I'm studying perturbation theory in QFT and I stumbled on a conceptual problem.

My understanding of the interplay between LSZ reduction formula and the Gell-Mann & Low perturbation series is that:

The LSZ formula (disregarding for a moment the normalization issue) allows to write the matrix elements of the S-matrix in terms of correlation functions, which are Vacuum expectation values of time ordered product of fields. Namely: $$\langle \Omega|T[\phi(x_1)\phi(x_2)...\phi(x_n)]|\Omega\rangle$$

This vacuum, $\Omega$, to my understanding is the vacuum of the interacting theory.

Such correlation functions can be found by using Gell-Mann & Low formula which relates them to vacuum expectation values of incoming fields, namely:

$$\langle \Omega|T[\phi_{IN}(x_1)\phi_{IN}(x_2)...\phi_{IN}(x_n) \exp(-i\int H_1^{IN}(z)d^4z)]|\Omega\rangle$$

Again, for how I saw this formula derived on my teacher's notes and on Bjorken Drell's book this looks to be the same vacuum as before, the vacuum of interacting theory.

Am I right? Does this matter at all?

Now comes my issue: why do we use Wick theorem to express products of $\phi_{IN}$ fields in terms of normal ordered products? After all these are free fields and there should be no hope for the VEV to be zero on normal ordered products.

My guess, which looks to me nothing more than a desperate hand waving, is that those incoming fields are made to be equal to the real interacting fields at $-\infty$ and so it doesn't make sense to distinguish the two vacuum as eventually our interacting fields will coincide with the incoming ones at infinite times, which is where we set up the experiment and where we measure the outcome.

It seems to me that mine is a really poor understanding of the matter and I would be grateful if someone would help me to make it clear.

I'm studying perturbation theory in QFT and I stumbled on a conceptual problem.

My understanding of the interplay between LSZ reduction formula and the Gell-Mann & Low perturbation series is that:

The LSZ formula (disregarding for a moment the normalization issue) allows to write the matrix elements of the S-matrix in terms of correlation functions, which are Vacuum expectation values of time ordered product of fields. Namely: $$\langle \Omega|T[\phi(x_1)\phi(x_2)...\phi(x_n)]|\Omega\rangle$$

This vacuum, $\Omega$, to my understanding is the vacuum of the interacting theory.

Such correlation functions can be found by using Gell-Mann & Low formula which relates them to vacuum expectation values of incoming fields, namely:

$$\langle \Omega|T[\phi_{IN}(x_1)\phi_{IN}(x_2)...\phi_{IN}(x_n) \exp(-i\int H_1^{IN}(z)d^4z)]|\Omega\rangle$$

Again, for how I saw this formula derived on my teacher's notes and on Bjorken Drell's book this looks to be the same vacuum as before, the vacuum of interacting theory.

Am I right? Does this matter at all?

Now comes my issue: why do we use Wick theorem to express products of $\phi_{IN}$ fields in terms of normal ordered products? After all these are free fields and there should be no hope for the VEV to be zero on normal ordered products.

My guess, which looks to me nothing more than a desperate hand waving, is that those incoming fields are made to be equal to the real interacting fields at $-\infty$ and so it doesn't make sense to distinguish the two vacuum as eventually our interacting fields will coincide with the incoming ones at infinite times, which is where we set up the experiment and where we measure the outcome.

It seems to me that mine is a really poor understanding of the matter and I would be grateful if someone would help me to make it clear.

EDIT: I just read for the first time about Haag's theorem and I suspect my question may be tangentially related to it.