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I've always seen the standard interpretation and axioms of QM as in some way tricky on a philosophical level. They state the collapse of the wave function is caused by the measurement.

3.b If $A$ is an observable with eigenvalues $a_k$ and eigenvectors $|k\rangle$ $(A|k\rangle = ak|k\rangle$), given a system in the state $|\psi\rangle$, the probability of obtaining $a_k$ as the outcome of the measurement of $A$ is $p(a_k) = |\langle k | \psi \rangle|^2$. After the measurement the system is left in the state projected on the subspace of the eigenvalue $a_k$ (wave function collapse)

source: https://ocw.mit.edu/courses/nuclear-engineering/22-51-quantum-theory-of-radiation-interactions-fall-2012/lecture-notes/MIT22_51F12_Ch3.pdf, page 15

I've known the general consensus is that the collapse of the wave function is a manifestation of a more general process called decoherence. So in the standard interpretation, the wave function seems to collapse due to the environment, but it really collapses due to the measurement. Maybe because in the first experiments of the physicists of the 20th century the measurement apparatus caused the collapse of the wave function, so it might have seemed the best way to define those axioms..

Why don't we use the decoherence process as an axiom somehow? In this way, we might avoid some misinterpretation of quantum mechanics and have a deeper understanding of the subject.

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    $\begingroup$ "Wave function collapse" is not part of QM. It is only part of some interpretations of QM (in particular, the Copenhagen interpretation). The fact that this interpretation is used in a lot of pop-science writing about QM doesn't make it an essential part of QM - to quote David Mermin, "just shut up and calculate!" Note: AFAIK there is no so-called "standard interpretation" of QM - it works perfectly well as a theory of physics with no "interpretation" at all. $\endgroup$
    – alephzero
    Commented Sep 18, 2018 at 21:24
  • $\begingroup$ For what it's worth, I'm a physicist and I think about quantum mechanics exactly as you've suggested here. The "conventional" way of talking about state collapse is old-fashioned and obviously no longer sufficient to elegantly explain the experiments we've done in modern times. $\endgroup$
    – DanielSank
    Commented Sep 19, 2018 at 4:24
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    $\begingroup$ @alephzero Do away with the word "collapse" if it bothers you that much but the standard fact of physics that "after the measurement, the system is left in the state projected on the subspace of the eigenvalue $a_k$" is a part of Quantum Mechanics (in fact, lies at the very heart of it). And the OP is simply asking whether this fact can be derived as a theorem if we somehow turn the decoherence process into an axiom (if and however it is possible). And in all honesty, for a process in which something suddenly reduces to one of its components, collapse seems a pretty good name to me. $\endgroup$
    – user87745
    Commented Sep 19, 2018 at 5:53
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    $\begingroup$ Also, as far as I am aware, in order to actually shut up and calculate, there is only one way to do it which can actually make correct predictions--it is the standard QM available in all the widely accepted textbooks--all of which necessarily involve the axiom stated by the OP. And really, the so-called Copenhagen interpretation is the only one that comprises of precisely these axioms and nothing more or less--it is the standard QM. All the other versions either try to reduce or add axioms--making their version necessarily something (at least) just slightly different than the standard QM. $\endgroup$
    – user87745
    Commented Sep 19, 2018 at 6:00
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    $\begingroup$ "I've known the general consensus is that the collapse of the wave function is a manifestation of a more general process called decoherence." No, sorry, that's really wrong. Decoherence says that the dead and the alive cat cannot interfere any more. It does not tell you that only one is present. Without the collapse postulate, it is unclear why a measurement has a unique result. $\endgroup$
    – Luke
    Commented Sep 20, 2018 at 15:15

2 Answers 2

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Many people have many different ways of stating the axioms of quantum mechanics, including some formulations that look nothing like the typical presentation in a freshman physics textbook. Not worrying too much about completeness and perfect rigor, a pretty standard way of formulating such a set of axioms would be:

  1. Wavefunctions exist in a Hilbert space.

  2. The time evolution of the wavefunction is unitary.

There is nothing about Copenhagen or wavefunction collapse, and nothing like that is needed. If you want wavefunction collapse, you're free to add that as an axiom. For a treatment in this style, including optional Copenhagen stuff, see Carroll and Sebens, "Many Worlds, the Born Rule, and Self-Locating Uncertainty," https://arxiv.org/abs/1405.7907 .

Decoherence is not an additional hypothesis to be added on to the standard axioms of quantum mechanics. Decoherence simply results from the axioms. For comparison, the standard axioms of arithmetic imply that 2+2=4. You could add an axiom saying 2+2=4, and the system would still be consistent, but the additional axiom would be superfluous.

An interesting paper on this type of thing is Allahverdyan, Balian, and Nieuwenhuizen, "A sub-ensemble theory of ideal quantum measurement processes," 2017, https://arxiv.org/abs/1303.7257 . They make a toy model in which you can see decoherence and pick out processes that look very much like the wavefunction collapse of the Copenhagen interpretation. One thing that comes out naturally is that there are various time scales involved. Since the Copenhagen interpretation is normally described in terms of instantaneous collapse, one possible point of view is that Copenhagen is just an approximation, which can fail to hold in the real world.

The most austere versions of the Many-Worlds interpretation ("MWI-lite") are immune to this kind of attack, since they only posit postulates 1 and 2 above, which everyone agrees on anyway. (This is the point of view advocated by Carroll and Sebens.) In this point of view, we never even talk about things like "worlds" or "branching" of worlds. If, on the other hand, you want a more baroque version of MWI ("MWI-heavy") in which we talk about these things, then MWI-heavy is probably only an approximation to the exact results of standard quantum mechanics, for the same reasons that CI can't be expected to be exact.

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    $\begingroup$ IMO it is highly inappropriate to never mention the serious issues with the Everrettian version of QM (the MWI-lite)--in particular, there is no definite scientifically agreed upon success in deriving the Born rule from the Everrettian version. To wit, Steven Weinberg just very recently commented that no one has really succeeded yet in deriving the Born rule in this version: youtu.be/mBninatwq6k?t=12m55s $\endgroup$
    – user87745
    Commented Sep 19, 2018 at 5:30
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So my question is: why don't we use the decoherence process as an axiom somehow?

The third axiom you quote is necessary in relating the mathematics to real numbers measurable in the laboratory. It is responsible for describing the atomic spectra, one of the main reasons quantum mechanics had to be invented.

Decoherence might be shaped into an axiom, with too much mathematics, not a one to one relation to observables. It is a "theorem", and as in pure mathematics a theorem can become an axiom and the axiom than becomes a theorem proven from the axioms, but one chooses the simplest form for axioms.

Imo the term "collapse" is unfortunate, turning the wavefunction into a balloon. It just means that an instance has been sampled from a probability distribution. The term "collapse" tries to describe, is an unfortunate shorthand, for the fact that after a measurement, the boundary conditions of the quantum mechanical problem have changed, a new wavefunction will describe the system, not the old one. If a particle decays, the decay products are described by a new wavefunction. If an excited atom emits a photon, ditto. To "observe" at a particle level, means to "interact" and that introduces new boundary conditions.

So imo the parenthesis should not have been there, as it carries too many misconceptions.

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  • $\begingroup$ The point of the question (obviously) seems to know whether the quoted third axiom can be derived without explicitly assuming it to start with if we instead take decoherence as an axiom. In response to that, this answer simply adds that decoherence might be shaped into an axiom but is ambiguous as to whether the quoted third axiom will follow from that as a necessary consequence or not. $\endgroup$
    – user87745
    Commented Sep 19, 2018 at 5:36
  • $\begingroup$ Decoherence--if made an axiom--would necessarily be a statement about physics and thus would relate maths to the real world. Thus, in principle, one might expect that the quoted third axiom can follow from that. This answer doesn't address this point at all--which should be the core part of an answer to the question asked IMO. $\endgroup$
    – user87745
    Commented Sep 19, 2018 at 5:38
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    $\begingroup$ @DvijMankad All I am saying is that "if decoherence is a mathematical effect of using the quantum mechanical equations consistently, with the axiom as it is, decoherence could be made an axiom and the third postulate turned into a theorem" in a very complicated mathematically way I am not willing/able to explore. The way in euclidian geometry axioms and theorems can be interchanged. The third axiom as it stands is about physics, it is about the spectra of atoms. $\endgroup$
    – anna v
    Commented Sep 19, 2018 at 5:46
  • $\begingroup$ Decoherence is a mathematical fact about solutions of the Schrödinger equation, and it makes very little sense to posit that this should now become an axiom. $\endgroup$
    – Luke
    Commented Sep 20, 2018 at 15:16
  • $\begingroup$ @Luke iThat is the point of my answer, that one uses as axioms the simplest mathematical forms $\endgroup$
    – anna v
    Commented Sep 20, 2018 at 16:41

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