# [time 196] Re: [time 193] Re: [time 192] one more addition to Re: Prugovecki's time

Wed, 7 Apr 1999 12:12:44 +0900

Dear Stephen,

Just on technical points...

----- Original Message -----
From: Stephen P. King <stephenk1@home.com>
Sent: Wednesday, April 07, 1999 3:46 AM
Subject: [time 193] Re: [time 192] one more addition to Re: Prugovecki's time

snip

> > > there is more to this that we have
> > > covered so far. Remember that length is not an absolute invariant!
> > > I would like to discuss how the equivalence principle is modeled in LS
> > > theory. By the way, Prugovecki talks about rigged Hilbert spaces on page
> > > 446. ("Gel'fand space"!).
> >
> > What he writes around page 446 are quite elementary things well-known to
> > mathematicians. How do you intend to utilize those?
>
> "...in the simple case of a single nonrelativistic quantum particle of
> zero spin, one conventionally writes:
> <x'|x> = \delta^3(x - x'), <x'|k> = (2\pi)^-2/3 exp(ik . x'), x, x',
> k \elements of R^3 (3.1)
>
> neither the \delta functions nor the plane waves [eigenfunctions as
> considered conventionally], are Lebesgue square-integrable functions
> [L^2(R^n)], so they do not belong to the Hilbert space with the inner
> product defined in (3.1) For that reason, von Neumann (1932) avoided the
> use of \delta "functions". Eventually their mathematical nature was,
> however, totally clarified by L. Schwartz (1945). The mathematically
> correct treatment of the objects in (3.1) was subsequently supplied by
> the theory of rigged Hilbert spaces (Gel'fand et al., 1964, 1978) as
> well as that of equipped Hilbert spaces (Berezanskii, 1968, 1978) These
> mathematical frameworks pinpoint the objects in (3.1) as elements of
> eigenfunction expansions - and not as eigenvectors of Hilbert space
> operators."
> Is this included in your thinking?

Yes, of course. This is a common sense of usual mathematicians.

These are the frameworks that
> Prigogine uses in his work!
>
> There is also the matter of "equivalence classes".

Equivalence classes is the notion that the freshmen have to learn at first
when they enter a university. It is one of indispensable knowledges for anyone
who wants to learn something about math and narure.

Prugovecki writes on
> pg. 447:
> "...the generic element of \H is not a single function, but rather an
> equivalence class of almost everywhere (in the Lebesgue sense) equal
> functions, which are such that one can change the value of any one of
> these functions \Psi(x) at any given point x without leaving the
> equivalence class -namely, in physical terms, without changing the
> quantum state vector. "
>
> I had stated in [time 188]:
> > Yes, Peter and I discussed this for a while. It appears that the
> > subject-object relation is symmetrical. There is a wonderful thing that
> > happens when we consider an LS as a subject as a singleton set A and the
> > other LSs that it is near to as the singleton's complement A^c. If we
> > think of A^c as a finite number of LSs that can somehow be reduced to a
> > singleton by some particular observation by A, by symmetry, would we not
> > expect that A becomes many neighboring yet distinct LSs? As one fuses,
> > the other fissions, many -> one | one-> many ... Does this make sense?
> > There exists a mathematical way of saying this but I do not remember it
> > now. :(
>
> It is the fact that I am not stating explicitly the "mathematical way
> of saying this" that, I think, is the reason I am just making noise
> here... :( The role of equivalence classes is very important!

Yes, it is so important that one usually does not refer to the usage of it,
i.e. people frequently use it without mentioning...

Hitoshi

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