What type of regulation is being employed?

Question

As already mentioned in this post. In the context of QFT, the kernel of integration for the overlap of a field configuration ket, $| Phi rangle$ with the vacuum $|0rangle$ in a free theory is given by (See: S. Weinberg's Vol. 1 QFT Foundations Ch. 9.2)
$$	mathcal{E}({bf x},{bf y}) = frac{1}{(2pi)^3} int {rm d}^3{bf p}, e^{i{bf p}cdot({bf x}-{bf y})}sqrt{{bf p}^2 + m^2}tag{1}label{eq:kernel}$$
which can be shown to algebraically match the following expression by abusing the Basset Integral for index $nu = -1$.
$$mathcal{E}({bf x},{bf y}) = frac{1}{2pi^2} frac{m}{r} frac{rm d}{{rm d} r} left( frac{1}{r} K_{-1}(m r) right)quad text{for}quad |{bf x - y}| = r tag{2}label{eq:kernel2},$$
where $K_{-1}$ denotes a modified Bessel function of the second kind. It is clear that the integration in Eq.~eqref{eq:kernel} is divergent, while Eq.~eqref{eq:kernel2} is not, so some sort of regularization happened in between these steps. Does anybody know which technique one could use to formalize the relation between the two?

Valter Moretti · Accepted Answer

If you want to use the standard Fourier transform of functions instead of distributions (see @AccidentalFourierTransform comment), you can regularize as follows,
$$mathcal{E}({bf x},{bf y}, t_x,t_y) := mathcal{E}({bf x},{bf y}, t,t)$$
for
$$mathcal{E}({bf x},{bf y}, t_x,t_y) = w-lim_{epsilon to 0^+} frac{1}{(2pi)^3} int {rm d}^3{bf p}, e^{i[{bf p}cdot({bf x}-{bf y}) - p^0(t_x-y_y-iepsilon)]}sqrt{{bf p}^2 + m^2}$$
where $p^0:= sqrt{{bf p}^2 + m^2}$ and $w$ denotes the weak limit: first integrate against a smooth compactly support (or Schwartz) function of $({bf x},{bf y}) in mathbb{R}^6$ and next take the limit. Using that procedure one sees that it is equivalent to directly integrate the smooth function against the integral kernel in the right-hand side of (2).   Hence (2) is an identity in the sense of distributions in ${cal S}'(mathbb{R}^6)$.

What type of regulation is being employed?

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