From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Geometric Quantization Date: 8 Aug 2000 21:55:11 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8m8klo$bn2@gap.cco.caltech.edu>, Toby Bartels wrote: >See here, John: the standard example of phase space >is the cotangent bundle of a previously given configuration space. >Now, do we recover configuration space as a subspace of this? >No! we recover configuration space as a *quotient* space of this. Well, if I felt like arguing, I'd say: we can do it either way! The projection p: T*M -> M exhibits the configuration space M as a quotient space of the phase space T*M. And the zero section s: M -> T*M exhibits the configuration space as a subspace of the phase space. I won't deny that the projection p is somehow more "fundamental" than the zero section s, but they are both perfectly natural in the category-theoretic sense, and they are both interesting. >So, I would define a coLagrangian quotient space of a Poisson manifold >to be a maximal quotient space satisfying the property that, >if f and g are functions on the alleged Lagrangian quotient space, >then {f,g} = 0 when f and g are interpreted as functions on phase space. >(A not necessarily maximal quotient space with this property >would be a coisotropic quotient space.) I haven't thought about these concepts before, but they are bound to be important - probably people have studied them already. Let me just remind everyone that it was squark who brought up the subject of Lagrangian submanifolds. They are interesting things, but they are not precisely what I was trying to talk about - so don't blame me if you don't like them! I was trying to talk about geometric quantization. So I was trying to talk about the all-important notion of a "polarization". This is what we need to take our Hilbert space of sections of the line bundle L -> X, where X is our phase space, and chop it down to get a smaller Hilbert space. The Hilbert space of *all* square-integrable sections is way too big. So we only look at sections whose covariant derivative vanishes in certain directions. Which directions? That's what the polarization tells us! So what's a polarization? Well, the simplest case is a real polarization. Given our symplectic manifold X, a real polarization assigns to each point x a Lagrangian subspace L_x of the tangent space T_x(X). To explain what this means, I had to say what a Lagrangian subspace of a symplectic vector space is... so I did that, and gave some examples. But I never got around to finishing the definition of a real polarization! Here it is: we want Lagrangian subspaces L_x of the tangent spaces T_x(X), these subspaces should vary smoothly with x, and they should fit together in a nice way: they should be "integrable". That means they are the tangent spaces of submanifolds that foliate X. (And yes, it follows that these are Lagrangian submanifolds... whoops, so they *are* somewhat relevant here.) Let's do an example, quick! Let X be the cotangent bundle T*M - the most traditional example of a phase space. Then at each point x of X we let L_x be the space of "vertical" tangent vectors: those that project down to zero when we apply the projection p: T*M -> M. It's easy to check that L_x really is a real polarization, and the submanifolds foliating X are just the cotangent spaces of M. That's good, because when we do geometric quantization, we want our Hilbert space to consist of sections of our line bundle L -> X that are covariantly constant along these submanifolds! And what do those amount to in this example? Just functions on the configuration space M! So we've got the usual Schrodinger recipe for quantization.... Toby will enjoy pondering this and seeing how it fits in with the yoga of quotient spaces versus subspaces... I won't deprive him of this pleasure by doing it for him here. Okay, enough of that for now. Now let's go back to some linear algebra! We've got a symplectic vector space V. And Toby muses: >[...] there must be some reason why >(at least when phase space is a vector space) >we can talk about subspaces instead of quotient spaces. >As we mathematicians know, many physicists love to >pretend quotient spaces are really subspaces, >since subspaces are easier to visualise. >This is OK when there are theorems to justify it, >and I guess we must have them here. Yes: the symplectic structure on V gives a map from V to V*, which is actually an isomorphism. And this allows us to identify subspaces with quotient spaces. It's all very much like the case of a real inner product space, but with a few twists. Let me mention one that will delight the mathematicians in the crowd (and nobody else): Suppose you start with a symplectic vector space V. We get a map V -> V* given by v -> w(v,.), and this is an isomorphism. Using this isomorphism we can transfer the symplectic structure on V to one on V*. And now we can play the game all over again! We get a map V* -> V**, which turns out to be an isomorphism. Composing these isomorphisms V -> V* and V* -> V**, we get an isomorphism V -> V**. Now there happens to already be a famous isomorphism from V to V**, so we can ask: did we just get the same one? And the answer is: NO! Instead, we get -1 times that famous isomorphism!!! This minus sign makes symplectic vector spaces a bit subtler than real inner product spaces, where the same sort of trickery gives the usual famous isomorphism from V to V**. And eventually, pondering this, one sees that it's all part of the analogy: bosons: symplectic vector spaces :: fermions : real inner product spaces The difference between bosons and fermions is just a sign.....