A problem in deriving the interaction field equation for photons

[Qubix]

Well, I was doing a problem the other day, inserting the complete interaction QED Lagrangian into the Euler-Lagrange equations, in order to obtain the field equation governing the interacting electromagnetic field, with the Dirac field. The problem is that, by doing so, I got the equation, but also a 1/4 factor in front. Any idea why this is ? (too lazy to latex the whole thing).

$${\frac{-1}{4}} (∂^{α} ∂_{α} A^{μ}) = e {\overline{\Psi}}{\gamma}^{μ}{\psi}$$

 

[Bill_K]

Probably because you wrote the Lagrangian as - (1/4)F_μνF_μν, and when you varied this with respect to A_μ,ν you forgot that it appears four times.

 

[buoyant]

and the answer will be of the form $\partial_\mu F^{\mu\nu} = ej^\nu$ the right hand side is called Dirac current which correspond to the source term of Maxwell's equation