# Klein gordon equation

• May 15th 2008, 09:57 AM
ppyvabw
Klein gordon equation
I am being really thick here

I have this wave equation, the massless klien gordon equation

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

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

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

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

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

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

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

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

Hope peeps understand the question.
• May 16th 2008, 03:38 AM
mr fantastic
Quote:

Originally Posted by ppyvabw
I am being really thick here

I have this wave equation, the massless klien gordon equation

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

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

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

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

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

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

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

My question is, is $\displaystyle \overline{\phi}(p)$ 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 $\displaystyle \overline{\phi}(p)$ is arbitary.