Dirac's identity

Question

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

Answer 1

Your identity is in fact the expression for the fundamental solution of the linear wave equation in (1+3) dimensions. This should be in most textbooks on electrodynamics or intro to quantum field theory.

Google also tells me:

• http://www.math.ucsd.edu/~lindblad/110b/l18.pdf
• http://www.math.ucla.edu/~ralston/266b.1.08w/Funwave.pdf
• page 15 of http://www.math.princeton.edu/~seri/homepage/courses/Analysis2008.pdf