Re: An Invitation to Quantum Mathematics

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

.

Relevant Pages

  • Re: why cant fields be quantized too?
    ... > you were born) researching about Quantum Field Theory, QCD, ... > will publish the 'corrected' enhanced version in Physics ... actually means in physics before claiming that QFT is wrong! ... > failure by physicists to find a mathematics based directly on ...
    (sci.physics)
  • Re: An Invitation to Quantum Mathematics
    ... mathematics do have substantial quantum analogues. ... classical/quantum correspondence which I appreciate. ... "angular momentum" is at the heart of the problem. ...
    (sci.math)
  • Re: An Invitation to Quantum Mathematics
    ... of non-analyticity or if information is more distributed in nature, ... mathematics do have substantial quantum analogues. ... classical/quantum correspondence which I appreciate. ...
    (sci.math)
  • Re: An Invitation to Quantum Mathematics
    ... It turns out that fundamental notions and results of classical ... mathematics do have substantial quantum analogues. ... classical/quantum correspondence which I appreciate. ...
    (sci.math)
  • Re: Questioning the defintions of set and element.
    ... In formal set theory the notions of "is a set" ... Wolfram says it's a multiset. ... a definition, in mathematics, is a FORMAL statement. ... He noted that in modern theories primitive terms are undefined. ...
    (sci.math)