page, how does setting [itex] A=\bar{A}+\mathcal{A} [/itex] achieve this?OK I will leave it at those questions for now, and see how it goes...thanks for any help

 

[eljose79]

Hi there,

I'll try my best to answer your questions about ch78 of Srednicki's QFT: background field gauge.

1) Yes, you are 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). This is because the combined transformation is simply the sum of the individual transformations. And you are also correct in thinking that the invariance of the ghost Lagrangian under this combined transformation does not depend on the specific form of the D's, as long as they transform the same way. This is because the ghost Lagrangian only depends on the transformation properties of the fields, not on the specific form of the transformations.

2) In ch21, Srednicki defines the quantum action as the sum of all 1PI diagrams with external propagators replaced with fields. This is shown in Eq. (21.10) where the quantum action is written as a sum of all diagrams with external propagators replaced by fields. The reason why the quantum action is not gauge invariant is because we had to fix a gauge in order to do the path integral. This means that the path integral is only invariant under the gauge transformations that leave the gauge fixing condition unchanged. Any other gauge transformation will change the path integral, and therefore the quantum action.

3) Yes, (78.15) is true all the time. This is because it represents the gauge transformation of the background field, which is always invariant under \delta_{G+BG}. The reason why we set A=\bar{A} in the quantum action is because we want to eliminate the redundant degrees of freedom in the path integral. This is achieved by choosing a specific gauge fixing condition, which in this case is A=\bar{A}.

4) In (78.22), we are replacing external propagators with external fields because we want to eliminate the redundant degrees of freedom in the path integral. By setting A=\bar{A}+\mathcal{A}, we are essentially choosing a specific gauge fixing condition that eliminates these redundant degrees of freedom. This is explained in more detail in the previous pages.

I hope this helps clarify your questions about the background field gauge. Let me know if you have any further questions or need more clarification.