Why Shift ##z_0## by ##-i\epsilon## in Non-Convergent Integrals?

[Jufa]

Homework Statement

I encounter a divergent integral when computing a commutator of two fields in quantum field theory homework

Relevant Equations

$\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})$

I am asked to compute $[\phi(x), \phi^\dagger(y)]$ , with
$\phi = \int \frac{dp^3}{(2\pi)^3}e^{-ipx}\hat{a}(\vec{p})$ and with z=x-y a spacelike vector. And show that this commutator does not vanish, which means that for this non-relativsitic field i.e. with $p^0 = \frac{\vec{p}^2}{2m}$ causality is violated.

After two straightforward steps I get to this

$[\phi(x), \phi^\dagger(y)] = \int \frac{dp^3}{(2\pi)^3}e^{-i \frac{\vec{p}^2}{2m}z_0}e^{i\vec{p}\cdot\vec{z}}$

It is self-evident that this integral does not converge and, therefore, I have been suggested to shift $z_0$ by adding to it a quantity $-i\epsilon$. This makes the integral convergent and allows one to take the limit $\epsilon \rightarrow 0$ in the end, which works perfectly.
But why are we allowed to do this? To me it seems that the integral I am asked to compute is just (I would end the problem here and say that it is not zero, just what I was demanded to prove) divergent and that this trick allows to compute not the requested integral but a different one.

I need some help on this.

Thanks in advance.

 

[MathematicalPhysicist]

I believe this trick is justified in Distribution theory, a mathematical theory which was invented by Laurent Schwartz.
https://en.wikipedia.org/wiki/Distribution_(mathematics)