Derive the Feynman rules for for a complex scalar field.
[itex]L=\partial_\mu\phi^\dagger\partial^\mu\phi +m^2\phi-\lambda/4 |\phi|^4[/itex]
The Attempt at a Solution
I wrote the generating functional for the non-interacting theory
[itex]Z_0[J]=Z_0exp(-\int d^4xd^4yJ^\dagger (x) J(y) D_F(x-y)[/itex]
And I think I can use this to calculate the correlation functions directly, I just don't understand exactly how the presence of antiparticles change the Feynman diagrams/rules. I guess charge has to be conserved at all vertices, but I don't explicitly see that condition (I see overall charge conservation). Is this the only change in the Feynman rules? The propagators for both seem the same, and each vertex still gives [itex]-i\lambda\int d^4z[/itex].
These pictures contribute to different 4 point functions, but do they contribute the same term to their respective sums? Also, does the presence of anti particles change the calculation of symmetry factors?