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Connes: Physics to Number Theory

  1. Nov 11, 2004 #1


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    today Alain Connes (with co-author Matilde Marcolli)

    Physics to Number Theory via Non-Commutative Geometry, Part II

    Part I got a big play on SPR, we should know something about this.
    Maybe only a little. But something.

    Part One of "Physics to Number Theory" was

    I cant do more than flag these two papers. If this thread goes anywhere
    it will have to be by other people having comments to make about non-commutative geometry applied to physics.
    Last edited: Nov 11, 2004
  2. jcsd
  3. Nov 11, 2004 #2


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    a must read

    I've had a preliminary look at this paper: EVERYONE should
    read this. Universality for renormalisation appears
    to be properly worked out.

    Also: on page 8 " ... relevant physical quantities, including the
    coupling constants, share this implicit dependebnce on the scale...."

  4. Nov 11, 2004 #3


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    dont wait for help or companionship Kea,
    comment some more

    on this board it is not considered bad to post two or three in a row
    as you think of more to say. I hope very much you will comment some more---and other people may help too
  5. Nov 12, 2004 #4


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    Connes and Marcolli have defined a new object they call Q-lattices. I have only glanced oved it, I wonder if it is a generalization of Eratosthenes's sieve. Remember that this sieve is basically to scale an integer lattice and to map it over itself. After considering all the possible scalings, the sites not receiving any element are the prime numbers.
  6. Nov 12, 2004 #5


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    It will take me a long time to understand this paper...but
    a few more immediate thoughts
  7. Nov 12, 2004 #6


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    The Riemann-Hilbert problem is important in soliton theory -
    ie. the inverse scattering method - which was generalised
    to a quantum inverse scattering method. It was the study of
    the Sine-Gordon equation in this context (by Kulish and
    Re****ikhin) that led to the discovery of quantum groups
    by physicists. And of course, quantum group Hopf algebras
    have a great deal to do with knots (and Category Theory,
    which is what I'm trying to convert you all to)....so it's
    very nice to see this tie in by good mathematicians.

    The Hopf algebras they discuss have been studied by the
    causal set people (Markopoulou and others). More lattices.
    This is because lattices are basic to the structure of a
    Heyting algebra - that is, the intuitionistic logic of a topos.

    A good understanding of RepG (which I'm not claiming to have)
    relies on thinking of it in terms of abelian group objects
    in the functor category 'Set to the G', which is a topos.

    Now, from the perspective of String theory one might ask:
    How does one understand multidimensional extensions
    of this rich algebraic structure underlying the standard
    model? My guess (actually, I'm not really guessing) is that
    this paper demonstrates quite clearly that higher category
    theory is absolutely essential. Moreover, this provides the
    link to LQG.

  8. Nov 13, 2004 #7
    The Riemann-Hilbert problem was solved some years ago:

    http://homepage.ntlworld.com/paul.valletta/PRIME GRIDS.htm

    The webpage was left 'unfinished' for historical reasons, but you bet your bottom dollar 2005 would be a very significant year to post it's completion!

    LoOk ClosEly, do not take note of its 'apparent' clumsyness :biggrin:
  9. Nov 13, 2004 #8


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    Just a precision: the Riemann-Hilbert problem, so called because it comes from the list of Hilbert's problems, is different from the conjectures on Riemann's Zeta function.
  10. Nov 14, 2004 #9


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    look at this

  11. Nov 15, 2004 #10

    Identifying the Prime Vacuum INITIAL condition, within string theory has been just an hobby for some(self included :rolleyes: ).

    Quote from paper linked:In this context it
    is demonstrated that the trivial state, with V (q) = E = 0, is identified with the
    self–dual state under phase–space duality. These observations suggest a more general mathematical principle in operation. In physical systems that exhibit a duality structure, the self–dual states under the given duality transformations correspond to critical points

    The non-dynamic (static) Transformations that string-DUAL-theory are trying to incorperate will obviously lead to the asking of this: Will string theorists eventually admit defeat in their 'Vacuum Solutions' of phase evolution for generalized string worldlines, by the creation of a model that is on a par with the Wave-Particle interpretation, a model infact that goes 'BOTH-WAYS'..the correct way, and the in-correct way!

    Critical points can mean many things.
  12. Nov 15, 2004 #11


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  13. Nov 15, 2004 #12

    Really interesting that your first link reveals Osher, I take it that this is the same person as here:http://www.superstringtheory.com/forum/dualboard/messages11/770.html

    Which brings 'me' back to here:http://www.superstringtheory.com/forum/dualboard/messages9/136.html

    I recall Osher with some great respect, I only wish that I could find out if he/she is ok..I Hope so.

    Great links, I do not need to express the fact that my original website, with some very interesting images and text, was demolished by persons unknown, ok no big deal, but I am working towards a '05' deadline, in honour of Einstein and his miracle year :biggrin:
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