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I learned that a quantum system has an "overall state", a vector in a Hilbert space. That Hilbert space can be decomposed in a basis of "basic states". For example if in the Universe there is only a free electron with its own spin, and I pick an axis $z$, there is an infinite basis of "position+spin", let's say $|(0,0,0);1/2>, |(0,0,0);-1/2>, |(0.1,0,0);1/2>\ldots$

I cannot obviously write all possible positions, but that was to give the idea. Some $\psi (\vec{r},s)$ contains the coefficients for all the elements of this basis and is given. This is the "overall state" of my "one-electron Universe"

I was said that there is "measurement" that makes the $\psi$ collapse to one of the "basic states" and can give the observer the result (position + spin in the $z$ axis)

Since a measurement involves an interaction with some sort of detector I figured out that I would need QFT (the possibility to have multiple particles and interaction) to understand measurement and add a detector into my Universe. I thought that the dynamic of the interaction between the particles of the detector and the electron could act in a way that the $\psi$ collapsed (I hoped that even in QFT there was something analogous to the original $\psi$ of the electron).

Now I couldn't study QFT but from what I grasped it is much more complicated and the reason of the collapsing of the wave function is not explained by an "interaction". The wave function itself is subjective and depends on the knowledge of the observer.

So I am lead to believe that QFT does not contain anything that explains the collapse of the wave function. Is it true?

I know that I presented a rather subjective view. I hope that they are shared by many and that an answer that points out my misunderstandings could help others. I think that many people in the quest for a more "classical" explanation of QM resort to thinking that when they would see how the interactions work "really" so as to describe a fully fledged measurement equipment (photons, QED, QCD...) they would figure that out.

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The main difference between QM and QFT w.r.t measurements etc. is the kind of question you ask. Usually in QFT you simply start your calculation assuming a plane wave of incident particles and determine the probablility of scattering in different directions.

You do this to learn something about the interactions of the particles, not about the state the particle is in. You infer the interactions that took place from classical measurements of momentum and Energy, not from any idealized quantum measurement.

In order to describe the evolution of one specific particle, you would need to convolute your results for plane wave states with the wave function of your particle. Still, once you have determined the state your particle ends with, the wave function collapses and this process is not described dynamically by the theory, but put in by hand.

tl/dr: In QFT the measurement process is still not described, but the state of the system is also not the object of interest.

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Wave function collapse works the same way in QFT as it does in any other quantum theory. If you get new information about the state of a physical system, you have to change the wave function to account for this information.

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