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Expansion of the solution if interacting KG equation

  1. Oct 1, 2015 #1

    ShayanJ

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    Gold Member

    The exponentials ##\phi_p(x)=e^{ipx} ##(where ## px=p_\mu x^\mu##), are solutions of the free KG equation ## (\partial_\mu \partial^\mu+m^2) \phi =0##. I can expand the solutions of the interacting KG equation ## (\partial_\mu \partial^\mu+m^2)\psi=V\psi ## in terms of solutions of the free KG equation ## \psi(x)=\int e^{ipx} \pi(p) d^4p ##.
    But because there is an interaction, it seems to me that the evolution of ## \psi(x) ## should be different from a free field and so I think the function ##\pi## should also be a function of time. But because we want to think covariantly and also because it seems a natural generalization to let the field have different expansion coefficients in different events of space-time, I think we should let ## \pi ## depend on all four coordinates, i.e. we should write ## \psi(x)=\int e^{ipx} \pi(p,x) d^4p ##
    Is this the right way of thinking about it?
    Thanks
     
  2. jcsd
  3. Oct 2, 2015 #2
    It seems an advanced level question requiring tensors.
    Please make the question simple if possible!
     
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