Green's function for Klein-Gordon equation in x-space

[RedSonja]

I'm trying to derive the x-space result for the Green's function for the Klein-Gordon equation, but my complex analysis skills seems to be insufficient. The result should be:


\begin{eqnarray}
G_F(x,x') = \lim_{\epsilon \rightarrow 0} \frac{1}{(2 \pi)^4} \int d^4k\frac{e^{-ik\cdot(x-x')}}{k^2-\frac{m^2c^2}{\hbar^2}+i\epsilon}
= \left(-\frac{1}{4\pi} \delta (s^2) + \frac{mc^2}{8 \pi \hbar s} H_1^{(1)} \left(\frac{mc^2}{\hbar} s \right) \right) \:\Theta (s)
- \frac{m}{4\pi^2 s} K_1 \left(i\frac{mc^2}{\hbar} s \right) \:\Theta(-s))\\
\end{eqnarray}

with $s^2 = c^2(t-t')^2-(\vec{x}-\vec{x}')^2$, the source point $x'$, $\Theta(s)$ the Heaviside stepfunction, $H_1^{(1)}(x)$ the Hankel function of the first kind, and $K_1(x)$ the modified Bessel function of the second kind.

I changed to spherical polar coordinates and did the $\phi$ and $\theta$ integrals first. From there I've tried several approaches, but I always end up with a complex integral of a multivalued function:

\begin{equation}
\int \frac{f(z)}{\sqrt{z^2 \pm a^2}} dz
\end{equation}

From residues I get zero when $f(z)$ is an exponential function, but that can't be right?

Both my quantum field theory and many-particle books work in k-space and I haven't found the derivation here.

Would someone be kind enough to guide me throught the correct procedure for the different steps of the Fourier transform of the Feynman propagator?

Thanks for your time!

 

[andrien]

Have you tried greiner quantum electrodynamics.I think the x-space green function is derived there(but perhaps for spin 1/2 but that will lead you the way)

 

[Bill_K]

It ain't easy!

G(x,x') = (2π)^-4∫e^ip·(x-x')/(p² + m²-i0) d⁴p

Making use of the integral identity

∫₀^∞ e^-i(ξ-i0)s ds ≡ -i/(ξ-i0)

we obtain

G(x,x') = i/(2π)⁴ ∫₀^∞ ds ∫exp{-i[(p²+m²-i0)s - p·(x-x')]} d⁴p

Now complete the square in the exponent and use the Gaussian integral,

∫_-∞^∞ e^iax2 dx ≡ √(π/a) exp{(i a/|a|)(π/4)}

G(x,x') = (4π)^-2∫₀^∞ s^-2 exp{-i[m²s - (x-x')²/4s]}ds

From there, in a few more steps it reduces to Schlafli's integral representation of the Bessel function.

 

[RedSonja]

Cheers, Bill_K!
I just knew there had to be a neat trick for this!

 

[schieghoven]

There may be some useful comments amongst this fairly ancient thread

https://www.physicsforums.com/showthread.php?t=217846

(probably my own posts in it were too terse to be much help.) There were some references mentioned which might be more helpful. Bogoliubov & Shirkov definitely does it. There was also a little book by Feynman that had them in an appendix.

 

[Bill_K]

Thanks for calling attention to that! In fact, the 3rd Edition of "The BS Book" can be read online here.

Basically what they do is to reduce the Green's functions to a different integral representation of the Bessel functions, namely integrals of the form ∫_-∞^∞ e^{iz sinh t} dt.