Let me start by quoting John Baez.s Week 221 last paragraph.(adsbygoogle = window.adsbygoogle || []).push({});

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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# Re: Some Interesting Mathematical and Physical concepts

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