From: baez@rosencrantz.stcloudstate.edu () Subject: Re: Geometric Quantization Date: 31 Aug 2000 03:03:49 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8ofvoj$2sd8$1@mortar.ucr.edu>, Toby Bartels wrote: >John Baez wrote: >>It used to confuse the bejeezus out of me that "symplectic group" >>was used to mean two completely unrelated things: the group of >>real matrices that preserve a symplectic structure on a real vector >>space, and the group of quaternionic matrices that preserve an inner >>product on a quaternionic vector space. >>Eventually I realized that these were not unrelated at all! They >>are two real forms of the same complex simple Lie group, and there >>really is a profound conceptual connection between symplectic >>structures and quaternions that's responsible for this "coincidence". >OK, then you'll have to explain how this combines with >the much more prosaic explanation for the coincidence that I know. Okay, but let's see what your explanation is first. >Specifically, if Sp_{2n}\K for \K = \R,\C,\H is the group of >automorphisms of a nondegenerate symplectic form on \K^{2n}, >while Sp(n) is the group of automorphisms of a Hermitian form on \H^n, >then the relationship is that Sp(n) = U(2n) intersect Sp_{2n}\C. Hmm. This is perfectly true, but I wouldn't call it an "explanation" - you state that Sp(n) = U(2n) intersect Sp_{2n}\C, but you don't say why it's true! The proof that this is true is where we can see the profound conceptual connection between symplectic structures and the quaternions. Here's how I think about this stuff. Suppose H is an irreducible unitary representation of some group. And supppose H is is isomorphic to its dual. Then we get following amazing fact: There is a conjugate-linear isomorphism j: H -> H, and EITHER: A) j^2 = 1 OR B) j^2 = -1. In case A): j is a "real structure", since it makes H into the complexification of the real Hilbert space {x in H: jx = x}, which is also a representation of our group. In case B): j is a "quaternionic structure", since it makes H into a quaternionic Hilbert space, which is also a representation of our group. Why? Well, acting on H we have the usual complex number i, and also this operator j, satisfying i^2 = j^2 = -1, ij = -ji. Case A) happens precisely when we have a nondegenerate symmetric bilinear pairing f: H x H -> C which gives an intertwining operator from H tensor H to C. In other words, when H is equipped with an "orthogonal structure". Case B) happens precisely when we have a nondegenerate antisymmetric bilinear pairing f: H x H -> C which gives an intertwining operator from H tensor H to C. In other words, when H is equipped with a "symplectic structure". All these facts are easy to prove so I leave them to you. The first step is to realize that a nondegenerate pairing f: H x H -> C is the same as an isomorphism g: H -> H* is the same as a conjugate-linear isomorphism j: H -> H. The second step is to write H tensor H as the symmetrized tensor square plus the antisymmetrized tensor square and use Schur's Lemma all over the place. You can find proofs in Adams' "Introduction to Lie Groups" and also in my paper "Higher-Dimensional Algebra II: 2-Hilbert Spaces". This stuff is incredibly important! We've got a marvelous analogy: unitary : orthogonal : symplectic :: complex : real : quaternionic with (amusingly) the case of representations on *complex* Hilbert spaces being the basic one, and the real and quaternionic cases showing up as the two ways a representation can be isomorphic to its dual! >(BTW, I got these formulae from pages 98,99 of Fulton & Harris. >The same page says SL_n(\H) is the matrices "with real determinant one". >See below.) Oh-ho! I should read that! So maybe it's all very simple. >What has Doug Sweetser been up to lately, anyway? >(We need to put him in the acknowledgements for our paper.) He sent me an email in response to my remark about quaternionic determinants, so he's still alive and kicking. >>>I read something about Jordan algebras once, but I forgot about them. >>Good. Just don't forget the 26-dimensional one. >There's only 1? This has to do with string theory, right? Heh. First of all, I screwed up: I meant the 27-dimensional one! When you classify simple Jordan algebras, it's a bit like classifying simple Lie algebras: you get a bunch of infinite families and a few exceptional ones. In the case of Lie algebras you get 5 exceptions, which are all related to the octonions. In the case of Jordan algebras you get just ONE, which is the algebra of 3 x 3 self-adjoint octonionic matrices equipped with the product {A,B} = AB + BA. It's called "the exceptional Jordan algebra". It's 27-dimensional since you get 3 x 8 degrees of freedom for the off-diagonal entries and 3 for the on-diagonal ones. So why did I write "26"? Well, this has been on my mind lately: the group of automorphisms exceptional Jordan algebra is the exceptional Lie group F4, and the smallest nontrivial representation of this group is 26-dimensional. Why? Well, F4 acts on the exceptional Jordan algebra, so it has a 27-dimensional representation, but the identity matrix is preserved by all automorphisms, so this splits into a 1-dimensional trivial rep and a 26-dimensionsal rep. Anyway, seeing the number 26, one immediately hopes that this stuff is related to string theory. Is it? I think so, but I've been terribly lazy, so I haven't yet found out the details. I should read this paper: E. Corrigan, T. Hollowood, The exceptional Jordan algebra and the superstring, Commun. Math. Phys. 122 (1989) 392. and also a few others on related themes, all listed in reference 11 of Joerg Schray and Corinne A. Manogue, Octonionic representations of Clifford algebras and triality, http://xxx.lanl.gov/abs/hep-th/9407179. >You may recall my belief that time is imaginary. >Thus, while most momenta are imaginary, energy is real. Hmm. I'll have to see how that fits into the grand scheme I've been contemplating lately. All this algebra should really fit perfectly with everything we know about physics... we just need to think hard.