The Born rule -- 100 years ago and today

Question

Referee this paper: arXiv:2502.08545 by Arnold Neumaier

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

(Is this your paper?)

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Abstract: Details of the contents, and formulations of the Born rule changed considerably from its inception by Born in 1926 to the present day. Based to a large extent on little known results from the recent books 'Coherent Quantum Physics' by Neumaier and 'Algebraic Quantum Physics', Vol. 1 by Neumaier and Westra, this paper traces the early history 100 years ago, its generalization (essential for today's quantum optics and quantum information theory) to POVMs 50 year ago, and a modern derivation from an intuitive definition of the notion of a quantum detector. Also discussed is the extent to which the various forms of the Born rule have, like any other statement in physics, a restricted domain of validity, which leads to problems when applied outside this domain.

Comments

Haven't had a chance to read the paper yet, but from the contents I was curious what happened in 1970, and thereby encountered the claim that POVMs offer a generalization of the Born rule! This is quite interesting, I never heard this before (but then I haven't thought much about POVMs). This makes me curious about a lot of things:

Apparently weak measurements are described by POVMs. Long ago, Howard Wiseman tried to ground Bohmian mechanics in weak measurements, does this mean POVMs offer a new perspective on Bohmian mechanics too? 

Do POVMs have any special relationship to that other generalization of quantum mechanics, the Gell-Mann-Hartle-Omnes consistent/decoherent history? (edit: And how about Weinberg's generalization of QM, in which density matrices rather than state vectors are fundamental?) 

What about the measurement theory of relativistic QFT or string theory? What do POVMs look like there? How are they related to von Neumann algebras? (which have recently been of interest in quantum gravity)

Sorry for bringing up so many tangents! 

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@MitchellPorter: 

1. weak measurements: No - Bohmain: No 

2. decoherent history: Yes 

3. rel.QFT : no easy answer.

Please start new threads for the second and third question, so that there is some structure in the discussion.