The easiest way to understand the measurement problem is through a simple checklist of questions. Every presentation of any quantum “theory” uses a “wavefunction” to describe the physical state of a system. This is as true in QFT or quantum gravity as it is in basic quantum mechanics. Now we ask 2 questions:
Question 2 regards the evolution of the wavefunction (or quantum state) in time. Is that evolution always governed by Schrödinger’s equation (and is therefore linear, deterministic and unitary) or does it under some circumstances “collapse”? According to Bohmian mechanics, the wavefunction never collapses. According to a “collapse” or “objective reduction” theory the wavefunction does collapse. Theories with collapses make slightly different predictions than theories with no collapse. Most physicists deny that there is any real, physical collapse of the wavefunction.
If you assert that the wavefunction is complete and deny that it ever collapses, then you have a Many Worlds theory. In such a theory, for example, Schrödinger’s cat does not end up either just alive or just dead: rather, having started with one cat the experiment ends up with two cats: one alive and the other dead.
So there are really only three types of “interpretation of quantum theory” or better only three possible types of physics: “Hidden” variables, collapse, and Many Worlds.
Most physicists deny that they hold a hidden variables theory, and deny that they hold a collapse theory. and deny that they hold a Many Worlds theory. But that just means that most physics are confused. and QFT does not help this situation one bit.
My personal take (I call it “personal” because it may not represent the thoughts of a majority of physicists, though I am sure that at least a sizable minority share my view): It is an entirely artificial, apparent paradox. It arises because of how the measurement is envisioned. We envision a quantum system that just evolves in its own merry way, until suddenly, like some deus ex machina, the universe changes as the classical measurement apparatus appears out of thin air, changing the boundary conditions of the system. Yet we are surprised that suddenly, we have a discontinuous jump from a pre-measurement state that is a superposition of eigenstates to a post-measurement state that is an eigenstate with respect to the quantity being measured. This discontinuous jump is not described by the rules of quantum mechanics and it is not a unitary evolution of the system, so we have a problem on multiple fronts: the “measurement problem”.
Imagine instead making the classical measurement apparatus part of the system all along. Even if the quantum system that you are modeling is initially not interacting with the apparatus, the presence of the apparatus (e.g., represented by a Lagrange multiplier) is incorporated in the description in the form of a boundary condition. In this case, the wavefunction’s evolution will be unitary, even as it is confined to an eigenstate when it interacts with the apparatus. Of course it makes the theory manifestly non-local, since it implies that the wavefunction somehow “knows” about a future interaction with the apparatus as it evolves. But that should hardly come as a surprise, given what we know about Bell’s inequality, for instance: sure, quantum mechanics is manifestly non-local. But in this description, there is no wavefunction collapse, no non-unitary evolution, no “measurement problem”. And there is still no classical nonlocality: though the theory is manifestly non-local, it cannot be used to communicate classical information from the future to the past.
I think this is a little easier to conceptualize in QFT than in QM (QFT certainly helped in my case) but no, this does not mean that the “measurement problem” does not exist in QFT. That is to say, if we envision (incorrectly, in my opinion) the measurement by allowing the measurement apparatus to appear suddenly, “out of thin air”, we have a problem in both QM and QFT.
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