One of the main causes which leads to ill-defined loop integrals in Quantum Field Theory is that the variables of a Field Theory, $\varphi(x)$ for instance, are Quantum Fields which are governed by creation and annihilation operators which fulfill commutation relations which contain distributions
$$[ a(\mathbf{k}), a^\dagger(\mathbf{k}')] = \delta(\mathbf{k} - \mathbf{k}').$$
Therefore the Quantum Fields $\varphi(x)$ have to be considered as distributions and products of distributions, as they appear in the Lagrangian are not properly defined.
On the other hand when the path integral approach is used it is said (see Peskin& Schroeder, or Srednicki, or A.Zee's QFT in a nutshell) that the expression
$$Z(J) = \int D\varphi \exp{i \int d^4 x\left[\frac{1}{2}((\partial \varphi)^2 -m^2\varphi^2) +J\varphi\right]}$$
consists of ordinary fields which don't have operator character (they are not quantized), therefore do not contain distributions. However, upon evaluation the path integral of the free scalar theory as written above we already get (see A.Zee for instance):
$$Z(J) = {\cal{C}} \exp{ \left[-\frac{i}{2}\int \int d^4x d^4 y J(x) D_F(x-y) J(y)\right]}$$
where $D_F(x-y) $ is the scalar Feynman propagator which is well known to have distribution character.
How is it possible that from an apparent non-distributive approach (ansatz) one ends up with distributions? Even worse, for an interactive theory we get the whole series of Feynman-diagrams including those with loops which are infinite if they are not regularized.
So what is the underlying cause that also in the path integral approach we get results which only make sense if they are regularized? Is it the path integral measure $D\varphi$ which is mathematically not properly defined which leads to the same problems as those experienced in QFT based on second quantization?
