Fourier-transform of a delta-function of a square of the space-time interval What is the general framework for calculating expressions such as
$$
I = \int d^4x e^{-i(px)} \delta(x^2)
$$
where $x^2=x_0^2-\vec{x}^2$?
The problem here is that the delta-function carves out not a single point $x_\mu = 0$, but rather a sub-space $|x_0| = |\vec{x}|$.
Link to a paper or even a book would be appreciated.
 A: Recall the identity
$$
\delta(g(x)) = \sum_{i} \frac{\delta(x-x_i)}{\big|g'(x)|_{x=x_i}\big|}
$$
where $g(x_i) = 0$. Now the delta function in the integral above has the form
$$
\delta(x^2) = \delta(x_0^2 - \textbf{x}\cdot\textbf{x})
$$
The order of integration appears to be simplest if we integrate about $x_0$ first. So we rewrite the measure in the integral as
$$d^4x \rightarrow dx_0d^3x \rightarrow dx_0(r^2dr\;d\cos(\theta) d\phi)$$
so that $$\textbf{x}\cdot \textbf{x} = r^2$$ and
$$e^{-ip\cdot x} = \exp(-i(p_0x_0 - |\textbf{p}|r\cos(\theta)).$$ We now have something that we can work with
$$
I = \int_{-\infty}^{\infty}dx_0\int_0^{\infty}dr\;r^2 \int_{1}^{-1}d\cos(\theta) \int_{0}^{2\pi}d\phi\;\delta(x_0^2-r^2)\exp(-i(p_0x_0 - |\textbf{p}|r\cos(\theta))) \\
= 2\pi\int_{-\infty}^{\infty}dx_0\int_0^{\infty}dr\;r^2\int_1^{-1}d\cos(\theta)\;\times \frac{\delta(x_0-r) + \delta(x_0+r)}{2r} e^{-ip_0x_0}e^{i|\textbf{p}|r\cos(\theta)} \\
= 2\pi \int_0^{\infty}dr\;r^2 \int_{1}^{-1}d\cos(\theta)\frac{1}{r}\cos(p_0r)e^{i|\textbf{p}|r\cos(\theta)}
$$
I hope you can take it from here.
