[QFT] Feynman rules for self-interacting scalar field with source terms

[iibewegung]

I'm not sure if this is the right place to post a graduate level course material,
but I have a question about perturbative expansion of the 2n-point function of a scalar field theory.

Homework Statement

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)?

Homework Equations

Lagrangian density for real (no charge) field:
$$\mathcal{L} = \frac{1}{2}(\partial^\mu \phi)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^2 - \frac{\lambda}{4!} \phi^4 + J\phi$$

For complex (charged) field:
$$\mathcal{L} = \frac{1}{2}(\partial^\mu \phi^*)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^* \phi - \frac{\lambda}{4!} (\phi^* \phi )^2 + J\phi^* + J^* \phi$$
BTW, this charged-field case has two sets of arrows (charge and momentum direction).

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)

 

[turin]

First I will say that I never really did understand these symmetry factors. But, since no one else wants to attack this, maybe you would like to discuss. As far as the arrows that you're talking about, you shouldn't have them for an uncharged field; I don't believe that this has anything to do with whether you are drawing them in momentum space or position space.

The way I think of Feynman diagrams is that they are sort of in Limbo between position and momentum space. On the one hand, we all love to go to momentum space, because our friends the plane waves live there. However, any vertex in your diagram actually represents a position. You get "a momentum conserving delta" because you "sum over all positions" for these vertices, but, when you draw the diagram, you just draw a blob for one typical point in space time for each vertex. This also happens to external points that become external lines, but for some reason no one puts blobs at the end of their external lines. Actually, I know the reason. Because your piece of paper isn't infinitely long, and those points are in the (approximately) infinite past and future.

BTW, I think P&S have a very unsatisfactory way of getting at the Feynman rules, even when they derive them from the action. Unfortunately, I can't think of one source that does a good job in itself. Have you looked at Srednicki?

 

[olgranpappy]

I'm not sure if this is the right place to post a graduate level course material,
but I have a question about perturbative expansion of the 2n-point function of a scalar field theory.

Homework Statement

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

either.

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).

no. the arrows don't matter.

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)?

yes.

Homework Equations

Lagrangian density for real (no charge) field:
$$\mathcal{L} = \frac{1}{2}(\partial^\mu \phi)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^2 - \frac{\lambda}{4!} \phi^4 + J\phi$$

For complex (charged) field:
$$\mathcal{L} = \frac{1}{2}(\partial^\mu \phi^*)(\partial_\mu \phi) - \frac{1}{2} m_0 ^2 \phi^* \phi - \frac{\lambda}{4!} (\phi^* \phi )^2 + J\phi^* + J^* \phi$$
BTW, this charged-field case has two sets of arrows (charge and momentum direction).

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)