Seventh Post in the BAUT RST Forum
Doug
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The seventh post in the BAUT forum follows. These posts are a continuation of the RST thread in the “Against the Mainstream” forum of BAUT.
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Hmmm. I’m not sure how to interpret the silence. I know this is primarily an astrophysics/cosmology forum, but the science of astrophysics and cosmology rests on the foundation of particle physics. The whole idea of big bang, cosmic inflation, nucleosynthesis, etc, is developed on the basis of the standard model and general relativity theories, so these fundamental sorts of things are hopefully seen as vital here. Again, the trouble with physics right now is that these two theories must be treated separately, as if the quantum and the continuum were unrelated, though we know that they are not, and this conflict emerges most clearly in today’s concepts of cosmology and astrophysics.
However, to make my point a little more explicit and to explain things a little further, I’ll refer to John Baez’s characterization of Geoffrey Dixion’s work in Week 59 of his online mathematics tutorials:
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| Originally Posted by Baez Dixon is convinced that the details of the Standard Model of particle interactions can be understood better by taking certain mathematical structures very seriously. There are very few algebras over the reals where we can divide by nonzero elements: if we demand associativity and commutativity, just the reals themselves and the complex numbers. If we drop the demand for commutativity, we also get a 4-dimensional algebra called the quaternions, invented by Hamilton. If in addition we drop the demand for associativity, and ask only that our algebra be “alternative”, we also get an 8-dimensional algebra called the octonions, or Cayley numbers. (I’ll say what “alternative” means in “week61”.) Clearly these are very special structures, and also clearly they play an important role in physics… or do they? |
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| …there are not too many places in physics yet where the octonions reach out and grab one with the force the reals, complexes, and quaternions do. But they are certainly out there, they have a certain beauty to them, and they are the natural stopping-point of a certain finite sequence of structures, so it is natural for people of a certain temperament to believe that they are there for a reason. Dixon makes a passionate case for this in the beginning of his book. |
Quote:
| The octonions show up in many contexts in superstring theory, but few would say this theory is “based” on the octonions. A better description might be that string theorists are forced into using special algebraic structures to get things to work, and these special structures have an unnerving habit of being related to the octonions. Of course, this is far more interesting than if they’d *set out* to incorporate the octonions into their work. Geoffrey Dixon has an extension of the Standard Model which makes use of all four normed division algebras: reals, complexes, quaternions and octonions: Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Kluwer, Dordrecht, 1994. Not enough people have looked at this stuff carefully, but if one does, one finds interesting relations to the SU(5) and SO(10) grand unified theories, which show how the patterns underlying the symmetry breaking in these theories are related to division algebras. Sometime I’d like to write about this. You can read more about Dixon’s ideas here: http://www.7stones.com/Homepage/Alge…/algebra0.html There is also a bit more lurking in my seminar notes on Clifford algebras, spinors, the Standard Model and the SU(5) grand unified theory: http://www.math.ucr.edu/home/baez/qg-spring2003/ though I didn’t manage to cover enough of the really cool stuff. |
The main mathematical challenge of this structure is best described by the loss of its elements’ properties, as the dimension of the hypercomplex numbers increases; that is, in the words of one commentator on Baez’s site:
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| The real numbers are *not* of characteristic 2, so the complex numbers don’t equal their own conjugates, so the quaternions aren’t commutative, so the octonions aren’t associative, so the hexadecanions aren’t a division algebra. |
In Week 211, Baez writes:
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| Bott periodicity is all about how math and physics in n+8-dimensional space resemble math and physics in n-dimensional space. It’s a weird and wonderful pattern that you’d never guess without doing some calculations. It shows up in many guises, which turn out to all be related. The simplest one to verify is the pattern of Clifford algebras. |
Quote:
| Originally Posted by “John Baez” There are some spooky facts in mathematics that you’d never guess in a million years… only when someone carefully works them out do they become clear. One of them is called “Bott periodicity”. A 0-dimensional manifold is pretty dull: just a bunch of points. 1-dimensional manifolds are not much more varied: the only possibilities are the circle and the line, and things you get by taking a union of a bunch of circles and lines. 2-dimensional manifolds are more interesting, but still pretty tame: you’ve got your n-holed tori, your projective plane, your Klein bottle, variations on these with extra handles, and some more related things if you allow your manifold to go on forever, like the plane, or the plane with a bunch of handles added (possibly infinitely many!), and so on…. You can classify all these things. 3-dimensional manifolds are a lot more complicated: nobody knows how to classify them. 4-dimensional manifolds are a lot more complicated: you can prove that it’s impossible to classify them - that’s called Markov’s Theorem. Now, you probably wouldn’t have guessed that a lot of things start getting simpler when you get up around dimension 5. Not everything, just some things. You still can’t classify manifolds in these high dimensions, but if you make a bunch of simplifying assumptions you sort of can, in ways that don’t work in lower dimensions. Weird, huh? But that’s another story. Bott periodicity is different. It says that when you get up to 8 dimensions, a bunch of things are a whole lot like in 0 dimensions! And when you get up to dimension 9, a bunch of things are a lot like they were in dimension 1. And so on - a bunch of stuff keeps repeating with period 8 as you climb the ladder of dimensions. (Actually, I have this kooky theory that perhaps part of the reason topology reaches a certain peak of complexity in dimension 4 is that the number 4 is halfway between 0 and 8, topology being simplest in dimension 0. Maybe this is even why physics likes to be in 4 dimensions! But this is a whole other crazy digression and I will restrain myself here.) |
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