Stephen P. King (stephenk1@home.com)
Tue, 06 Apr 1999 14:46:35 -0400
Dear Hitoshi,
Hitoshi Kitada wrote:
snip
> > It is the same as your in the sense of equivalence. Note that how we
> > think of a theory is also subject to the axioms of LS theory, and our
> > minds are independent LSs. :)
>
> Is your intention to explain everything? If so, I have to say "I think it is
> impossible."
I am causing you to lose patience with me. :( I am sorry. I do not
expect to be able to "explain everything", it is, as you say impossible,
but we can have a paradigm that allows us to think consistently about
any particular finite situation. This is like having "thumb rules"... :)
snip
> > Do we say that there is a spacetime proper to *each* LS and,
> > symmetrically, an LS proper to each spacetime?
>
> There is no symmetry between LS and space-time at least in LS theory. LS
> theory assumes the existence of space-time for each LS. Probably, the problem
> you raise here is that we have a common space-time. My point in LS theory is
> that that is only an assumption that cannot be verified, and that it is a
> thing that we assume. This is the reason why I impose axioms 4 and 5. This
> assumption remedies people from their "common" universe being chaos. But note
> that, at the deepest level, there is no assurance that we can have a "common"
> universe.
Do we agree that a spacetime is the "object" of an LS's "subjective"
perspective? If so, would it not follow that an arbitrary spacetime is
identified with the centers of mass of LSs that can observe each other?
Could we then expect that these LSs can be sublocal systems of a single
LS?
> This is another way of
> > saying that for every "subject" there is an "object" and for every
> > "object" there is a "subject".
>
> There are many objects for every LS.
Yes, they make up the possible observables of that LS. They are defined
as the centers of mass of independent LSs. Upon thinking further, maybe
we need to look more carefully at what we mean by "independent".
> > >Second, here is another implicit assumption. The particles' "correlation"
> > >cannot be known unless the correlation is observed or measured. If no one
> > >observes the system, the correlation cannot be known, or is forgotten. The
> > >forgotten correlation cannot be traced further. One needs to start to measure
> > >other systems similar to that one, in order to reproduce the observation.
> > >In this sense, the word "observation" has an implicit assumption behind it that the
> > >_different_ observations could be identified if one's memory tells one that
> > >the situation looks the same as before ("before" in the time coordinate of the
> > >one's, and "one" can be a set of observers, e.g., a set of modern physicists,
> > >that has a time coordinate which might have begun in the 16th century or so
> > >(with Galileo Galilei and/or others?)).
> >
> > Could it be that the act of measurement/observation *is* a mapping of
> > correlations,
>
> Mapping from where to where? Namely what are the domain and range of your
> mapping?
That is a good question! We think of the vector bundle X x R^6. (Are
not both infinite in themselves?) Quoting from your paper: "with X being
the observer's reference frame and R^6 being the unobservable inner
space(-time) with in each observer's local system", we can think of the
"images" of centers of mass of the reference frame as the domain (range)
and the "images" of the LS's internal configurations of its inner
space(-time) as the range (domain). I am saying that there is a symmetry
between the domain and range, since the mappings here are invertible,
and I use the word "images" to denote the fact that what is observed is
*not* the object or center of mass, it is the observer's "version" of
it. When we look at the world we must realize that what one individual
observes is *not* exactly what another observes. There is just a
correlation between such.
> > >I distinguish this difference of observations. Thus local systems have
> > >different Hilbert spaces as in axiom 1, even when they have common particles.
> >
> > >In other words, the local system is the notion that describes the object of
> > >observation, that is different from time to time (time is again the time of
> > >the observer) and from observer to observer.
> >
> > When we say "spacetime" does this not assume, at least, that such is
> > what an LS observes of other LSs? I think we agree that there is more
> > than one spacetime since each LS's observations make up such. But if the
> > subject-object relation is symmetric,
>
> Subject-object relation is not symmetric, because the outside of an LS
> consists of an infinite number of particles that cannot be reduced to an LS
> which has a finite number of particles, while inside an LS there are only a
> finite number of particles.
I understand this point, but I think that I am not understanding your
argument. When we say that a particular LS X has a finite number of
particles, do we also think of these "particles" as what the sublocal
systems of X would observe as each other's centers of mass? Is not each
of the sublocal systems a complete LS Y : Y =/= X, in its own right,
with its own sublocal systems that observe each other as centers of
mass, and so forth? There is obviously an infinite regress here! This,
to me, is not a bad thing! I may be not understanding some of your basic
assumptions... :(
That there are an infinite number of particles outside of an LS, we
agree. Any given LS can only observe a finite number of these at any
single moment due to the fact that it has only a finite number of
internal states, but would it not have, if its "life span" were
infinite, the ability to observe an infinite number of particles, just
not at once?
> > 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? These are the frameworks that
Prigogine uses in his work!
There is also the matter of "equivalence classes". 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! Peter
discusses them in his paper...
Onward to the Unknown,
Stephen
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