Flow of charge on fermion propagator

[spaghetti3451]

The momentum-space fermion propagator in the free Dirac theory is given by

The arrow on the fermion propagator is said to represent the flow of charge.

How can we derive this statement quantitatively from the Dirac Lagrangian?

What is the quantitative form of the charge being referred to here?

 

[Vanadium 50]

How can we derive this statement quantitatively from the Dirac Lagrangian?

You can't. The arrow is a mnemonic device so you get the right wavefunctions on the external lines. The propagator doesn't care if its for a fermion or an antifermion, but the better mnemonic is "make sure the arrows follow a path" and not "make sure the arrows follow a path, except where you don't need to".