From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Higgs found? Date: 13 Sep 2000 08:00:43 GMT Newsgroups: sci.physics.research Summary: [missing] In article , Aaron Bergman wrote: >In article <8pl406$4q3$1@nnrp1.deja.com>, squark@my-deja.com wrote: >> Actually Connes managed to interpret the Higgs as a gauge boson using >> an extension of usual space-time to new, non-commutative dimensions. >IIRC, it's actually just a noncommutative Z_2. A noncommutative Z_2 - now *that's* something mathematicians would like to hear about! :-) Seriously, if all you want is the Higgs, Connes doesn't need full-fledged noncommutative geometry: he just needs a "two-sheeted" spacetime of the form R^4 x Z_2. I.e.: two copies of ordinary Minkowski spacetime, about 10^{-16} centimeters apart from other. The algebra of functions on this spacetime is still commutative! It's just his generalized definition of "connection" that allows the two sheets to talk to each other, with the Higgs as the part of the connection that lets you compare the value of a field on one sheet to its value on the other sheet. He got the Higgs this way in his very simplest model, "Model I" of his "Essay on Physics and Non-Commutative Geometry". When you want the whole standard model, that's when you need some noncommutative geometry: you have to take the algebra of functions on R^4 and tensor it with a finite-dimensional noncommutative algebra. >Unfortunately, the model has trouble with mass relations or something >like that, I think. Really? There are a bunch of versions of this model, which give different relations between masses of particles. The first ones were obviously "wrong", and only meant to illustrate the general idea; he has been working to improve them, and I didn't think the newest ones were experimentally ruled out. They are, however, becoming more and more complicated. >It's too bad, really, especially as the model actually gives almost exactly >the standard model gauge group. I'm not sure what you mean here. The simplest models (the 3 presented in his old "Essay on Physics and Noncommutative Geometry") have gauge group either U(1) x U(2) or U(1) x SU(2), and it's exciting how this pops out so naturally. The more sophisticated models include the SU(3) for the strong force, but the problem (to my mind) is that this is put in more or less "by hand", namely by cleverly choosing the finite-dimensional algebra mentioned above. In other words, it seems to me that at this stage, his model no longer puts out more than he puts in. A nice description of one of Connes' more recent particle physics models can be found in Daniel Kastler's essay "Noncommutative Geometry and Basic Physics", which appears in the book _Geometry and Quantum Phyics_. Here he tensors the algebra of functions on R^4 by the noncommutative algebra C + H + C[3] where C is the complex numbers, H is the quaternions, and C[3] is the 3x3 complex matrices. These three summands correspond to the three factors in the Standard Model gauge group U(1) x SU(2) x SU(3). The stuff about Z_2 is gone in this model. Moreover, in this model one needs to input a bunch of numbers to describe the quark masses and weak mixing of the quarks - just as in the Standard Model. Kastler also mentions an interesting attempt to simplify things by taking the noncommutative algebra to be the semisimple quotient of U_q(sl(2)) where q is a third root of unity. This algebra turns out to be C + C[2] + C[3] Whether this is profound or a coincidence remains to be seen. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Higgs found? Date: 15 Sep 2000 22:29:39 GMT Newsgroups: sci.physics.research In article , Aaron Bergman wrote: >In article <200009130552.e8D5qvD07526@math-cl-n03.ucr.edu>, >baez@galaxy.ucr.edu (John Baez) wrote: >>Aaron Bergman wrote: >> >Unfortunately, the model has trouble with mass relations or something >> >like that, I think. >> Really? There are a bunch of versions of this model, which give >> different relations between masses of particles. >I read it in hep-th/9701078. I haven't actually seen it myself. It also >talks about a "disturbing Fermion doubling". Connes' original "Essay on physics and non-commutative geometry" was published in 1990. This contained 3 different models. Then came two papers by Connes and Lott, "Particle physics and noncommutative geometry" and "The metric aspect of noncommutative geometry", in 1990 and 1991. In his 1995 paper "Noncommutative geometry and reality", Connes made some major changes in his theory. He made a bunch more changes and included gravity in his 1996 paper "Gravity coupled with matter and the foundation of noncommutative geometry". The last big step that I know about was his paper with Chamseddine, "The spectral action principle". I don't know which if any models has a "disturbing fermion doubling. In 1997, Daniel Kastler and others used this latest version of Connes' theory to predict the Higgs mass. They give a prediction of 182 +/- 17 GeV. >>>It's too bad, really, especially as the model actually gives almost >>>exactly the standard model gauge group. >> I'm not sure what you mean here. >You're right. Looking back at the paper, I misread it -- it went from >the two point model to the Standard Model so fast that I missed the >transition. You gotta keep an eye on these grand unification shell games. Look away for a minute and they'll be acting like they "derived" the Standard Model gauge group and so on - when in fact they slipped this stuff in by hand!