Multiplying distributions for QFT My understanding is that UV divergences arise due to improperly handling the product of distributions. In what sense is it "improper"? And how does its proper handling relate to the notion of renormalization freedom?
 A: Unlike functions, distributions don't have a multiplication everywhere.
To define where we can multiply, first we must define wavefronts. The wavefront of a distribution $s$ at a point $p$ is all the directions in which the Fourier expansion of the distribution does not decay exponentially. We write it as
$$Wv(s)[p].$$
Then two distributions $s,t$ multiply at a point when
$$(Wv(t)[p])' = - Wv(s)[p].$$
This is known as Hormander's Criteria. Here the apostrophe is the set complement.
Then handling:

*

*The multiplication of distributions properly via the analysis of their wavefronts

*The extension of the distribution to the interaction point via the analysis of their scaling degrees

leads to a well-defined rigorous theory with renormalisation freedom. This for example is used in Locally Covariant pQFT (perturbative QFT).
A: It means that the slightest anomaly in the convergence of Integral could cause the divergence(It can be through energy from an infinitesimal distance) which should be renormalized and regularized to have the resolution of the ultraviolet completion(now it is finite).
