There are several reasons given in the literature, why UV infinities arise in QFT in the first place. My problem is putting them together, i.e. understand how they are related to each other. So... UV divergences arise and thus we need to renormalize, because: We have infinite number of degrees of freedom ín a field theory. (From this perspective, the infinites seem inevitable.) We multiply fields to describe interactions, fields are distributions and the product of distributions is ill-defined. We neglect the detailed short-wavelength structure of scattering processes, and the infinites are a result of our approximations with delta potentials. (From this point of view, the UV divergences aren't something fundamental, but merely a result of our approximation method. ) We are dealing with non-self-adjoint Hamiltonians. (This is closely related to the 3. bullet point. From this perspective an alternative to the "awkward" renormalization procedure would be the "method of self-adjoint extension".) Are these reasons different sides of the same coin? And if yes, how can we understand the connection between them?