- #1

- 300

- 22

## Main Question or Discussion Point

I was reading about the classical Klein-Gordon propagator here: https://en.wikipedia.org/wiki/Propagator#Relativistic_propagators

Basically they are looking for ##G##, that solves the equation

$$(\square _{x}+m^{2})G(x,y)=-\delta (x-y).$$

So they take the Fourier transform to get

$$\left(-p^{2}+m^{2}\right)G(p)=-1.$$

Then they say that we need to add an ##i\epsilon## and when we take the inverse Fourier transform we get

$${\displaystyle G(x,y)={\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{p^{2}-m^{2}\pm i\varepsilon }}~.}$$

I don't quite follow the derivation. First I would like to ask if the Fourier transform even makes sense when it has poles in it and we can't do an inverse Fourier transform to get back the original function? Also how can we just add ##i\epsilon##, maybe it will change the result significantly? In the Wiki article it goes on to talk about how the sign of ##\epsilon## gives different propagators. So ##\epsilon## has a significant effect, how can we just add it?

Basically I am asking for a more rigorous derivation of the propagator, why the Fourier transform still works when poles are involved and where exactly the ##\epsilon## comes from.

Basically they are looking for ##G##, that solves the equation

$$(\square _{x}+m^{2})G(x,y)=-\delta (x-y).$$

So they take the Fourier transform to get

$$\left(-p^{2}+m^{2}\right)G(p)=-1.$$

Then they say that we need to add an ##i\epsilon## and when we take the inverse Fourier transform we get

$${\displaystyle G(x,y)={\frac {1}{(2\pi )^{4}}}\int d^{4}p\,{\frac {e^{-ip(x-y)}}{p^{2}-m^{2}\pm i\varepsilon }}~.}$$

I don't quite follow the derivation. First I would like to ask if the Fourier transform even makes sense when it has poles in it and we can't do an inverse Fourier transform to get back the original function? Also how can we just add ##i\epsilon##, maybe it will change the result significantly? In the Wiki article it goes on to talk about how the sign of ##\epsilon## gives different propagators. So ##\epsilon## has a significant effect, how can we just add it?

Basically I am asking for a more rigorous derivation of the propagator, why the Fourier transform still works when poles are involved and where exactly the ##\epsilon## comes from.