| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
31 May 2005 08:13:24 AM |
| Object: |
Quantum Algebra: Where It Went Wrong and Right |
From Osher Doctorow
COPYRIGHT NOTICE
Quantum Algebra: Where It Went Wrong and Right
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
Quantum Algebra, more specifically Local Quantum Physics and Local
Quantum Field Theory (LQP, LQFT respectively), began with a mistake,
which was Rudolf Haag's collaboration with and association with Werner
Heisenberg. From that the algebraic physicist Haag obtained an unusual
intuition for an algebraist, namely that he needed to reformulate
quantum theory algebraically, but in a way related to the Heisenberg
Uncertainty Principle. Notice that I did not indicate "with algebraic
topology or algebraic geometry," since those had not yet been developed
significantly. Topology had just been launched by Poincare of France
around the turn of the century (the beginning of the 20th Century), and
there was "always" considerable borrowing and imitation and
collaboration between Germany and France and considerable borrowing and
imitation between Germanic nations and Italy. So Haag in his algebraic
physics quest could draw from topology and Italian geometry, which John
von Neumann of Hungary and later the USA had already borrowed into the
elevation of Hilbert Space concepts which we now know as C*-algeba and
W*-algebra and which are key to Quantum Algebra. John, as I have
described him before, had one foot in Creative Genius and one foot in
Ingenious Imitation.
The big surprise comes of all places from the University of Texas
Austin, where A. Bohm and colleagues have discovered the deficiencies
of Hilbert Space although some of them have "papered over" the
deficiencies by regarding the repairs as more or less complete instead
of as the beginning.
Almost equally surprising if not equally is that C*-algebra and
W*-algebra are in the direct line of application of Functional Analysis
which is not algebraic as much as analysis and differential equation
oriented with heavy fundamentals from topology and geometry (sometimes
more indirectly for the latter). Functional Analysis and
Probability-Statistics are the most applications-oriented and
real-world oriented branches/offshoots of analyis while still
containing an approximately equal and high level of abstraction
(especially probability and statistical decision theory and dependence
theory).
The result of all this is that today Quantum Algebra is quite close to
Probable Influence (PI) Theory except for a "tiny" difference: Quantum
Algebra has at least until very recently been obsessed with Hilbert
Space and that kept it backward until very recently just as Quantum
Logic's similar obsession has resulted in even worse damage - the
almost complete destruction of its discipline.
What makes Quantum Algebra and PI so similar is their emphasis on
Causation and Locality. In fact, Marleen and my first paper in 1980
was "On the nature of causation," which launched PI.
I'll try to continue this relatively soon.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Algebra: Where It Went Wrong and Right |
31 May 2005 08:32:09 AM |
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From Osher Doctorow
If Hilbert Space is too restrictive, then what is left? Well, Banach
Spaces in Functional Analysis and Probability Spaces in Probability for
one thing so to speak. There are also Rigged Hilbert Space, Lattices
and Chains of Hilbert and Banach Spaces that A. Bohm of U. Texas and
his colleagues have pursued so much.
The U.K., the most reality-oriented and practical of the European
nations except for Italy which is about equal in those regards, is a
"gold mine" for Functional Analysis and Probability. The USA is not
far behind or maybe equal.
The essential difference between Hilbert Space and the more general
Banach Spaces which contain Hilbert Space as a subset is the lack of
requirement for inner products in general Banach Spaces. So the formal
notion of "orthogonal" or "perpendicular" is in general absent from
Banach Spaces. This turned off von Neumann and others from more
general Banach Spaces, but in fact Probability spaces also do not in
general contain notions of orthogonal and yet they do quite well in
orthogonal and more general situations related to curved spacetimes via
notions like dimensions and phases which are more general than
orthogonality.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Algebra: Where It Went Wrong and Right |
31 May 2005 09:01:16 AM |
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From Osher Doctorow
So where did Locality and Causation come from?
Albert Einstein's insight in his General Relativity (GR) was that
spacetime is Locally Special Relativistic with Newtonian as a subtype.
Causation? David Hume of the U.K. was a good a starting point as any,
though like Rudolf Haag in a sort of opposite way. Hume tore into the
nature of Causation to such a degree that one could say of his theory
that Correlation was all that was left, but not quite. There was also
time and Coincidence, and we now study the latter in probability. Take
away the Coincidences, throw in time, and Correlation becomes
Causation. That is what Probable Correlation, P(A<-->B), does.
But the U.K. had other reasons to be interested in Causation. Their
decision to resettle large populations in their former Colonial period
turned out to be pivotal in transfering deep parts of their
civilization to their former colonies or protectorates incluing the
USA, Australia, and Canada. Their decision to go the way of
Parliamentary Democracy, beginning although far from fully developed in
the 1200s, proved pivotal in the beneficial developments of their
history including their victory in WWII together with the USA's
victory. Their decision to break with the Catholic Church via Henry
VIII was less well planned and more debatable, but it did infuse them
with a sense of accepting new ideas just as Catholic Italy kept its
Renaissance emphasis on new ideas from the origin of the Renaissance in
its Catholic Chuch. Similarly for the U.K. and USA decisions to
elevate women to approximate equality with men at least in voting,
which helped create many new allies for Parliamentary Democracy.
Not being gifted with great agriculture unlike France and Italy, the
U.K. had to think hard to survive. Nations with abundant "wild food"
or abundant agriculture can afford to drown their strategies in wine or
beer. Not so the U.K.
I'll try to continue soon.
Osher Doctorow
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