Re: Some Interesting Mathematical and Physical concepts
- From: Kwok Man Hui <kmhui@xxxxxxxxxxxxxxx>
- Date: Fri, 14 Oct 2005 07:10:58 +0000 (UTC)
Let me start by quoting John Baez.s Week 221 last paragraph.
Begin quote
But, someday I should really explain the ideas behind the baby "abelian"
case of the Langlands philosophy in simpler terms than Frenkel does here.
The abelian case goes back way before Langlands: it's called "class field
theory". And, it's all about exploiting this analogy, which I last
mentioned in "week218":
NUMBER THEORY COMPLEX GEOMETRY
Integers Polynomial functions on the complex plane
Rational numbers Rational functions on the complex plane
Prime numbers Points in the complex plane
Integers mod p^n (n-1)st-order Taylor series
p-adic integers Taylor series
p-adic numbers Laurent series
Adeles for the rationals Adeles for the rational functions
Fields One-point spaces
Homomorphisms to fields Maps from one-point spaces
Algebraic number fields Branched covering spaces of the complex
plane
End quote.
This is indeed a very beautiful but a static picture. I remember a late
Chinese scholar mentioned that the western philosophers always focuses on
objects, but the old Chinese wisdom is always on the relation between
objects and how they convert to each other.
I want to put forward another kind of analogy if it is possible.
Elementary Real Analysis Homotopic/Algebraic Theory
Real numbers Some objects
Inequality It can compare any two objects' size.
Limit It can tell one object approaches
another object
Continuity It can tell successive objects are
not deviated from their
immediate neighborhood
Differentiation It can tell the "slope" of
successive objects
Integration (We know there is motivic
integration, but it may not fit
the overall picture.)
Differential forms (We know there is n-gerbes, but,
again, may not fit the overall
picture.)
I hope you can see that all these kind of concepts are to capture the
ubiquitous devil, which is everywhere, but nowhere can it be caught. The
devil is the change, the perpetual change, which is dormant in each object
or group of objects.
If one day there is another "David Hilbert" appears on the face of this
planet, and he can synthesize all the current fragmentary math concepts.
And, there comes another "Albert Einstein" to start building a new quantum
mechanics in which "observable is relative to another observable" is
achieved. (Maybe like C. Rovelli idea in Relative Quantum Mechanics or in
another paper about observables of quantum spacetime, which mentioned
"observables built upon observables". Frankly, as an adherent of
"perpetual change", I really don't buy the no topology change in Loop
Quantum Gravity because metric can be bent or stretched, so any
measurement of length or area or volume is a relative measurement even in
pure gravity condition. Area, and volume should be quantizable but can't
be fundamental. Maybe they have never advocated that whatever quantizable
is fundamental in their theory if they try to defense above criticism. I
don't see LQG have conceptually demonstrated the relativity concept I put
forward here or the one in C. Rovelli papers though C. Rovelli he helped
build the LQG.)
I found out on some websites that in some European institutes, researchers
are trying to put quantum mechanics on a categorical footing, but I much heavy
machinary they can bring in or invent to succeed the goal, and how much it
will look like my intend theory.
Second, when will Konsevich will prove his speculation that his
deformation quantization theory will lead to a new class of quantum field
theory. If he can do that, I may expect that a "dynamical" quantzation
scheme may become realizable not in a distant future.
I can foresee that if you buy this kind of concepts, it will lead you
to some astounding philosophical and physical conclusions about quantum
spacetime and its ramification in understanding the nature of the
fundamental four forces. I save this premature thought for the
future occassion.
Finally, since Godel's Theorem always backs me up, I am granted to say
that there is always room to add some possible math concepts to the current
ones. Whether it is possible or not, it just depends on our imagination
and talents. With the dominance of string theory in academic, it is very
unlikely that the current talented group of people will think outside the
box and even in the coming few generations.
Charles Hui
"I leave my thought in a region of spacetime for the physicists of future
generations."
.
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