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Phase and rescale transformation of k-g field

  1. Sep 20, 2014 #1

    ChrisVer

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    Gold Member

    We have the KG Lagrangian density:
    [itex] \mathcal{L}= \partial_{\mu} \phi \partial^{\mu} \phi^* - m^2 \phi \phi^*[/itex]

    Suppose instead of taking [itex]\phi \rightarrow e^{ia}\phi[/itex] with [itex]a \in \mathbb{R}[/itex] we take [itex]a \in \mathbb{C}[/itex].
    Then is it true to say that [itex]\mathcal{L}'= e^{-2~ Im(a)} \mathcal{L} [/itex] and so gives the same equations of motion?
    What is to happen if I also allow for local transformations [itex]a \equiv a(x)[/itex]? I guess this again depends on the KG Lagrangian:

    [itex] |\partial_\mu \phi|^2 \rightarrow \Big( e^{ia} \partial_\mu \phi + i e^{ia} \phi (\partial_\mu a) \Big)\Big( e^{-ia^*} \partial^\mu \phi^* - i e^{-ia^*} \phi^* (\partial^\mu a^*) \Big)[/itex]
    [itex]=e^{-2 ~Im(a)} \Big( \partial_\mu \phi \partial^\mu \phi^* + i \partial_\mu a (\phi \partial^{\mu} \phi^* -\phi^* \partial^{\mu} \phi) + \phi \phi^* \partial_\mu a \partial^\mu a \Big) = e^{-2 ~Im(a)} [D_\mu \phi) (D^\mu \phi)^*][/itex]

    The mass term is also going to give a [itex]e^{-2~ Im(a)}[/itex] as a common factor, so I can drop it out the Lagrangian? I think I cannot, because in this case the kinetic term of the E-L equations will act on this exponential as well...

    Where in this case the [itex]D_\mu[/itex] shall differ from the standard one, since it will have to give a complex vector field as a connection and additionally the vector's transformation is going to bring also some rescaling it it: [itex]A'_\mu \propto A_\mu - \partial(Im(a)) + i \partial(Re(a))[/itex]
    Is that correct?
     
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