Formalization (and meaning) of Heisenberg Cut In discussions around a recent  Stack answer which covered axiomatic quantum mechanics, Bell's Theorem and Random Fields (related to axiomatic QFT) the phrase "Heisenberg Cut" was used. The basic question is what does this mean?
Having reviewed this discussion I can provide some background to set a context for the question. In (Von Neumann) QM a distinction is made between :  OBSERVER [Cut] QUANTUM OBJECT
This simplistic distinction has been critized later in that article and elsewhere such that the term "Heisenberg Cut" has fallen into disuse. If one wishes to make distinctions in a QM experiment then here is a larger proposal:
ENVIRONMENT [Cut1] QUANTUM OBJECT [Cut2] CLASSICAL INSTRUMENT [Cut3] OBSERVER [Cut4] O-O
here O-O is any hypothetical observer-observer as discussed in the "Wigner's Friend" concept.
So has the concept of "Heisenberg Cut" been re-introduced in modern axiomatic QM/QFT and if so what is its formalisation (ie how can one make experimental or Bell Theorem-like deductions from it?)
 A: The term "Heisenberg cut" has only been used in the philosophical discussions about the character of quantum mechanics. It's the classical-quantum boundary that should be placed somewhere in between the observed quantum objects and observer's perceptions.
It's never a mistake to place the cut closer to the observer - to treat a larger set of phenomena using the machinery of quantum mechanics because at the end, quantum mechanics applies everywhere. On the other hand, it could be a mistake to treat some systems or their properties classically.
The "minimal" location of the Heisenberg cut - one that treats a maximum fraction of the world classically - may be calculated by decoherence. In modern quantum mechanics, decoherence is what defines the classical-quantum boundary - but the term "Heisenberg cut" is rarely used for this boundary in modern physics.
A: I take the heart of the Heisenberg cut to be the way we calculate probabilities and expected values in QM. For elementary QM, for some measurement described by an operator $\hat M$, the expected value in a state described by a density matrix $\hat\rho$ is given by the trace $\left<\!M\right>=\mathsf{Tr}\left(\!\hat M\hat\rho\!\right)$. What we put in the $\hat\rho$ is what is in our model universe. The measurement operator describes our measurement apparatus, which is not in the model universe, but instead describes how our measurement apparatus gets information out of the model universe. There's an almost-symmetry between the ways $\hat M$ and $\hat\rho$ appear; it's almost as if there's a measurement apparatus universe as well as the model universe. Different measurement apparatuses can affect each other in the measurement apparatus universe without changing the model universe, which is called measurement incompatibility (did I just create an interpretation of QM? Do I know this one? I guess it's too glib, sadly.)
Edit: We can extend the mathematics in many different ways, but one deserves mention because it's of great practical value. We can introduce transformations $\hat T_i$ that operate between the preparation apparatus and the measurement apparatus, in which case we have $\left<\!M\right>=\mathsf{Tr}\left(\!\hat M\hat T_n\cdots\hat T_2\hat T_1\hat\rho\!\right)$. We can at any time say that $\hat M\hat T_n\cdots\hat T_5$, say, or any other part of this list (without changing the order), is our measurement.
Anyway, to some extent we can move stuff from the model universe into the measurement apparatus universe and vice versa, although we may have to get into technical stuff like POVMs to do it. Once we go to quantum field theory, there's a tight relationship between the measurement apparatus universe and the model universe, because we use the same Lego blocks to build measurements and states.
The separation into states and measurements is absolutely fundamental in QM. It's how the relationship between Hilbert spaces and experimental results works, which causes trouble when people want to do cosmology, with everything in the model universe. Although I hadn't previously thought about calculating where one should put the separation, I can see that if one chooses a particular accuracy that one wants one's model of an experimental apparatus to achieve relative to one's real apparatus, that might put a limit on where one can put the Heisenberg cut. I'm not sure, however, that one can't always improve the sophistication of one's description of a measurement, particularly if one is willing to go to POVMs. I suppose, however, that putting people inside your model universe is always going to be in the realm of toy models. The separation into states and measurement famously comes under the microscope in Bell's article `Against "Measurement" '.
Incidentally, I see you went to Willem de Muynck, who is perhaps a little idiosyncratic, but I've often found his a good counterpoint of view.
A: in addition to what Lubos said about QM, the philosophy of axiomatic QFT is to construct/describe QFTs without any reference to concepts of classical physics. As far as I know there is not even a concept for a "classical limit" in axiomatic QFT. Especially "macroscopic classical" devices like detectors are modeled in AQFT via an observable, that is a selfadjoint operator, and that is a pure quantum concept. Therefore I don't think that you'll find a formalization of something like the "Heisenberg cut" in axiomatic QFT.
As for the interpretation of the measurement process etc., this is usually left to the philosophical interpretation of QM, from a pure philosophical point of view there is no conceptual difference of the interpretation of QM and that of QFT, which is the reason why most people working on this topic concentrate on the technically much simpler QM.
