From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.physics
Subject: Poisson brackets, symplectic geometry, quantization - and everything!
Date: 25 Jan 93 02:37:32 GMT
Organization: University of California, Riverside

Micheal Weiss writes:

> I'm emailing you the question, but (should you have time to respond) you
> should probably post the answer, since I'm sure I'm not the only one who'd
> be interested.

Sorry that I have been rather busy with guests and slow to reply.
 
> I'm trying to work out the connections between two things you've written,
> and Dirac's famous Poisson bracket equation:
>
>     uv - vu  =  ih/2pi [u,v],     where [u,v] is the P.B. of u and v
>
> You wrote at one time:
>
> Note that there IS a functor from the symplectic category to the
> Hilbert category, namely one assigns to each symplectic manifold X the
> Hilbert space L^2(X), where one takes L^2 w.r.t. the Liouville measure.
> Every symplectic map yields a unitary operator in an obvious way.
> This is called PREQUANTIZATION.  The problem with it physically is that
> a one-parameter group of symplectic transformations generated by a
> positive Hamiltonian is not mapped to a one-parameter group of unitaries
> with a POSITIVE generator.  So my conjecture is that there is no
> "positivity-preserving" functor from the symplectic category to the
> Hilbert category.)
>
> On another occasion:
>
> In the case of bosons, the classical phase space is a symplectic vector
> space (i.e., equipped with a nondegenerate antisymmetric bilinear pairing).
> When we have picked a complex structure, the antisymmetric pairing is
> supposed to become the imaginary part of a (complex) inner product on V,
> which then can be completed to obtain a complex Hilbert space.
>
> (I've silently elided phrases about the fermionic case in this quotation.)
>
> I have the feeling all three statements (Dirac's P.B. equation,
> prequantization, and the equation Im(inner product)=symplectic form) all
> belong to the same circle of ideas, but I don't quite see the details.

Indeed they do belong to the same circle of ideas - but alas, the circle is
rather big, and (typical of circles) it's hard to know where to start!  First
let me say that there is a vast lore concerning prequantization, and then 
quantization, of classical mechanical systems whose phase space is a 
symplectic manifold.  Prequantization is easy (and I've described most of
it above!), but quantization is hard.  Note that in prequantization one cooks
up a Hilbert space by taking L^2 of the phase space.  This is "twice as big 
as it should be," since in the simplest kinds of quantization one uses L^2
of the *configuration* space, which is a manifold of half the dimension.
So the trick is to find a Hilbert space that's "half as big" lurking inside
L^2 of the phase space.  When one does this correctly, your Hamiltonian
(which started life as a postive function on the classical phase space) should
somehow correspond to a positive operator on the Hilbert space.

There are two basic approaches:

1)  If your phase space (= symplectic manifold) is the cotangent bundle of
a manifold (the configuration space), you can use L^2 of the configuration
space.  This is the simplest approach and is used in freshman quantum 
mechanics.  

2)  If your phase space is not just a symplectic manifold but actually
a Kaehler manifold (a complex manifold with an inner product on each tangent
space whose imaginary part is the symplectic structure), you can take
the holomorphic L^2 functions on phase space.  

Both these approaches work fine for the LINEAR case in which your phase space 
is just C^n (or, basically the same, the cotangent bundle of R^n).  Moreover
both generalize the INFINITE-DIMENSIONAL linear case, which is what 
comes up in quantum field theory, and is the subject of
my book with Segal and Zhou.  Approach 1 (the "real wave representation")
and approach 2 (the "complex wave representation" aka "Bargmann-Segal 
representation") give isomorphic answers.   There is a 3rd equally good approach
in this case, more algebraic and less geometrical, the "particle
representation" aka "Fock space".  This is most often used in quantum 
field theory by people who just want to calculate the answers.  The
real wave representation is nicer for constructive quantum field theory.
The complex wave representation illuminates some other features, namely:
for prequantization all we needed was the symplectic structure (in the
finite-dim case), but for quantization we need the complex inner product 
of which the symplectic structure is the imaginary part.  In other words,
quantization requires making a *choice*.  Now in the finite-dimensional case
this choice turns out not to matter much - that is, given two different
inner products having the same imaginary part, we get two different complex
wave representations, but they turn out to be isomorphic.  In the infinite
dimensional case this fails, and the choice of inner product really matters
a lot!  This is why such things as picking the right complex structure
(to get an inner product from a symplectic form) are so important.

Here's another way of putting it that might clarify the relation to
Dirac's equation

     uv - vu  =  ih/2pi [u,v],     where [u,v] is the P.B. of u and v

Say we are given a symplectic vector space V with symplectic
form omega.  We say that a "Weyl system over V" is the following:

1)  A Hilbert space K
2)  A real-linear map phi from V to self-adjoint linear operators 
on K ("field operators") such that

[phi(v),phi(w)] = i omega(v,w).

Here I am eliminating the hbar and also ignoring the analysis problems
in taking commutators of unbounded self-adjoint operators.   See our
book for the way to do it right!  A Weyl system is a kind of quantization
(or prequantization) of the classical system whose phase space is V.
The Stone-von Neumann theorem says that (modulo those analysis problems!)
there is a unique irreducible Weyl system over a finite-dimensional 
symplectic vector space, and all the rest are direct sums of this.
This theorem breaks down in infinite dimensions, which is part of why
QFT is so hard.

A Weyl system only notices the symplectic structure of V.  If however
V is really a complex Hilbert space (with the imaginary part of its inner
product as a symplectic form), we can define the "free boson field over V"
to be a Weyl system with 

1) a unit vector v in K ("the vacuum") which is a cyclic vector for
the operators phi(v) (so all states can be obtained from the vacuum by hitting
it with products of field operators and taking linear combinations and limits
thereof).
2) a unitary representation Gamma of the unitary operators of V on K, such 
that a) v is invariant under Gamma, b) Gamma is "positive", i.e.
for any self-adjoint A on V with A >= 0, we have dGamma(A) >= 0, and
c) Gamma(g)phi(v)Gamma(g^{-1}) = phi(gv).

Part 2 makes the symmetries of the classical system V act as symmetries
of the quantum Hilbert space K in a nice way.  The big theorem is that
for every Hilbert space V there is a *unique* (up to isomorphism) free
boson field.  But note that this depends on the inner product on V, not
just on the symplectic form.




Newsgroups: sci.physics