Question

I have been told that a quick way to get the Feynman rules from a Lagrangian is to take an interaction term, forget about the fields and multiply an i. This works perfectly for example for QED but I wonder if there is an analogous non rigorous quick way to obtain the feynman rules of for example non abelian gauge theories.

Answer 1

This works perfectly for example for QED

No, it doesn't. You can get the interaction vertex, but the photon propagator is still ill-defined. In order to get everything up-and-working you have to introduce an additional gauge-fixing term in the Lagrangian:

Only with this additional (unphysical and non-gauge-invariant) term can you obtain Feynman rules by 'looking' at the Lagrangian. There is a beautiful mathematical approach (Faddeev-Popov method), which is usually used to show that this gauge-fixing term can be added.

No to the N/A gauge theories. First, because the curvature is now modified by an additional Lie algebra commutator

we get two extra terms in the Lagrangian

which give us two interaction vertices (3-valent and 4-valent).

The propagator is ill-defined, though. Performing the Faddeev-Popov procedure, we get the gauge-fixing term (just like in the Abelian case), but we also get new dynamical fields -- Faddeev-Popov ghosts. These are unphysical and can be thought of as aspects of specific gauge fixing.

They interact with gauge bosons via the 1-boson + 2-ghost interaction vertex. More details (including expressions for propagators and vertices) can be found in almost any QFT textbook (e.g. Peskin-Schreder).