Understanding how to derive the Feynman rules out of the path integral

[nrqed]

Well, once we include tadpoles, $7$ more contributions to $\mathcal{O}(\lambda^4)$ arise (just as one more arose for $\mathcal{O}(\lambda^2)$). Source: Srednicki QFT, chapter 9

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Good! One comment: in particle physics, people often omit the tadpoles. The reason for this is that they always disappear once one renormalizes. The reason is that the integrals over the loop momenta (that flow in the tadpoles) do not contain the external momenta. In the examples here, we then get that the tadpoles are just (divergent ) factors multiplying the two point functions, factors which are independent of the external momenta. After renormalization, these terms get canceled completely. However, when working in condensed matter systems (and at T not equal to zero), these tadpoles do not cancel and are therefore important.

Anyway, I know this is a technical point but I thought I would point it out.

[Moderator's note: An additional off topic comment has been deleted.]

 

[JD_PM]

I am not sure if you are still interested in getting the rules from the path integral, but the key point is that when you square you will have to apply
\begin{align*}
\frac{(i)^2}{4}\int d z \, dw \, dx \, dy \,\left(\frac{\delta}{\delta\phi(z)}\right)\Delta_F(z-w)\left(\frac{\delta}{\delta\phi(w)}\right)
\left(\frac{\delta}{\delta\phi(x)}\right)\Delta_F(y-x)\left(\frac{\delta}{\delta\phi(y)}\right)
\end{align*} on the exponential.

Please let us discuss the up-to-second-order result

\begin{align*}
&\exp\left(\frac{i}{2}\int d^d x \int d^d y \frac{\delta}{\delta \phi(x)}\Delta_F(x-y) \frac{\delta}{\delta \phi(y)}\right)\times \exp\left(i \int d^d z \left( -\frac{\lambda}{3!}\phi^3+J\phi \right)\right)\Big|_{\phi=0} \\
&=1+\frac{i}{2}\int d^d x \int d^d y\,(i(J(x))\Delta_F(x-y)(iJ(y)) + \\
&-\lambda \int d^d z \int d^d w \Delta_F (0) \Delta_F (z-w) \\
&-\frac{1}{4}\int d^d x \int d^d y \int d^d z \int d^d w \Delta_F(x-y) \Delta_F(z-w) J_x J_y J_z J_w + \mathcal{O}(\Delta_F^4)
\end{align*}

I conceptually understood that there are three diagrams for the $2-$function.

My question now is: how could we establish the Feynman rules out such result so that we can obtain such three diagrams algorithmically/systematically?

 

[JD_PM]

Hi @vanhees71 , I just wanted to ping you because I thought you might be interested in #33

 

[vanhees71]

Hi @vanhees71 , I just wanted to ping you because I thought you might be interested in #33

I think I've to do the calculation myself first. That needs a bit of time. I'm not sure whether this approach is in any way simpler than the standard approach though.