From: baez@galaxy.ucr.edu (John Baez)
Subject: Re: Geometric Quantization
Date: 14 Aug 2000 21:05:01 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <8n6qfc$ffe$1@nnrp1.deja.com>,  <squark@my-deja.com> wrote:

>Another note about polarizations: the integrability of the subspace
>field, i.e. the existance of a corresponding foliation by Lagrangian
>submanifolds is important. 

Yes indeed!

NOTE ON TERMINOLOGY:

Folks often use the term "distribution" for what you're calling a 
"subspace field".  To be precise: a "distribution" on the smooth
manifold X is a subspace V_p of each tangent space T_p X, such that 
V_p depends smoothly on p.   

Of course this is horrible terminology, since "distribution" means at 
least 2 other quite different things in mathematics.  The term "subspace 
field" is much better - except for one big problem: it reminds me of Star 
Trek.

We say the distribution V is "integrable" if there is a foliation of X by
submanifolds such that V_p is the tangent space of the submanifold 
containing the point p.  

So in standard jargon, a real polarization is an integrable Lagrangian
distribution.  Or in your (better) terminology, a real polarization is
an integrable field of Lagrangian subspaces.  

>Otherwise, we cannot expect our space to be
>very big. We have the torus example we discussed a while ago, for
>instance, where the irrational slope lines lead to polarizations which
>give rise to a one dimensional space - for obvious reasons.

Hmm, but in this example the subspace field *is* integrable!   

In other words, there *is* a foliation here, namely the foliation 
of the torus by lines of irrational slope.  

The problem is not a lack of integrability.  I think the main problem
is that the space of leaves of this foliation is non-Hausdorff when 
we give it the quotient topology.

>Now I
>remember reading Allan Connes' "Noncommutative Geometry", and in the
>very beginning, he had this example of the spaces of leaves of a
>foliation. When the "foliation" - whatever he called it - is not
>integrable in the above sense, the "quotient space" is very bad
>behaved. 

Ah, see, you're using the words a bit wrong here, but you've got
the right idea.  We have an integrable field of subspaces, so we 
have a foliation, but the quotient space (the space of leaves) is 
non-Hausdorff.  

And one good thing about noncommutative geometry is that it gives 
a nice way to construct a noncommutative version of the "space of 
leaves" in this context, which is better than the actual space of
leaves.  

>Now, it may be Connes even invented the concept in
>the context of geometric quantization, I simply wouldn't remember that
>as I didn't know anything about geometric quantisation then.

Were you British back then?  I note a certain vacillation in your
spelling here.  :-)

Anyway, I don't recall Connes trying to relate this stuff to geometric
quantization, though it's certainly related to quantum theory.

By the way, I *will* eventually get around to telling you how 
observables get quantized in the context of geometric quantization.
Actually, I'll start by explaining how they get "prequantized".
But not in this post!