I'm not sure if this is the right place to post a graduate level course material,(adsbygoogle = window.adsbygoogle || []).push({});

but I have a question about perturbative expansion of the 2n-point function of a scalar field theory.

1. The problem statement, all variables and given/known data

First, the question:

In which space (position or momentum) is the topological distinctness of Feynman diagrams decided?

Why I'm asking:

I've been trying to calculate the symmetry/multiplicity factors for a Feynman diagram, but depending on which space (position or momentum) I use these rules, I get different symmetry factors.

eg. In the real-field case, the Feynman diagrams in position space have greater multiplicity than those in momentum space (since the arrows make each propagator distinct from its reverse).

Restatement of the question:

When finding the multiplicity of each Feynman diagram, should I stick to the symmetries I see in the position space (and ignore the momentum space arrows)?

2. Relevant equations

Lagrangian density for real (no charge) field:

[tex]\mathcal{L} = \frac{1}{2}(\partial^\mu \phi)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^2 - \frac{\lambda}{4!} \phi^4 + J\phi[/tex]

For complex (charged) field:

[tex]\mathcal{L} = \frac{1}{2}(\partial^\mu \phi^*)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^* \phi - \frac{\lambda}{4!} (\phi^* \phi )^2 + J\phi^* + J^* \phi[/tex]

BTW, this charged-field case has two sets of arrows (charge and momentum direction).

3. The attempt at a solution

This stuff is basically from Peskin&Schroeder (Ch.4.4), where it tells me to "divide by symmetry factor" in the last step of the Feynman rules. I wasn't sure the symmetry factors they're talking about are the same in both position and momentum spaces.

(and if they are, in which space I should count them)

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# Homework Help: [QFT] Feynman rules for self-interacting scalar field with source terms

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