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>,  <squark@my-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.