Re: An Invitation to Quantum Mathematics


I'm just guessing here, but it seems like this is getting into an area of
abstract algebra based on the ideas of Von Neuman et al, numbers and
operators as objects, etc.

But I dont think that it answers the question.

Of course, one would need to know some abstract algebra, which I dont, and
then one would wonder why the solution to the central riddle of the entire
cosmos rates about 20 hits on Google.

There is a very simple explanation of whichway information which can be
stated in one sentence as follows. Length itself is probabilistic, particles
are merely probabilistic deformations of that probabilistic manifold i.e.
they are just probabilities in the first place, hence when chopped into
trivially sized fragments they remain existent in the form of a
"probabilistic entity" due to conservation, when uncertainty is destroyed so
is the triviality and the particle must come back due to conservation of
(ahem) "energy".

If you know of an abstract algebra explanation of whichway info please post
a ref.


And you know something, speaking of "Harris Escher Bach", these silly
trivialistic ideas really are trivial. A theory based on trivia is just
that - trivial. However, existence of a trivial is "indeterminate", and so
validity of such a theory is not negated but merely "indeterminate" - which
is exactly what one would expect physics to look like if Godel is to be
believed. A theory of physics which is known to be inherently crippled, and
it is exactly as it should be, because the universe is not determined. HAH !
The mind reels at the prospect, but how could anything else possibly be
true.



Properties which hold for Pisier Spaces, can be Google up by searching
for "Pisier Space".

J.S.Harris Fan schreef:

Does this allow you to model whichway information experiments ?


"Mpilot" <mobilepilot@xxxxxxxxx> wrote in message
news:1163760232.916931.120490@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
For all examples of Hilbert Spaces, their dual quantum spaces turned
out to be different from the initial spaces. Piser showed that amound
all quantum Hilbert spaces there is exactly one behaving in complete
correspondence with the "classical" Riesz theorem: for this space its
canonical bijection is a completely isometric conjugate linear
operator. Thus, such quantum space plays in quantum mathmatics the
same
role as the usual Hilbert Space does in the classical functional
analysis.

markwh04@xxxxxxxxx schreef:

Mpilot wrote:
Quantum Mathematics is the mathematical apparatus of quantum
mechanics.
What is the essence of this mathematical ideology ?

It is...

* Classical Mathematics deals [sic] exclusively with spaces of
functions and
its main structure is the uniform norm.
* Quantum Mathematics deals with the spaces of operators and the
main
structure is the quantum norm.

Not any of these. There is a Hilbert space formulation for Classical
Mechanics, as well as Quantum Mechanics; or any other physical
theory
whose observables form a C*-algebra.

There is a phase space representation for quantum mechanics, with
states represented by *positive* phase space distributions. These
distributions are Gaussian convolutions of the Wigner functions
representing the states and can alternatively be characterized as
transition probabilities with respect to the coherent states. Hence
if
W is a mixed state with density matrix rho, then <p,q|rho|p,q> =
rho(p,q) is the phase space density corresponding to W, where |p,q>
is
the coherent state for phase space point (p,q). This function
rho(p,q)
is the Gaussian smear of the Wigner function W(p,q). Though W(p,q)
is
not positive definite, rho(p,q) will always be.

Another way of saying this is that quantum "distributions" in phase
space (that is: Wigner functions) are inverse Gaussian convolutions
of
phase space densities.

So classical mathematics applies to quantum mechanics, quantum
mathematics applies to classical mechanics.

There is even a classical analogue of the celebrated Naimark
theorem.
The only difference is that the "Naimark" extension of a classical
system is generally a quantum system (which, after the fact,
justifies
using quantum mechanics techniques even for classical physics)




.

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