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Noncommutative geometry?

  1. Oct 24, 2005 #1
    How new is this subject of noncommutative geometry? I tried googling it, but few info comes out and there is not alot of books about it either.

    What is this subject about exactly and is it going to be something major?
     
  2. jcsd
  3. Nov 10, 2005 #2
    Hi, I think the subject is around 30 years old (it might be even more). Intuitively, non commutative geometry is a strict algebraic theory that allows one to generalize Riemannian manifolds. Connes remarked that such a structure (actually, you have to restrict yourself to manifolds with a spin structure if I remember correctly) can be fully characterized by the *commutative* C* algebra of C^infty functions equipped with a derivative operator. Now, you can ask yourself what ``geometry´´ you get when you allow the C* algebra to be non commutative. At that point you can use the GNS representation theorems wich say that such non abelian C* algebra can be represented in terms of bounded operators on some Hilbert space. This gives you a link with quantum mechanics and one could hope to get quantum gravity out in this way. If you want references: search on Connes first.

    Cheers,

    Careful
     
  4. Nov 10, 2005 #3

    matt grime

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    Non-commutative geometry is a blanket term: Connes definition, if he even has such a thing as 'a definition' would not agree with, say, an algebraic geometer's idea.

    The first thing you should ask yourself is: do i know what commutative geometry is? If so then it is relatively easy to see what 'non-commutative' geometry is: geometry without the restriction of commutativity. How you relax that criterion would I suspect depend upon whom you asked.
     
  5. Nov 10, 2005 #4
    Indeed, and I gave one which is used by physicists (and which I remember to have read from a paper Connes has written for physicists). More abstract stuff can be found on webpages of Lieven Lebruyn and Michel Van den Bergh.
     
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