On page 74 of Timo Weigand's QFT notes, right at the top, the following equality is used:
$$\left[(2\pi)^4\delta^{(4)}(p_f-p_i)\right]^2=(2\pi)^4\delta^{(4)}(p_f-p_i)(2\pi)^4\delta^{(4)}(0) \tag{2.167},$$
however I cannot figure out why this should be the case. I can't even see where to begin, could someone more enlightened please explain this step?
The square of the Dirac delta distribution does not make mathematical sense, cf. e.g. this Phys.SE post.
However, physicists often implicitly imply that the spacetime is a finite (large) box. Then momentum eigenvalues become discrete, integrals are replaced by sums, Dirac delta distributions are replaced by Kronecker delta functions, etc., cf. e.g. this Phys.SE post. In this regularization, one can make sense of $\delta(0)$ and eq. (2.167).
