**Hitoshi Kitada** (*hitoshi@kitada.com*)

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

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 197] Re: [time 196] one more addition to Re: Prugovecki's time"**Previous message:**Ben Goertzel: "[time 195] Re: [time 190] Re: [time 187] Re: one more addition to Re: Prugovecki's time"**Maybe in reply to:**Ben Goertzel: "[time 190] Re: [time 187] Re: one more addition to Re: Prugovecki's time"**Next in thread:**Stephen P. King: "[time 197] Re: [time 196] one more addition to Re: Prugovecki's time"

Dear Stephen,

Just on technical points...

----- Original Message -----

From: Stephen P. King <stephenk1@home.com>

Cc: Time List <time@kitada.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
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*> > > theory. By the way, Prugovecki talks about rigged Hilbert spaces on page
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*> > > 446. ("Gel'fand space"!).
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*> >
*

*> > What he writes around page 446 are quite elementary things well-known to
*

*> > mathematicians. How do you intend to utilize those?
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*>
*

*> "...in the simple case of a single nonrelativistic quantum particle of
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*> zero spin, one conventionally writes:
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*> <x'|x> = \delta^3(x - x'), <x'|k> = (2\pi)^-2/3 exp(ik . x'), x, x',
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*> k \elements of R^3 (3.1)
*

*>
*

*> neither the \delta functions nor the plane waves [eigenfunctions as
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*> considered conventionally], are Lebesgue square-integrable functions
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*> [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
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*> use of \delta "functions". Eventually their mathematical nature was,
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*> however, totally clarified by L. Schwartz (1945). The mathematically
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*> correct treatment of the objects in (3.1) was subsequently supplied by
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*> the theory of rigged Hilbert spaces (Gel'fand et al., 1964, 1978) as
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*> well as that of equipped Hilbert spaces (Berezanskii, 1968, 1978) These
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*> mathematical frameworks pinpoint the objects in (3.1) as elements of
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*> 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
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*> equivalence class of almost everywhere (in the Lebesgue sense) equal
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*> functions, which are such that one can change the value of any one of
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*> these functions \Psi(x) at any given point x without leaving the
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*> equivalence class -namely, in physical terms, without changing the
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*> quantum state vector. "
*

*>
*

*> I had stated in [time 188]:
*

*> > Yes, Peter and I discussed this for a while. It appears that the
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*> > subject-object relation is symmetrical. There is a wonderful thing that
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*> > happens when we consider an LS as a subject as a singleton set A and the
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*> > other LSs that it is near to as the singleton's complement A^c. If we
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*> > think of A^c as a finite number of LSs that can somehow be reduced to a
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*> > singleton by some particular observation by A, by symmetry, would we not
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*> > expect that A becomes many neighboring yet distinct LSs? As one fuses,
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*> > the other fissions, many -> one | one-> many ... Does this make sense?
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*> > There exists a mathematical way of saying this but I do not remember it
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*> > 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

**Next message:**Stephen P. King: "[time 197] Re: [time 196] one more addition to Re: Prugovecki's time"**Previous message:**Ben Goertzel: "[time 195] Re: [time 190] Re: [time 187] Re: one more addition to Re: Prugovecki's time"**Maybe in reply to:**Ben Goertzel: "[time 190] Re: [time 187] Re: one more addition to Re: Prugovecki's time"**Next in thread:**Stephen P. King: "[time 197] Re: [time 196] one more addition to Re: Prugovecki's time"

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