Re: An Invitation to Quantum Mathematics


Mpilot wrote:
Will you please discuss Jx, Jy, and Jz ?

If we define Information (bits) as the most elementairy parts in our
Universe, we can ask the question whether information resides in points
of non-analyticity or if information is more distributed in nature,
which relates to the wave-partice duality of light.

If we define your Jz the axis of points of "non-analyticy", Information
as your Jy axis and Physical entropy S as your Jx-axis, we can define a
Dynamic Theory of Information and Entropy.

Opinions on the Physical Nature of Information have tended to be
contradictory. One view is that information is inherent to points of
non-analyticity ("vitual particles"), whereas others consider
information to be more distributed in nature. Such considerations are
akin of the wave-particle duality, with the question of which of the 2
complements best characterses information.

Timothy Golden BandTechnology.com wrote:
Mpilot wrote:
Let me try to clarify the above given statements.

It turns out that fundamental notions and results of classical
mathematics do have substantial quantum analogues. You can say that
these classical notions represent a small classical part of a huge
quantum iceberg. To comprehend all of this iceberg we must replace
functions lying in the foundation of the notions (results, methods,
problems) with operators. The question is how to perform such
quantization in practice for a concrete notion taken from some area of
mathematics. Often it is not clear in advance what to do and different
people can give you different suggestions.

However, some conformity has been established. For instance, the book
by Connes is especially impressive. It is a main source for quantum
mathematics.

Let me concentrate on the theory of normed spaces. There are no other
normed spaces, but function spaces. Thus every normed space coincides
with some space of bounded functions endowed with the uniform norm.
Being spaces of functions automatically become spaces of operators.

The essential new phenomena of quantum mathematics appear when we move
from lineair operators to multilineair operators. In principle, the
relations between quantum and classical functional analysis are similar
to those between quantum and classical physics. On one hand, the things
in classical science (notions, facts, methods) have meaningfull quantum
analogues, which allow to better understand their classical prototypes.
On the other hand, quantum science comes across essentially new
phenomena not encounted in classical sience.

Timothy Golden BandTechnology.com wrote:
Mpilot wrote:
Quantum Mathematics is the mathematical apparatus of quantum mechanics.


What is the essence of this mathematical ideology ?

We can say quantum mathematics emerges from the classical mathematics
after replacing functions by operators. The outstanding role of
functions in classical mathematics with the pointwise commutative
multiplication is passed in quantum mathematics to operators with their
non-commutative multiplication (composition).

The following 2 statements serve as a "guide to action":
* Classical Mathematics deals 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.

Will you please discuss Jx, Jy, and Jz ?

-Tim

I probably would not pass a test on the multilinear function's norm
space but I'm getting some of the gist. And you are addressing the
classical/quantum correspondence which I appreciate.
When a physicist is concerned about a free particle in one of these
situations and comes up with these complex probability distributions
and they have already expressed the particles behavior in terms of
momentum and position what right have they to reintroduce in angular
momentum atop this free point particle's trajectory? I fail to see the
classical interpretation of this. It seems to me more like attributing
additional degrees of freedom in a flase way.

-Tim

No. They are angular momentum.
I Believe the simpler form is just L in the standard developments, but
they go to J after introducing spin. I hope to understand how they
develop this theory. I am entirely open to not understanding the
problem and so my question may be phrased inadequately. The phrase
"angular momentum" is at the heart of the problem. As I go over the
treatments I see momentum equations developed and then angular momentum
equations developed. I am left with the impression that they have
superposed these two to get results and I question the validity of
that, particularly in light of the correspondence principle which you
have a keen awareness of.

-Tim

.

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