Geometric Quantization (Part 3)

 

Okay, I’ll stop warming up and actually do something. I’ve secretly been trying to convince you that it’s not utterly insane to start geometric quantization with something other than a symplectic manifold—since, after all, geometric quantization normally starts with much more than a mere symplectic manifold. In Parts 1 and 2 I explained that we often start with a Kähler manifold together with a suitable line bundle. This is a big pack of data that includes a manifold that’s simultaneously symplectic, complex and Riemannian, all in a compatible way—but also more, namely the line bundle.

Now I’ll do a stripped-down, bare-bones version of geometric quantization. It starts with something that sounds quire different: a smooth complex projective variety. This is a smooth submanifold of that’s picked out by homogeneous polynomial equations.

In fact this is not as different as it sounds: if we give its Fubini–Study metric, any smooth projective variety becomes a Kähler manifold! We just restrict the Fubini–Study metric to

Furthermore, has a god-given line bundle over it. You see, any point in is really a line through the origin in , and these lines define a line bundle over This construction is so ridiculously tautological that people call the resulting bundle the tautological line bundle. This bundle has no holomorphic sections, but its dual has lots: every linear functional on gives one, and that’s how they all arise! Let’s call this dual bundle If we restrict to we get a holomorphic line bundle with enough sections to be useful in geometric quantization. Let’s call this bundle

Even better, and this line bundle have the whole laundry list of extra structure and properties that I was bemoaning last time. As I already said, is Kähler. Furthermore, is holomorphic. Better yet, there’s a god-given hermitian metric on coming from the standard inner product on (and thus its dual). So, as I explained near the end last time, we get a connection on called the Chern connection. And if I’m not horribly mistaken, the curvature of this connection gives the symplectic structure on at least up to those factors of and that I keep losing track of.

In short, we get everything we want to use in Kähler quantization starting from something very simple: a smooth complex projective variety!

So let’s formalize this a little. Let’s define two categories, and and make geometric quantization into a functor

I’m using a crude teletype font here because this is the first and crudest of a series of similar constructions. In fact I’ll restrain my desire to show off and do the simplest thing I can. My categories will be mere posets.

I’ll fix a number and let be the poset of smooth linearly normal subvarieties of ordered by inclusion. Don’t worry—I’ll tell you what ‘linearly normal’ means in a minute. I’ll let be the poset of linear subspaces of also ordered by inclusion.

You can think of any object of as a ‘space of classical states’, since as we’ve seen it’s a symplectic manifold of a very nice kind, bedecked with all the bells and whistles we need for geometric quantization. Similarly you can think of any object of as a ‘space of quantum states’, since as a subspace of it’s a Hilbert space.

Here’s how the functor works on objects. As we’ve seen, any smooth projective variety comes with a line bundle namely the restriction of the dual of the tautological line bundle to To quantize we form the space of holomorphic sections of

Now, this doesn’t instantly look like a subspace of But we can identify it with one as follows. Every holomorphic section of restricts to a holomorphic section of And when is linearly normal, every holomorphic section of arises this way! This is just the definition of ‘linearly normal’.

In this case, the space of holomorphic sections of is a quotient of the space of holomorphic sections of But I’ve already mentioned that the latter space can be identified with So, we’ve got a quotient space of But by the magic of linear algebra, this corresponds to a subspace of And that’s

This sounds complicated! Luckily, when you unravel it all you get a much simpler description, pointed out by Allen Knutson in a comment to Part 1. For any variety there’s a smallest subspace such that sits inside the projective space We have

We can take this as the definition of if we want. This simpler description makes it blatantly obvious that

So, is a functor.

Now, I said in Part 1 that there’s also a reverse process going from the quantum realm back to the classical. This is even simpler: it’s ‘projectivization’, and it gives a functor

It works like this. An object is a linear subspace of This gives a projective variety Since this variety is linearly normal, we can define

Sorry for the subtle difference in fonts… but it’s okay to use subtly different fonts for two things that are the same, right?

Again, it’s obvious that

so is a functor.

Moreover, quantization and projectivization are adjoint functors! In fact quantization is the left adjoint of projectivization. This simply means that

And this is obvious: a projective variety sits inside the projectivization of some linear subspace iff the smallest linear subspace whose projectivization contains is contained in

(Mathematicians have a tendency to say something is ‘obvious’ before reeling off a string of words no normal person would ever say.. All I mean is that no deep facts are required to see this.)

Even better, this adjunction has a special extra property:

for all Again this is obvious: the smallest linear subspace whose projectivization contains the projectivization of is

Thanks to this extra property we say is a reflective subcategory of A more familiar example of this situation is how is a reflective subcategory of The forgetful functor has a left adjoint, namely ‘abelianization’, with the special property that if we take an abelian group, forget that it’s abelian and think of it as a group, and then abelianize it, we get back the same abelian group we started with. (At least this is true up to natural isomorphism. But when we projectivize and then quantize we get back the same subspace exactly: that’s because the categories and are mere posets, where all isomorphisms are identity morphisms.)

So, we’re seeing that just as abelian groups are especially nice groups, and we can ‘abelianize’ any group to make it nice, quantum state spaces are especially nice classical state spaces, and we can ‘quantize’ any classical state space to make it nice!

This is all very pretty, I think. However, it would be nice to remove some of the ridiculous restrictions I’ve imposed so far, like only considering projective varieties sitting inside a fixed projective space and only allowing inclusions as morphisms between these. That’s what I’d like to do next.

I should also note that while I’ve muttered words like ‘Hilbert space’, ‘symplectic struture’, ‘Kähler manifold’ and ‘hermitian line bundle’, my construction of the functors

did not use the inner product on at all! That’s why I was talking about the dual space So, all these structures involving inner products are not an intrinsic part of what’s going on—at least, not yet! But they can be slapped on at will, simply by endowing with an inner product: any inner product, for example its standard one.

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• Part 1: the mystery of geometric quantization: how a quantum state space is a special sort of classical state space.

• Part 2: the structures besides a mere symplectic manifold that are used in geometric quantization.

• Part 3: geometric quantization as a functor with a right adjoint, ‘projectivization’, making quantum state spaces into a reflective subcategory of classical ones.

• Part 4: making geometric quantization into a monoidal functor.

• Part 5: the simplest example of geometric quantization: the spin-1/2 particle.

• Part 6: quantizing the spin-3/2 particle using the twisted cubic; coherent states via the adjunction between quantization and projectivization.