Proof of Gelfand-Maurin Nuclear Spectral Theorem?

strangerep

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...

zhupihou

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.

strangerep

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.

dextercioby

I couldn't find the <incompletenes/inaccurate> statement/footnote either.

zhupihou

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?

strangerep

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?