# 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. – Willie Wong Jun 21 '12 at 9:00
• I added my final answer to the question – Prelude Jun 21 '12 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! – Willie Wong Jun 22 '12 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 – Prelude May 13 '14 at 4:04