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Unity tensor

  1. Jan 11, 2009 #1

    Imagine a spatially 2d world. The electromagnetic field could be richer here, because you could add to the Lagrangian L an additional term (known as the Chern-Simons Lagrangian)

    [tex]L_{CS} = \epsilon_{0}\frac{\kappa}{2}\epsilon^{\alpha \beta \gamma}(\partial_{\alpha}A_{\beta})A_{\gamma}[/tex]​


    [tex]\epsilon^{\alpha \beta \gamma}[/tex]

    denotes the completely antisymmetric unity tensor in a world with 2 spatial dimensions (and time) and [tex]\kappa[\tex] is a coupling constant. In order to be suitable as part of the total Lagrangian, [tex]L_{CS}[\tex] must be a Lorentz scalar. Explain why the Chern-Simons expression is indeed a scalar. Why is the action generated by [tex]L_{CS}[\tex] guage-invariant?

    3. The attempt at a solution

    I must admit, I'm rather confused by this one, and haven't done much work with the unity tensor before (I've only just begun playing with tensors, really).

    I was hoping to learn something by trying this problem, but haven't got anywhere with it yet, and any help would be greatly appreciated.
  2. jcsd
  3. Jan 11, 2009 #2
    that sounds like an interesting problem. Firstly, it's obvious that L is either a scaler or pseudo-scaler. You just need to rule out that last possibility. Parity transformation maps
    [tex]u_0 \rightarrow u_0[/tex]
    [tex]u_1 \rightarrow -u_1[/tex]
    [tex]u_2 \rightarrow -u_2[/tex]

    it suffices to show that L is invariant under this transformation. As for gauge invariance, this is pretty much the transformation:
    [tex]A_\alpha \rightarrow A_\alpha + \partial_\alpha V[/tex]
    for scaler V. If L changes by a divergences under this transformation, you get gauge-invariance.
  4. Jan 12, 2009 #3
    Thank you :-)

    Would I be right in thinking that a covariant dot product can be formed in the above (and these are invariant) - another way of looking at it (?)

    Regarding the guage invariance, are you proposing simply to substitute that transformation into the above expression and see if I can kill off all the [tex]\partial_\alpha V[/tex] terms, somehow (?)


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