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Chern-Simons and massive A^{\mu}

  1. Jan 7, 2012 #1
    Hi, I am struggling with a problem in field theory:
    We are looking at a Chern-Simons Lagrangian describing a massive A field:
    [itex]L = -\frac{1}{4} F^{\mu\nu}F_{\mu\nu}+\frac{m}{4}\epsilon^{\mu\nu \rho}F_{\mu\nu}A_{\rho}[/itex]
    I find those field equations:
    [itex]\partial_{\mu}F^{\mu\lambda}=-\frac{m}{2} \epsilon^{\lambda\mu\nu}F_{\mu\nu}[/itex] and now I need to show that F satisfies the Klein-Gordon equation: [itex](\Box+m^2)F_{\mu\nu}=0[/itex] using the EL equations, but after a time playing with both equations, I still can't prove that.
    Last edited: Jan 7, 2012
  2. jcsd
  3. Jan 7, 2012 #2


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    It's normal here to introduce the dual of the field strength

    [tex]\tilde{F}^\lambda = \frac{1}{2} \epsilon^{\lambda\mu\nu}F_{\mu\nu}.[/tex]

    Then taking the divergence of your equation of motion above yields [itex] \partial_\lambda \tilde{F}^\lambda = 0,[/itex] while multiplying by an appropriate epsilon yields the KG equation for [itex]\tilde{F}^\lambda[/itex].
  4. Jan 7, 2012 #3
    Thanks a lot. I did manage to find the Klein Gordon equation for the dual, then it's just a matter of applying another LeviCivita to find it for the "regular" form. ;)
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