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I have this wave equation, the massless klien gordon equation

[tex]\partial_{\mu}\partial^{\mu}\phi(x)=0[/tex]

where the summation over [tex]\mu[/tex] is over 0,1,2,3

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

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

where [tex]\overline{\phi}[/tex] is the weighting function.

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

[tex]p_{\mu}p^{\mu}[/tex] which equals zero. (mass shell constraint), thus satisfying the equation of motion.

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

Hope peeps understand the question.