Second Order Correction to the Perturbative Approximation of the Transition Amplitudes in RQM I am studying Relativistic Quantum Mechanics from my professor's notes. When calculating the second order perturbative correction to the transition coefficient $T_{fi}$* in a scattering process by a potential of finite duration, the following term appears
$$\sum_{n\neq i}\int_{-\infty}^\infty dt e^{i(E_f-E_n)t}\int_{-\infty}^tdt'e^{i(E_f-E_n)t'}.$$
My professor notes that the $dt'$ integral does not make sense unless we interpret it as an expression of the form
$$\lim_{ε\to 0^+}\int_{-\infty}^tdt'e^{i(E_f-E_n-iε)t'}.$$
I understand why the integral expression does not make sense as well as why the interpretation does, but I would really appreciate any physical reasoning for acting so, any indication as to what might gone wrong in the formulation of the derivation**
and more than anything, any mathematical arguments for why this is alright (for example something indication that this is the only way of giving meaning to such troublesome expressions).
*With this I mean the long time value the coefficient of the the f state of a wave function which purely on the i state before the potential is turned on.
**Which is not presented here, but I hope that it is a standard one and that it is more or less given away (to someone familiar with the subject) by the notation and the results. If not, please ask me to clarify.
 A: I struggle with this concept as well, although I have found some comfort in casual functions.
You're given
$$\int_{-\infty}^{t}dt' e^{i w t'} =\int_{-\infty}^{\infty}dt' \theta(t-t') e^{i w t'} = e^{i w t}\int_{-\infty}^{\infty}dt' \theta(-t') e^{i w t'}  = e^{i w t} \tilde{f}_{-}(w)$$
which calls for the fourier transform of a theta function, which I'm preemptively calling $\tilde{f}_{-}(w)$, to be defined below. In classical analysis such an integral does not exist, so I will appeal to distribution theory, defining both $f(t') = 1$ and $f_{-}(t')=\theta(-t') f(t')$, where $f_{-}(t')$ is called the advanced part of $f(t')$. Causality comes in to play when splitting functions into advanced and retarded parts, which have poles exclusively above and below the real axis, respectively.
Then take a laplace transform
$$F[z] = \int_{-\infty}^{\infty}dt' e^{i z t'} f_{-}(t')$$
which has nicer convergence properties due to the imaginary frequency $z$. Since $f_{-}$ is causal, one can show the relation
$$\lim_{\epsilon \to 0^+} F[w - i\epsilon] = \tilde{f}_{-}(w)$$
where the limit is a distribution limit and I have not bothered tracking overall signs. From here
$$e^{i w t} \tilde{f}_{-}(w) = e^{i w t}\int_{-\infty}^{\infty}dt' e^{i (w-i\epsilon) t'} f_{-}(t') = \int_{-\infty}^{\infty}dt' e^{i (w-i\epsilon) t'} f_{-}(t'-t) = \int_{-\infty}^{\infty}dt' \theta(t-t') e^{i (w-i\epsilon) t'}$$
which is the desired result given $w = E_f - E_n$. Hopefully this offers some insight into how what your professor has does is more than just inserting a convergence factor and is in fact necessary in order to preserve the causal structure of your theory.
