Do somebody knows anything about the Dirac's identity?
\begin{equation} \label{Dirac} \frac{\partial^2}{\partial x_{\mu}\partial x^{\mu}} \delta(xb_{\mu}xb^{\mu}) = -4\pi \delta(xb_0)\delta(xb_1)\delta(xb_2)\delta(xb_3) \end{equation}
here
$xb$, is the 4-vector $x-b$ in Minkowsky spacetime
$\delta$ is the Dirac delta function
$x_0 = -x^0, \quad x_1 = x^1, \quad x_2 = x^2, \quad x_3, = x^3$.
Do you know where can i find some material about it?
Thanks!
UPDATE:
Following Willie's link
i've understood that a solution for the linear wave equation $$ \square \psi(\mathbf{r},t) = g(\mathbf{r},t) $$ for a given $g(\mathbf{r},t)$ is $$ \psi = \int \int g(r',t')G(r,r',t,t')dV'dt' $$ where $$ G(r,r',t,t') = AG^+(r,r',t,t') + BG^-(r,r',t,t') , \qquad A + B = 1 $$ and $$ G^{\pm}(r,r',t,t') = \frac{\delta(t' - (t \mp | \mathbf{r} - \mathbf{r'} | / c))}{4\pi | \mathbf{r} - \mathbf{r'} | } $$
I think Dirac's follow from the solution of
$$ \square \psi(r,t) = \delta(\mathbf{r},t) $$
But i'm not sure of the details. Can you Willie help me?
Thanks