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Re: Some Interesting Mathematical and Physical concepts

  1. Oct 12, 2006 #1
    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":

    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

    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

    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
  2. jcsd
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