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Math Help - Klein gordon equation

  1. #1
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    Klein gordon equation

    I am being really thick here

    I have this wave equation, the massless klien gordon equation

    \partial_{\mu}\partial^{\mu}\phi(x)=0

    where the summation over \mu is over 0,1,2,3

    the general solution is a superposition of plane waves yes? i.e

    \phi(x)=\int d^4 p \overline{\phi}(p)exp(i p_{\mu}x^{\mu})

    where \overline{\phi} is the weighting function.

    When you susbsitute this back into the klein gordon equation you get down two factors of p, i.e

    p_{\mu}p^{\mu} which equals zero. (mass shell constraint)

    My question is, is \overline{\phi}(p) arbitrary? I don't really understand why this is so, let alone believe it.

    Hope peeps understand the question.
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  2. #2
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    Quote Originally Posted by ppyvabw View Post
    I am being really thick here

    I have this wave equation, the massless klien gordon equation

    \partial_{\mu}\partial^{\mu}\phi(x)=0

    where the summation over \mu is over 0,1,2,3

    the general solution is a superposition of plane waves yes? i.e

    \phi(x)=\int d^4 p \overline{\phi}(p)exp(i p_{\mu}x^{\mu})

    where \overline{\phi} is the weighting function.

    When you susbsitute this back into the klein gordon equation you get down two factors of p, i.e

    p_{\mu}p^{\mu} which equals zero. (mass shell constraint)

    My question is, is \overline{\phi}(p) arbitrary? I don't really understand why this is so, let alone believe it.

    Hope peeps understand the question.
    The K-G equation is linear and plane waves (with the mass shell constraint) are solutions to it. Therefore arbitrary linear combinations of plane waves are solutions to it. Therefore \overline{\phi}(p) is arbitary.
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