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A problem in deriving the interaction field equation for photons

  1. Dec 19, 2013 #1
    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).

    [tex] {\frac{-1}{4}} (∂^{α} ∂_{α} A^{μ}) = e {\overline{\Psi}}{\gamma}^{μ}{\psi} [/tex]
     
  2. jcsd
  3. Dec 19, 2013 #2

    Bill_K

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    Probably because you wrote the Lagrangian as - (1/4)FμνFμν, and when you varied this with respect to Aμ,ν you forgot that it appears four times.
     
  4. Dec 19, 2013 #3
    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
     
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