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Proof of Gelfand-Maurin Nuclear Spectral Theorem?

  1. Feb 22, 2009 #1

    strangerep

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    I want to study a detailed proof of the Nuclear Spectral Theorem
    (which underpins the use of Rigged Hilbert Spaces in modern QM
    to make the Dirac bra-ket formalism respectable).

    Most textbooks and papers refer to the old multi-volume series on
    generalized functions by Gelfand and Vilenkin, but I cannot borrow
    it locally and the price from Amazon is ridiculous.

    Does anyone know of proofs in other textbooks, or maybe from
    a (free) online source?

    Thanks in advance for any suggestions...
     
  2. jcsd
  3. Aug 16, 2010 #2
    The proof in Gelfand's Generalized Function vol 4 is incorrect (at least
    not complete), as pointed out by the translator of the English version.
     
  4. Aug 17, 2010 #3

    strangerep

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    Thanks for your comment! (That was indeed an unexpected and interesting first
    post in this forum, at least to me. :-)

    I now have a copy of the (English version of) Gelfand & Vilenkin vol4, but I cannot
    find where the translator says this. (I looked at the translator's notes near the
    beginning, but I couldn't find where he says this.)

    If you have a copy at hand, could you possibly give me a more specific page
    reference to where the translator says this?

    Thanks again.
     
  5. Aug 17, 2010 #4

    dextercioby

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    I couldn't find the <incompletenes/inaccurate> statement/footnote either.
     
  6. Aug 18, 2010 #5
    Hi.

    The trouble is on page 122 of vol 4 (I mean Gelfand-Vilenkin "Generalized Functions").
    At the bottom of that page, the translator expressed some concern
    "... it is not clear why..."

    As I read through the proof, this concern is serious, and I don't know how to fix it
    (this is not my field so I am far from being an expert, and it seems no one I know cares
    about rigged Hilbert space!).

    In fact, after a search online, there is a paper of G. G. Gould (J. London Math. Soc.
    43 (1968) 745-754) that claimed to have resolved this issue; but that paper is not
    so easy to read. On the other hand, apart from this issue the Gelfand book is user-friendly.

    Maybe you can ask some experts and update this?
     
  7. Aug 18, 2010 #6

    strangerep

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    Oh, thanks. I see it now.

    It sounds like you know more about this than I do. :-)

    I know what you mean. This is an unfortunate situation, since
    RHS theory silently underpins much of modern quantum theory.

    Rafael de la Madrid has, in recent years, written a number of papers
    trying to emphasize RHS (eg his tutorial paper quant-ph/0502053, and
    quite a few others), but these are mainly applications of RHS without
    giving details of the heavy proofs that underlie it.

    There's also this paper:

    M. Gadella & F. Gomez,
    "On the Mathematical Basis of the Dirac Formulation of Quantum Mechanics",
    IJTP, vol 42, No 10, Oct 2003, 2225-2254

    Gadella & Gomez give updated version of the spectral theorem(s) near the end,
    but not detailed proofs, afaict. But much of this paper is over my head, and
    I haven't yet had time to try and chase down the further references therein.
    If you haven't previously seen this stuff, I'd be interested to hear your comments.


    Thanks. I'll take a look at it when I get a chance.

    Yes, it's certainly better than Maurin's text which seems to contain many typos
    and/or errors. (Sometimes I'm not sure which is which.)

    I don't know many experts on this directly, but I'll try.

    BTW, what is your interest in RHS? Physics or maths?
     
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