Green's function & Klein Gordon

[Sekonda]

Hello,

I was wondering what the use in the Green's function for the Klein-Gordon equation was, I have listed it below:

$$\int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2-m^2}e^{ip\cdot(x-x')}$$

We find this gives an infinite result when the Klein gordon equation is applied to it and if x=x', what does this mean, i.e.

$$(\frac{\partial^2 }{\partial t^2}-\nabla^2+m^2)\int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2-m^2}e^{ip\cdot(x-x')}=i\delta^{4})(x-x')$$

Does this mean the particle described by the green's function in the first equation can only be found in one place when x=x'?

Also if we set x=x', the Green's function above turns to

$$\int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2-m^2}$$

Which is alike to the 2 point function first order correction of form

$$-\frac{i\lambda}{2}\int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2-m^2}$$

What is the relationship here? What use are the Green's function to the above equation? p is the four vector :

$$p=(E,\mathbf{p})$$

Thanks,
SK

 

[Sekonda]

I'm now aware that the first equation in my previous post is also a position space propagator for the Klein-Gordon equation; would anyone be able to explain to me why for this diagram

The correction to the 2 point function for this first order can be given as an integral over the 3 position space propagators associated with (x',x'), (x,x') and (x',y)?

Along with the -iλ/2 from the vertex (/symmetry factor = 1/2)

Thanks