# How to find the Green's Functions for time-dependent inhomogeneous Klein-Gordon equation?

I'm trying to find the Green's functions for time-dependent inhomogeneous Klein-Gordon equation which is : \begin{align*}‎‎ \left[ -‎ ‎\nabla ‎^2 + ‎‎‎‎\frac{1}{c^2} ‎‎\dfrac{\partial ^2}{\partial t^2} +‎ ‎‎‎\kappa ‎^2 ‎‎\right] ‎‎\psi(‎{‎\mathbf{r},t )}‎ = ‎‎‎‎\rho‎(‎\mathbf{r},t‎)‎ \end{align*}

It has been mentioned in the question that I can find the Green's functions : \begin{align*}‎ ‎‎&G_R(‎\mathbf{r} , t , ‎\mathbf{r'} , t') = ‎\dfrac{c}{8 \pi ^2 ‎\mathbf{R} i }‎\dfrac{d}{d‎\mathbf{R}} ‎\int_{- ‎\infty‎}^{+‎\infty‎} ‎‎\dfrac{e^{ i ‎\frac{R}{c} ‎‎\sqrt{q^2 - k^2 c^2}‎‎}}{‎\sqrt{q^2 - k^2 c^2}‎} ‎e^{ - iq (t - t')} ‎dq‎ \\‎ &G_A(‎\mathbf{r} , t , ‎\mathbf{r'} , t') = ‎-\dfrac{c}{8 \pi ^2 ‎\mathbf{R} i }‎\dfrac{d}{d‎\mathbf{R}} ‎\int_{- ‎\infty‎}^{+‎\infty‎} ‎‎\dfrac{e^{- i ‎\frac{R}{c} ‎‎\sqrt{q^2 - k^2 c^2}‎‎}}{‎\sqrt{q^2 - k^2 c^2}‎} ‎e^{ - iq (t - t')} ‎dq‎ \end{align*}‎ using the fourier transform, but when I use the fourier transform I don't gain the proper answer. The fourier transform which I use is the one which is generally given as : ‎\begin{align*}‎ ‎f(r) = ‎\dfrac{1}{‎\sqrt{2 \pi}‎} ‎\int_{- \infty}^{\infty} ‎e^{ik.r}‎\hat{f}‎(k) ‎dk ‎‎ \end{align*} but from this transform I cannot find $G_A$ and $G_R$.

Is there another transform which I should use to find the Green's functions?

Edit The Green's function which I wind up with is : ‎\begin{align*}‎‎ G_A(‎\mathbf{r} , t , ‎\mathbf{r'} , t')‎ = ‎‎‎‎\dfrac{1}{(2\pi)^4} ‎\int ‎d^3\mathbf{k} ‎dk' ‎‎\frac{1}{k^2} ‎e^{i\mathbf{k}.(‎\mathbf{r} - ‎\mathbf{r'}‎‎)}e^{ik'(t-t')}‎ \end{align*} which is not even similar to the answer given here!

• At a first glance the advanced and retarded functions look roughly correct. What is the result of your computation? And perhaps showing work may allow us to pinpoint where the problem is. Jun 21, 2012 at 9:00
• I added my final answer to the question Jun 21, 2012 at 16:11
• what you wrote down has no dependence on the mass $\kappa$, so it definitely not right. I think what you are missing is that after you take the Fourier transform in space-time, you should try to solve the equation $$( |\mathbf{k}|^2 -\frac{\tau^2}{c^2} + \kappa^2)\hat\psi(\mathbf{k},\tau) = 0$$ It looks to me that you just forgot that the mass term should be factored in. Also: be careful with your $\kappa$ and $k$! They are two different letters! Jun 22, 2012 at 9:18

I'll try to point out things I'm seem that are not fine.

• You are making confusion between $k$ and $\kappa$.
• There is no problem with your convention for fourier transform. It's only a little unconventional in this kind of problem (based on the books I used)
• You have to integrate! Note that the integral you found has $d^3{\bf k} dk'$ you must solve it. The common path is to integrate first $k'$, then put the integral in spherical coordinates and solve for the angles. By the end solve the radial integral.

Have you tried first to solve the $\kappa=0$ case? If no, you should do that.

I'll not reproduce here what's said in some books, instead I'll let you with some sources:

• I think the better reference is Hassani book on Mathematical Physics (Sec. 22.4.4), things are clearer there. The problem is that he doesn't solve the $\kappa \neq 0$ case.
• Another great reference (and which solves your problem) is Eleftherios Economou's book Green's Function on Quantum Mechanics Sec.2.2 (Look page 31, eq 2.63 for your specific case).
• Other references are Bogoliubov Shirkov &15 (which is canonical but a little bit confusing, in my opinion) and
• Morse Feshbach Chap 7.
• really useful. Tnx May 13, 2014 at 4:04