From: "Jeffrey Lee" Subject: Re: non commutative geometry Date: 2 Dec 2000 00:01:40 GMT Newsgroups: sci.physics.research Summary: [missing] Noncommutative geometry goes like this. 1. Reformulate geometry, topology and calculus in terms of algebraic statements about the C* algebra of functions on the manifold, subalgebras like the smooth algebra C^\inft (M) and the projective modules (over C(M)) of sections of vector bundles. This is possible for compact Hausdorff spaces because of the Gelfand-Naimark theorem with relates the topology with the algebra in a very tight way. There is an equivalence here. Now take the purely algebraic statements and drop the assumption that the algebra is commutative. The underlying space may not literally exist as space of points. One sort of just pretends the space exist by analogy with the commutative case. Its not an easy subject to explain but try looking at http://www.math.unam.mx/~micho/qgeom_m.html wrote in message news:9089ck$41g$1@nnrp1.deja.com... > Can someone explain what this is? From what I've read, it seems > that this type of geometry replaces the notion of 'point' > with some kind of cell and hence QFT's based on it don't have > certain divergences. Wasn't this one of the reasons why string > theory works as well. Are they really different approaches? Can > you have string theory with a non-commutative geometry? ============================================================================== From: Charles Francis Subject: Re: non commutative geometry Date: Sun, 3 Dec 2000 08:08:40 +0000 Newsgroups: sci.physics.research In article <9089ck$41g$1@nnrp1.deja.com>, thus spake tedsung6674@my- deja.com: >Can someone explain what this is? From what I've read, it seems >that this type of geometry replaces the notion of 'point' >with some kind of cell and hence QFT's based on it don't have >certain divergences. Wasn't this one of the reasons why string >theory works as well. Are they really different approaches? Can >you have string theory with a non-commutative geometry? String theory and non-commutative geometry are different types of approach. I don't think they have been combined. There are different approaches to non-commutative geometry, and with luck you will get an answer from John Baez, if only to tell you that things are not the way I think. But I will give you my pragmatic approach, which is rooted in the way in which we do measurements, rather than the abstract math, which he is much better at. Start off with the idea that geometry is the study of measurement of the world, and that to understand geometry we should think about how observers measure it. Each observer has a clock, which is, without loss of generality, the origin of his co-ordinate system. Each event is given a co-ordinate by measuring the time taken for light to travel to and fro the event. Now suppose that there is some fundamental minimum unit of time measured by the clock, and that all co-ordinates are integer multiples of that unit (these integer co-ordinates could be regarded as 'cells' in the sense I think you mean). The co-ordinate system is thus a lattice consisting of measured events and potential measured events based on that clock. When you do a Lorentz transformation you are switching to a set of measured events and potential measured events based on another clock. One lattice does not transform into the other, but a proscription is given according to which we can relate together predictions made in one lattice with predictions made in the other, and we require the same laws governing the proscription. Predictions are given as expectation values of hermitian operators, and because the lattices do not transform directly one into the other it is easy to see that the operator for position in one frame does not commute with the operator for position in another. That is essentially what I mean by non-commutative geometry. Because the introduction of a lattice also puts a bound on momentum, it does work out that field theories built on a lattice are finite. But you have to make a few more adjustments to remove the lattice dependency from Feynman rules to get a realistic theory. - -- Regards Charles Francis charles@clef.demon.co.uk ============================================================================== From: Thomas Larsson Subject: Re: non commutative geometry Date: 5 Dec 2000 17:59:29 GMT Newsgroups: sci.physics.research Jeffrey Lee wrote in message news:908ve1$pun$1@news.acs.ttu.edu... > Noncommutative geometry goes like this. > 1. Reformulate geometry, topology and calculus in terms of algebraic > statements about the C* algebra of functions on the manifold, subalgebras > like the smooth algebra C^\inft (M) and the projective modules (over C(M)) > of sections of vector bundles. This is possible for compact Hausdorff spaces > because of the Gelfand-Naimark theorem with relates the topology with the > algebra in a very tight way. There is an equivalence here. > Now take the purely algebraic statements and drop the assumption that the > algebra is commutative. > The underlying space may not literally exist as space of points. One sort of > just pretends the space exist by analogy with the commutative case. > Its not an easy subject to explain but try looking at > http://www.math.unam.mx/~micho/qgeom_m.html To use more low-brow language: a manifold may be identified with the set of maximal ideals of the commutative function algebra on the manifold. In local coordinates, a maximal ideal consists of the functions f(x) that vanish at x=x_0. They form an ideal, because f(x)g(x) also vanishes at x=x_0 for any g(x). So we identify this ideal with the point x_0. Noncommutative geometry (NCG) is now obtained by replacing the function algebra by a noncommutative algebra. Since such an algebra need not have very many ideals (none if it is simple), von Neumann called it "pointless geometry". Quantization can be thought of as endowing phase space with a noncommutative geometry; the pointlessness leads to Heisenberg's uncertainty relation. However, when Connes and others speak about NCG, I think they mean to replace the algebra of functions on configuration space with a noncommutative one. I have a problem to understand what happens to phase space in this process, which maybe somebody has the answer to. The problem is the following: Assume first that the algebra of coordinates in configuration space takes the form [q^i, q^j] = f^ij_k q^k, where f^ij_k are structure constants which vanish in the classical limit. Assume further that we still have the notion of a phase space. So we add momenta p_j. They can not satisfy [p_j, q^i] = \delta^i_j, because that would lead to f^ij_k = [p_k, [q^i, q^j]] = 0. It is possible to trivially construct a NCG by twisting the elements in phase space. So let q^i be *commutative* coordinates, with momenta p_j defined as above, and let x^a be an additional set of commutative coordinates. Now the new coordinates Q^i = q^i - g^ij_a p_j x^a, were g^ij_a are a set of constants, obey [Q^i, Q^j] = (g^ij_a - g^ji_a) x^a, [Q^i, x^a] = [x^a, x^b] = 0. So it appears that the Q^i define a NCG, but of course we know that it arose from the phase space of old-fashioned commutative geometry. My question is this: if the NCG is defined by a simple Lie algebra, then it is completely pointless. If it is defined by a nilpotent algebra, I think it can always be embedded in a phase space as above, so it isn't really deep either. So what kind of algebra defines a good NCG? Thomas ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: "pointless geometry" and QM Date: 24 Dec 2000 02:30:03 GMT Newsgroups: sci.physics.research Chris Hillman wrote: >The most elementary way to understand what "noncommutative geometry" is >all about is to consider some "geometric categories" (on the right) which >are -equivalent- to "algebraic categories" (on the left). A small table >might help: > > "Geometric category" "Algebraic category" > > --------------------------------------------------------------- > > compact Hausdorff space commutative semisimple > X unital C-* algebra A > > conts. maps norm decreasing algebra homs > X -> Y A -> B > > ---------------------------------------------------------------- > > Borel measure on X, mu positive linear functional on A > > measure preserving norm decreasing algebra hom > map X -> Y Yes, this is definitely a good way to get going on noncommutative geometry. Strictly speaking, the categories on the left are equivalent to the *opposite* of the categories on the right: all the arrows get reversed, e.g. a map f: X -> Y between spaces gives a map f*: C(Y) -> C(X) between their algebras of continuous functions. This is a nice way to keep track of the difference between "geometry-flavored" categories, where the objects are "spaces" of some sort, and "algebra-flavored" categories, where the objects are algebras of some sort. In fact I think Schanuel even gave a purely category-theoretic definition of what it meant for a category to be "geometry-flavored", but now I can't remember it.... And then, as you say, the trick is to take certain algebra-flavored categories and *define* their opposites to be categories of certain new generalized sorts of "spaces". Noncommutative geometry is a great example, where we invent "quantum topological spaces" as the opposite of the category of C*-algebras, but we can also get nice examples from algebraic geometry, where certain commutative rings with nilpotents wind up corresponding to "infinitesimal spaces" - e.g. the space with two infinitesimally separated points, which we've already discussed here on s.p.r.... And somewhere between noncommutative geometry and commutative algebraic geometry, there's supercommutative geometry. And a bit more general than that, but still far from general noncommutative geometry, there's braided-commutative geometry. For example, the algebra generated by x and y with the relation xy = qyx is braided-commutative, and we can think of it as the algebra of "functions" on a certain generalized "space", which reduces to the ordinary plane in the q -> 1 limit.