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As far as I know, there are two ways of constructing the computational rules in perturbative field theory.

The first one (in Mandl and Shaw's QFT book) is to pretend in and out states as free states, then calculating $$\left\langle i \left| T \exp\left(-i \int_{-\infty}^{\infty} H_{int} dt \right) \right| j \right\rangle $$ by the Wick theorem, blah blah blah. The problem is, field/particle always has self interaction, in and out states are not free states. Mandl and Shaw (rev. edi. p 102) then used a heuristic argument, that assuming the interacting is adiabatically switched on, $$H_{int}(t) \rightarrow H_{int} (t) f(t)$$
such that $f(t) \rightarrow 0$, if $t \rightarrow \pm \infty$.

One may regard this is hand-waving. In certain circumstance, such as the Gell-Mann Low theorem http://en.wikipedia.org/wiki/Gell-Mann_and_Low_theorem , the adiabatically switching can be proved even non-perburbatively.

The second approach, e.g. Peskin and Schroeder's QFT, is to start from correlation function, then use LSZ reduction to connect S-matrix and correlation function. In the correlation function, an epsilon prescription of imaginary time is used. $$ | \Omega \rangle = \lim_{T \rightarrow \infty ( 1 - i \varepsilon ) } ( e^{-iE_0 T} \langle \Omega | 0 \rangle^{-1} ) e^{-iHT} | 0 \rangle $$

where $\Omega$ and $0$ are the vacua of interacting and free theories, respectively.

My question is about comparing these two approaches. It seems to me, at the end of the day, that the final results of calculations carried out using both approaches are identical. One may say, LSZ reduction is more physical, since there is no switch on/off in nature. One may also say, time is a real number in nature. There is no imaginary time anyway. And adiabatic switching has potentially advantage in non-perturbative aspect.

Is there any deeper reasoning for comparing these two approaches? I am sorry if this is opinion-based question.

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The in- and out-states are free states, and the S-matrix definition of Mandl and Shaw is perfectly valid (with an appropriate notion of Texp). It is the one used in rigorous mathematical physics; see the treatise by Reed and Simon. It is also the one from which the LSZ formula is derived. It is the only way to define the S-matrix rigorously.

The $+i\epsilon$ prescription also has a rigorous justification through Cauchy's theorem, which allows one to rewrite certain integrals by deforming the integration path into the complex domain. and represent square integrable functions as boundary values of functions in a complex Hardy space.

The approximations made are instead in the way one justifies the use of the S-matrix to describe the effect of collisions at finite times. Here the idea is that the particles (more precisely the bound states of the Hamiltonian after separating out the center of mass motion) behave approximately free for a sufficiently short time (smaller than that to move a mean free path length), and the collision time is short compared to that case. If these conditions are satisfied it is justified to approximate the particles by free in-going particles (thought of coming from infinity), treating the collision as one extending from time $-\infty$ to time $+\infty$, resulting in free out-going particles at infinite times, that can be walked back (by the Moeller operator) to one at finite time significantly larger than the collision time but significantly shorter than needed to move a free mean path length.

Making this rigorous needs additional assumptions (e.g., conditions matching those in the collision area of a particle accelerator, or a dilute gas assumption), in which case one can derive error bounds.

Whenever these conditions are not satisfied the particle picture breaks down completely, due to Haag's theorem which says that the interacting Hilbert space cannot be the same as the asymptotic Hilbert space. Instead, one has to use a more heuristic quasiparticle (dressed particle) picture, as in solid state physics for phonons.

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