Stephen P. King (stephenk1@home.com)
Sat, 27 Mar 1999 09:49:37 -0500
Dear Hitoshi,
My initial assessment of the paper Quantum Norm theory... is a bit
incorrect. It is an attempt to construct a quantum theory on the set
\M(X) of all metrics on a set X. It attempts to model topological
variations of a spacetime foam. It is very interesting to me as it
explains many of the details involved very well. It is the possibility
of thinking of M_p black holes as LSs! But, the spectra of such
(involving the external center of mass issue) remains unexplored. :(
(that is where I think Weyl's work comes in...)
Hitoshi Kitada wrote:
>
> Dear Stephen,
>
> Thank you for your detailed explanation of your position. I understand now
> that you are trying to find a solution for the observable world, which is
> treated quite roughly and insufficiently in my papers. I agree that you are
> another discoverer of the notion of local systems in slightly different,
> philosophically formed statements in
> https://members-central.home.net/stephenk1/Outlaw/forbiden.html
> Thus I say not only that I have no objection with that you try to extend it to
> a more understandable one, but further I ask you to do so. As you say, we are
> a team, each one of whom should not be lacked. I just might be able to give a
> help when there appears any that can be given a mathematical formalism, but I
> state I am enjoying invaluable discussions developed here.
Thank you for your kindness. My mathematical abilities are very
limited. I think we need to formalize a more detailed explanation of how
LSs interact. I propose that we look at the way that a given LS's
subsets, which are independent on their own, observe each other's center
of mass and how corrections are computed when dealing with large
velocities/energies. I see that a hierarchical structure is involved but
I don't have a way to illustrate it here. Wheeler's discussions of how
the event horizon has a finite information capacity is a key! I am
thinking of this in a computational sense; LSs, like black holes, are
"black boxes" with respect to each other. This, I believe, is why Pratt
and Wegner's work is important to us! :)
> Best wishes,
> Hitoshi
>
> P.S. Your hint that a Superspace for X might be non-Hausdorff may be a clue
> for us:
"A topological space is 'Hausdorff' if any two disjoint points are
contained in disjoint neighborhoods." (pg. 1057 ibid.)If we weaken this
using Kosko's formalism, we could think of LSs as disjoint points that
are contained to a degree [0,1] in disjoint open balls, both embedded in
a Superspace X of indefinite metric (in the "in itself" sense, thus I am
considering that there is a "spacetime" equivalent to your \psi... "all
possible metrics simulataneously ). What I think is pertinent is that
LSs define computationally (via some form of bisimulation function)
their common spacetimes within which finite communications are
possible... (I hope that I got the details correct here. :) ) I am
exprapolating from: "It is well known (Van Mill 1989) that every
seperable metrisable space is homeomorphic to a subspace of the Hilbert
cube Q, namely the countable product of intervals [-1, 1] equipped with
the metric d(x, y) = \Sum_i 2^-i |x_i - y_i| (this is essentially the
Urysohn theorem)..." pg. 1056 ibid.
> >Dear Hitoshi,
> >
> >You may be interested in the following paper: "Quantum norm theory and
> >the quantization of metric topology" by C J Isham, Y Rubyshin and P
> >Renteln; Class. Quantum Grav. 7 (1990) 1053-74
> >
> >it is getting into our question of using a Superspace for X. The
> >possibility of its topology being non-Hausdorff is partcularly
> >intriguing! That would allow for the use of fuzzy subsethood! I do not
> >know your opinion of Kosko's ideas; most mathematicians do not like the
> >weakening of the "law of excluded middle" ... :)
> >
> >Later,
> >
> >Stephen
> >
> >PS, Have you thought about the "noise" interpretation of LSs? I
> >recommend: Gaussian Random Fields, K. Ito & T. Hida (eds.) Series on
> >probability and Statistics Vol. 1, The third NAgoya Levy Seminar, 15-20
> >Aug. 1990. World Scientific (QA 274.4.N34 1990)
Kindest regards,
Stephen
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