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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 23rd 2010

    I posted a question to Math Overflow here, but perhaps it makes sense to ask here too (heads-up to Urs): is there a mathematically respectable treatment of rigged Hilbert spaces to be found anywhere online?

    I’ve got a few responses on MO so far, but it’s hard not being near a good library where I can spend a few hours looking up all these references. If anyone knows how to develop rigged Hilbert spaces and the Gelfand-Maurin theorem rigorously, I’d love to hear about it. (This is straight mathematics, isn’t it? Why does it have to be so mysterious?)

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 24th 2010

    Have you thought at all about nonstandard analysis? I’ve never looked into it in detail, but I think there is a theorem in NSA about embedding a standard infinite-dimensional Hilbert space into an internal hyperfinite-dimensional one, so that you can essentially treat it as if it were finite-dimensional with all the well-behaved spectral theory etc. And when I read physics literature it usually seems to me that what they want to do is basically pretend that an infinite-dimensional Hilbert space is finite-dimensional.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 25th 2010

    That’s an interesting idea (and it seems to accord with another application, which is to study probability on hyperfinite probability spaces instead of using the usual measure-theoretic technology). But no, I haven’t thought about that yet. There seem to be several solutions for how to deal effectively with the spectral theory used in physics – the usual Stone-von Neumann theory, direct integrals of Hilbert spaces, rigged Hilbert spaces – all of which would be fun to labbify one day, and maybe eventually this NSA approach will be one of them. For now I’m going to work with rigged Hilbert spaces, since I now have some half-decent references.

    • CommentRowNumber4.
    • CommentAuthorYemon Choi
    • CommentTimeOct 25th 2010

    Poking my head furtively out of the trench: I'd have to look this up, but I had the impression that the NSA embedding of something like l^2 into a Hilbert space whose dimension is hyperfinite doesn't quite solve the problem of the messiness of operators on l^2. (Though one advert for these methods was in showing that any polynomially compact operator has a non-trivial invariant subspace.) Moreover, not all operators on Hilbert space do have well-behaved spectral decompositions...

    That said, for the discussion at hand, it could be that NSA works well (as per Todd's comments on hyperfinite probability spaces)

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 25th 2010

    I’ve also heard it said that NSA methods in functional analysis are mainly useful in studies of compact operators, and since nuclear operators are compact, it could indeed be that there are reasonable NSA alternatives to the concept of rigged Hilbert space. But I’m talking a bit over my head at the moment.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2010
    • (edited Oct 25th 2010)

    (heads-up to Urs)

    Would the style of

    S. Wickramasekara and A. Bohm, Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics (pdf)

    be close enough to what you are looking for? See their definition 1.1, the remarks below that and also the footnote.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2010

    this reference here looks good:

    • John Roberts, Rigged Hilbert spaces in quantum mechanics , Communications in Mathematical Physics, 3 (1966)

    I have added a pointer to the entry.