I have checked some Quantum Field Theory texts that include basic QED and they all include the Feynman rule that each vertex bring with it a factor of $$\pm i e \gamma^\mu$$ but I have yet to find a derivation of this rule. How can we start from the interaction term in the Lagrangian, $-e \bar{\psi}\gamma^\mu A_\mu \psi,$ and derive this rule?
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1$\begingroup$ In the time-dependent perturbation theory, we get $i$'s because we expand $e^{-iV_{I}t}$ in a Taylor series. $i$'s appearing in Feynman rules have exactly the same origin. $\endgroup$– higgsssDec 14, 2013 at 10:58
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1$\begingroup$ @higgsss I am interested in a derivation of the entire contribution $\endgroup$– Kevin DriscollDec 14, 2013 at 11:00
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1$\begingroup$ It is really the $i$ from $\exp(-iV_{int}t)$, or in the language of the path integral, the $i$ from $\exp(i\int d^{4}\mathcal{L_{int}})$. Any QFT textbook should have the derivation in where it introduces the interaction picture, perturbation theory, and eventually Feynman rules. $\endgroup$– higgsssDec 14, 2013 at 11:11
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2$\begingroup$ @higgsss As I mentioned, I consulted two textbooks before asking and while both had the rule, neither has a real derivation. They just pluck the rule out of the air. Of course these rules come from the action in the path integral and thus also from the interaction Lagrangian, but how, exactly? $\endgroup$– Kevin DriscollDec 14, 2013 at 11:14
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$\begingroup$ What books did you consult? Most QFT books do have a derivation. Could it be that you went directly to the parts about QED, while the reasoning behind diagrammatic perturbation theory where explained earlier in simpler settings (such as scalar $phi^3$ or $phi^4$ theory)? Are you ok with the derivation in those simpler situations? If so, you should explain what the problem with QED is, if not, check out those first. $\endgroup$– fqqDec 15, 2013 at 15:53
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1 Answer
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Try, for instance, section 9 of Srednicki. The way to do it is to replace the fields in the interaction Lagrangian by functional derivatives with respect to the sources, then write power series for the exponents. Take the first order contribution.
Then, use that you need to consider three-point functions where the fields are again replaced by functional derivatives.
Finally, you work out all these derivatives (either explicitly or making use of diagrammatic techniques) and see what numerical factor you end up with. Note that you'll have to use Grassman variables in your path integral because you're dealing with fermions.
