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A question about Lorentz invariance for Klein-Gordon field

  1. Sep 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Hi everyone, in Peskin & Schroeder, P36, the derivative part of KG field is transformed as eqn (3.3). But why does the partial derivative itself not transform?

    2. Relevant equations

    [tex] \partial_{\mu} \phi (x) \rightarrow \partial_{\mu} ( \phi ( \Lambda^{-1} x) ) = ( \Lambda^{-1})^{\nu}_{\mu} (\partial_{\nu} \phi ) (\Lambda^{-1} x) [/tex]

    3. The attempt at a solution

    I thought the transformation was

    [tex] \partial_{\mu} \phi (x) \rightarrow \Lambda^{\nu}_{\mu} \partial_{\nu} ( \phi ( \Lambda^{-1} x) ) =
    \Lambda^{\nu}_{\mu}
    ( \Lambda^{-1})^{\lambda}_{\nu} (\partial_{\lambda} \phi ) (\Lambda^{-1} x) ? [/tex]

    Thank you~
     
  2. jcsd
  3. Oct 21, 2011 #2
    Peskin & Schroeder are using an active transformation (not passive) which is why the transformed field is [itex]\phi ( \Lambda^{-1} x)[/itex] and not [itex]\phi ( \Lambda x)[/itex].
    Only the field is transformed - not the coordinate system. This means the Lorentz transformation you introduce should not be there. I assume you put that in there because you believed the the partial derivative should transform.

    Hope this helps!
     
    Last edited: Oct 21, 2011
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