Loop contributions of the Quantum Action

[bobloblaw]

As far as I understand the tree level diagrams of the quantum action in QFT give the complete set of diagrams for a give process. Do the loop diagrams of the quantum action have any physical significance?

I thought maybe that by summing up all the diagrams in the quantum action might take us back to the "normal" action but after some thought this doesn't seem to be the case.

 

[kurros]

Hmm, not sure I totally understand, but certainly the tree level diagrams are NOT the complete set of diagrams for a given process. Perhaps you mean that processes that do not have a tree level diagram don't happen, and that loop diagrams just modify the allowed tree-level processes? This is also not true; there are many processes that are forbidden at tree level, but allowed at 1 or 2 or more loops. Here is an example:

http://upload.wikimedia.org/wikipedia/commons/8/82/Kaon-box-diagram-with-bar.svg

Kaon oscillations are forbidden at tree level (in the Standard Model) but they can proceed at the 1-loop level like so. Since one-loop diagrams are suppressed by a higher power of the couplings than tree level it makes such processes more rare, but they wouldn't happen at all if only the tree-level processes were allowed. Many many such things happen in nature.

Not sure if this answers your question.

 

[bobloblaw]

I think we might be talking about different things. I'm not talking about the tree level Feynman diagrams derived from the standard lagrangian but the feynman diagrams derived from the diagrams the quantum action or quantum effective action.

So for example in \phi^3 the quantum action reads

<br /> S = -\frac{1}{2}\int \frac{d^dk}{(2\pi)^d}\phi(-k)\left(k^2+m^2-\Pi(k^2)\right)\phi(k) <br /> + \sum^{\infty}_{n=3}\int \frac{d^dk_1}{(2\pi)^d}\cdots\frac{d^dk_n}{(2\pi)^d}(2\pi)^d\delta^d(k_1+\cdots+k_n) \times V_n(k_1,\cdots,k_n)\phi(k_1)\cdots\phi(k_n)<br />

As Srednicki says:
"The quantum action has the property that the tree-level Feynman diagrams it generates give the complete scattering amplitude of the original theory"

So my question is do the loop diagrams derived from the quantum action have any physical significance?

Your point is interesting though because I hadn't seen any processes that were forbidden classically although it makes sense.

 

[kurros]

Hmm I see. Well actually I haven't read Srednicki's book, but from what I quickly read of it on google books just now it does indeed seem like this "effective action" he is computing takes into account quantum corrections in some way such that you only need the tree diagrams it generates. This is a very strange concept to me though and I am not sure how it can be true. This formulation of field theory seems unusual; at least, I haven't seen it before, sorry. It looks like the path integral formulation but with a strange twist.