quantum field theory - How can we derive the Feynman rule for the ordinary QED 3-vertex?

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?

Answer

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.