From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Geometric Quantization Date: 31 Jul 2000 19:14:12 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8ls7kl$j4d$1@nnrp1.deja.com>, wrote: >In article <8lprsq$gsn@math.ucr.edu>, > baez@galaxy.ucr.edu (John Baez) wrote: >> A few comments now, more later. Here we're talking about how to >> compute the dimension of the "Lagrangian Grassmannian": the manifold >> of all Lagrangian subspaces of a given symplectic vector space of >> dimension 2n. And we're blundering all over the place, but having >> a fair amount of fun in the process. >A naive question: what is a Lagrangian subspace? First: suppose we have a vector space V with a nondegenerate skew-symmetric bilinear map w: V x V -> R. Then we call V a "symplectic vector space" and call w the "symplectic structure". In this context, a vector subspace L of V is said to be "Lagrangian" if: 1) w(v,w) = 0 whenever v and w lie in L and 2) the dimension of L is half the dimension of V. Or if you prefer, we can replace 2) by the equivalent 2'): L is maximal, i.e., we cannot make L any bigger without losing property 1). In physics, the symplectic vector space V serves as the "phase space" of a classical system and the Lagrangian subspace L is a choice of something that can play the role of "configuration space". In other words, L is a choice of a maximal Poisson-commuting set of linear coordinate functions. This sort of choice becomes very important when we quantize. For example, if we took V to be 4-dimensional, with basis p1,p2,q1,q2 and the usual symplectic structure w(p1,q1) = w(p2,q2) = 1 (all others zero), we could take the Lagrangian subspace L to have the basis q1,q2 - this would be the usual choice of configuration space for the "position representation". Or we could take L to have basis p1,p2, as we do in the "momentum representation". Or we could take it to have basis p1,q2, or p2,q1. Or lots of other things! The space of all these choices is the "Lagrangian Grassmannian". Extra technojargon for the budding expert: a subspace with property 1) is said to be "isotropic", so a Lagrangian subspace is a maximal isotropic subspace.