From: baez@galaxy.ucr.edu (John Baez)
Subject: Re: Geometric Quantization
Date: 7 Sep 2000 02:13:15 GMT
Newsgroups: sci.physics.research
Summary: [missing]

In article <8oqa93$2hn8$1@mortar.ucr.edu>,
Toby Bartels <toby@math-cl-n03.math.ucr.edu> wrote:

>John Baez wrote:

>>Hmm.  This is perfectly true, but I wouldn't call it an "explanation"
>>- you state a relation between the two meanings of "symplectic group",
>>but you don't say why it's true!  The proof that it's true is where
>>we can see the profound conceptual connection between symplectic
>>structures and the quaternions.

>OK, but it's the prosaicness that really bothers me.
>If the proof of this shows a connection between symplectic structures and \H,
>then there must also be a connection between symplectic structures and \C
>in the proof of U(n) = O(2n) intersect Sp_{2n}\R.

There is.  

Here's the proof that U(n) = O(2n) intersect Sp(2n,R):

Take an n-dimensional complex vector space with an inner product.  The
real part of this inner product is an inner product on the underlying
2n-dimensional real vector space, so the group of real-linear
transformations that preserve *that* is O(2n).  The imaginary part of
this inner product is a symplectic structure on the underlying
2n-dimensional real vector space, so the group of real-linear
transformations that preserve *that* is Sp(2n,R).  The transformations
that preserve *both* lie in the intersection of these two groups, so
U(n) is the intersection of O(2n) and Sp(2n,R).

I deliberately skipped one crucial little step of the proof here;
maybe someone can spot it and fill it in.

But the moral is clear: the cool thing about a complex inner product
is that it's built from a *real inner product* and a *symplectic structure*.

It might be nicer to say the same thing with different terminology: a
unitary structure is built from an orthogonal structure plus a symplectic 
structure.  

(Here a "unitary structure" is my name for a complex inner product,
and an "orthogonal structure" is my name for a real inner product.)

As you know, I've rhapsodized endlessly over the profound implications
this has for physics!  A symplectic structure is just what the phase
space of a boson has, and just what need to write down the canonical
commutation relations.  An orthogonal structure is just what the phase
space of a fermion has, and just what we need to write down the canonical
anticommutation relations.  A complex Hilbert space has both!  To get
from  one to the other, we need a compatible complex structure.  And so
on, and so on... I wrote a 291-page book about this, so I won't go on.

(By the way, I just gave away the answer to the puzzle I posed earlier
in this post.  At least if you read carefully....)

(And if you dig a wee bit deeper, you'll see what the connection between
symplectic structures and C is - the one that Toby wanted.)
 
>IOW, if your profound relationship is the analogy
>unitary : symplectic :: \C : \H in a complex setting,
>then there should also be the analogy
>orthogonal : symplectic :: \R : \C in a real setting,
>giving a profound relationship between symplectic structures and \C.

Right.   To really see what's going on, it's good to see how
many concepts like this you can make sense of:

A complex structure on a real vector space.
A real structure on a complex vector space.
A quaternionic structure on a real vector space.

... and so on, for a total of 6 - or 9 if you wish to count
trivial structures like "a real structure on a real vector
space".  For example:

A complex structure on a real vector space is a real-linear
operator J with J^2 = -1.  (Think "multiplication by i".)

A real structure on a complex vector space is a conjugate-linear
operator K with K^2 = 1.  (Think "complex conjugation".)

A quaternionic structure on a real vector space is a pair
of real linear operators I, J with I^2 = J^2 = -1, IJ = -JI.

... and so on: I leave the rest for you!

And then repeat this, replacing the words "vector space" by 
"inner product space"!  You'll get lots of fun stuff.  You 
might even think about it all in terms of functors between
categories.  

>The complex case has to be the basic one,
>because only there does Schur's lemma apply in all its glory.

In the result I was talking about earlier, the complex case does
indeed seem basic.  This is how group representation people think
about it: first they classify irreducible *complex* representations of
a compact group, then they see which of them are self-dual, and then
they classify *those* into real and quaternionic reps.

>So, the reals have to do with bosons and the quaternions with fermions? 

I guess that's what the math gods are trying to tell us!  

>>we just need to think hard.  

>Isn't that always the way?  

Heh, I guess so.  The big picture is staring us in the face; 
we just need the wits and patience to see it.