Green function, determine $D(k)$ in $\,\,\,\,\, -(k^2-m^2)\left[ g^{\mu\nu}- \frac{k^{\mu}k^{\nu}}{k^{2}-m^{2}} \right]D(k)=i\delta^{\mu}_{\rho}$

Question

Given $g^{\mu\nu}=diag(1,-1,-1,-1)$ and $\delta^{\mu}_{\rho}$ the Kronecker delta.

I'm in the fourier space: $$-(k^2-m^2)\left[ g^{\mu\nu}- \frac{k^{\mu}k^{\nu}}{k^{2}-m^{2}} \right]D(k)=i\delta^{\mu}_{\rho}$$ I want to determine $D(k)$.

I make a hypothesis about the solution: $D(k)=Ak^{2}g_{\nu\rho}+Bk_{\nu}k_{\rho}$

How can I determine the coefficients A and B if I have only one equation and two coefficients.

Solutions. $$A=\frac{1}{k^2m^2-k^4} \,\,\,\,\,\,\, B=-\frac{1}{m^2(m^2-k^2)}$$

Answer 1

Since $$\left(g^{\mu\nu}-\frac{k^\mu k^\nu}{k^2-m^2}\right)\left(g_{\nu\rho}+Ck_\nu k_\rho\right)=\delta^\mu_\rho+k^\mu k_\rho\left(C\left(1-\frac{k^2}{k^2-m^2}\right)-\frac{1}{k^2-m^2}\right)=\delta^\mu_\rho$$simplifies to $\delta^\mu_\rho$ if $C=-\frac{1}{m^2}$,$$D(k)=-\frac{i}{k^2-m^2}\left(g_{\nu\rho}-\frac{1}{m^2}k_\nu k_\rho\right).$$