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    The separable Hilbert space formulation of quantum mechan... — Carmelics
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    The separable Hilbert space formulation of quantum mechanics is unsatisfactory.

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Von Neumann's spectral theorem requires rigged Hilbert spaces (Gel'fand triples) to handle continuous spectra rigorously, revealing separable Hilbert space as insufficient.
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    • 2.Dirac's bra-ket formalism—indispensable to quantum mechanics practice—is formally inconsistent within separable Hilbert space, requiring distributional extensions beyond it.
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    • 3.The mathematical framework physicists actually use presupposes structures (delta functions, plane waves) that only become rigorous in the larger rigged Hilbert space setting.
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    Reason for 2 of 2
    ?
    • 1.Superselection rules partition quantum state spaces into inequivalent sectors that cannot be represented within any single separable Hilbert space, as Haag's theorem demonstrates.
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    • 2.Algebraic quantum field theory, following Haag and Kastler, shows that inequivalent representations of the CCRs are physically distinct, making the separable Hilbert space choice underdetermined and arbitrary.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.The trace of the identity operator is infinite in an infinite-dimensional separable Hilbert space, preventing the definition of a correctly normalized a priori probability for measurement outcomes.
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    • 2.Unbounded observable operators give rise to domain problems.
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    Related

    Algebraic quantum field theory, following Haag and Kastler, shows that inequival...Dirac's bra-ket formalism—indispensable to quantum mechanics practice—is formall...Superselection rules partition quantum state spaces into inequivalent sectors th...The mathematical framework physicists actually use presupposes structures (delta...
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    The trace of the identity operator is infinite in an infinite-dimensional separa...Unbounded observable operators give rise to domain problems.Von Neumann's spectral theorem requires rigged Hilbert spaces (Gel'fand triples)...

    Source

    AI-extracted1/3 agreementValid
    SEP: qt-nvd
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    In the introduction to the first paper in the series of four entitled “On Rings of Operators”, Murray and von Neumann list two reasons why they are dissatisfied with the separable Hilbert space formulation of quantum mechanics. One has to do with a property of the trace operation, which is the operation appearing in the definition of the probabilities for measurement results (the Born rule), and the other with domain problems that arise for unbounded observable operators. The trace of the identi
    Extraction notes

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    Details

    Type
    claim
    Perspectives
    3 (2 for, 1 against)
    Edits
    1 edit