- #1
bartadam
- 41
- 0
I am being really thick here
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.
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.