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Why must all operators in QM be linear (and therefore able to be represented by matrices). What is the physical reasoning behind this?

Is it be possible that the non-unitary nature of quantum collapse could be described physically by a non-linear quantum theory where the coupling of a system to a measuring device (composed of a large (thermodynamic limit) number of particles) 'massages' the state into one of the observable eigenstates. (i.e. the large number of particles in the measuring device make any state but the observable eigenstates statistically impossible to be 'massaged' into)

I am (obviously) assuming that these suggestions are incorrect, but I think I could learn a lot about the nature of QM by understanding exactly why.

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  • $\begingroup$ Non-linear generalizations of QM have been explored, most notably in the work of Weinberg. The ultimate reason why this idea has failed is that it just doesn't work, or hasn't been working so far. Scientific theories are judged by their predictive power, which nonlinear QM models do not have. Perhaps at some point in the future we'll discover that some aspect of nature is best described by a generalization of QM to nonlinear QM, but this hasn't happened so far. $\endgroup$ Dec 16, 2021 at 15:55
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    $\begingroup$ What sort of answer do you expect to this question? What answer do you have for the classical (Hamiltonian) case to "Why are all physical quantities functions on the phase space?" $\endgroup$
    – ACuriousMind
    Dec 16, 2021 at 15:56
  • $\begingroup$ @Prof.Legolasov Is there a rigerous reason (e.g. from Bell's inequalities) that you can't have a physical collapse theory like this (which doesn't require a spontaneous collapse, just a coupling of a system to a large system)? This seems like such an obvious first thought, that I'm assuming it must have been ruled out rigerously $\endgroup$
    – Alex Gower
    Dec 16, 2021 at 15:58
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/1201/2451 and links therein. $\endgroup$
    – Qmechanic
    Dec 16, 2021 at 15:59
  • $\begingroup$ That is an interesting linked question, although it doesn't completely answer my question. Is the answer then simply 'it is possible that quantum collapse could be explained purely by non-linear dynamics without a spontaneous collapse time but experimental evidence currently does not support this'? $\endgroup$
    – Alex Gower
    Dec 16, 2021 at 16:04

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