I'm following Peskin, but don't understand how to rewrite position-space feynman diagram in momenum-space.

Suppose we are in $\phi^4$-theory and $\phi$ is a real scalar field. It would be instructive with an example. I suppose its possible to:

  • go from the analytical expression in position space to the analytical expression in momentum space
  • go from the feynman diagram in position space to the analytical expression in momentum space.

Can you show how to do this step by step? I can offer a feynman diagram to save time: Some feynman diagram from Peskin And here's the analyitical expression

$$ -\frac{(i \lambda)^2}{2 (4!)^2} \int d ^4 z\, \int d^4 w\, D_F(x-z) D_F(y-w) D_F(z-w)^3 $$


When you're proving it for the first time you apply the following identities by substitution: $$\begin{align}D_F(x-y) &= \int \operatorname{d}^4p \frac{\operatorname{e}^{-i p (x - y)}}{4\pi^2} \left(\frac{1}{p^2-m^2 + i\epsilon}\right)\\ \delta^4(x-y) &= \int \operatorname{d}^4p \frac{\operatorname{e}^{-i p (x - y)}}{4\pi^2}. \end{align}$$ The $i\epsilon$ is somewhat arbitrary, and gives the Feynman propagator.

Note that you can do this process at the level of the action/Lagrangian density integral, before you derive Feynman diagrams, short circuiting a lot of re-proving the same identities for different Feynman diagrams. Doing that requires using the substitution: $$\phi(p) \equiv \int \operatorname{d}^4 x \frac{\operatorname{e}^{ipx}}{4\pi^2} \phi(x),$$ in addition to the delta function one above.

When you get further along in your career you go straight to the momentum space propagators where each line gives a momentum space propagator (the quantity in the parentheses above), and the net 4-momentum flowing into each vertex is zero (conservation of momentum delta functions that arise from the space integrals of the exponentials).

  • $\begingroup$ It'd be great if you could walk through step by step, as requested:) $\endgroup$
    – Mikkel Rev
    Oct 8 '16 at 12:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.