Geometric Quantization (Part 4)

Last time I showed that geometric quantization could be made into a functor—and that this functor has right adjoint, ‘projectivization’, going back from the quantum realm to the classical. This was just a preliminary version of something that deserves to be polished up a lot. I’d also like to look at a bunch of examples of how this functor works, because they raise a lot of interesting questions.

I’m a bit torn between what order to do all this stuff: polish and then give examples, or give examples and then polish? I think I’ll do a little easy polishing today—very superficial, nothing deep—and then look at examples in my next post.

One reason to do this is to clear away some clutter. In Part 2 I sketched the usual version of Kähler quantization, which is fairly complicated. Last time I switched to a stripped-down version where these complications are removed… but tried to explain how it was ‘backwards-compatible’ with the usual story, by showing how all the complications could easily be restored if you want them. Unfortunately this makes things look more complicated than they are!

So let me present the same construction again, more cleanly.

For any n \ge 1, let \texttt{Class}_n be the poset of linearly normal varieties in \mathbb{C}\mathrm{P}^{n-1}, ordered by inclusion. Let \texttt{Quant}_n be the poset of linear subspaces of \mathbb{C}^n, also ordered by inclusion. Define the order-preserving map

\texttt{Q}_n \colon \texttt{Class}_n \to \texttt{Quant}_n

as follows: for any M \in \texttt{Class}_n let \texttt{Q}_n(M) be the smallest linear subspace of \mathbb{C}^n such that M is contained in the projectivization of that subspace. And define the order-preserving map

\texttt{P}_n \colon \texttt{Quant}_n \to \texttt{Class}_n

as follows: for any V \in \texttt{Quant}_n let \texttt{P}_n(V) be the projectivization of V.

Now for some remarks. I usually make fun of mathematicians who number remarks in their paper—when they’re at a party, do they number their jokes? But I feel like numbering these, so you can easily skip them if you want:

1) Last time I said that varieties in \texttt{Class}_n have to be smooth. This time I’ve dropped that condition! The reason is that last time I wanted to reassure skeptics by pointing out that our varieties could also be seen as Kähler manifolds, as traditional in geometric quantization. But when we do examples, it will also be fun to geometrically quantize non-smooth varieties, like those on top of these articles. (These pictures were drawn by Abdelaziz Nait Merzouk, and you can learn more by clicking on them.)

2) Everything I’m doing today would also work if we drop the condition that our varieties be linearly normal! However, when I geometrically quantize a projective variety M, I want to get the vector space of all sections of the dual of the tautological line bundle on M. The functors \texttt{Q}_n only do this job if and only if M is linearly normal. Otherwise they give a smaller vector space! So, I don’t think I want to drop linear normality.

It’s probably worth emphasizing that linear normality is an ‘extrinsic’ condition, depending not only on a variety but on its embedding in projective space. Later I will try a more ‘intrinsic’ approach to geometric quantization.

3) Everything I’m doing today would also work over an arbitrary field! I’m sticking to the complex numbers to keep physicists from thinking I’ve gone off the deep end. But it could be very interesting, at least for algebraists, to work more generally.

Okay, enough ‘remarks’—back to the main track. It’s a bit annoying to separately study geometric quantization for varieties in \mathbb{C}\mathrm{P}^{n-1} for each different n. We can get around this as follows.

For each n we have an inclusion

\mathbb{C}^n \to \mathbb{C}^{n+1}

sending each vector to the same list of numbers with a zero tacked on at the end. This gives an inclusion of posets

\texttt{Class}_{n-1} \hookrightarrow \texttt{Class}_n

So, we can take the colimit of the diagram

\cdots \hookrightarrow \texttt{Class}_{n-1} \hookrightarrow \texttt{Class}_n \hookrightarrow \texttt{Class}_{n+1} \hookrightarrow \cdots

or in more lowbrow terms, the union of all the posets \texttt{Class}_n, each included in the next, and get a big poset I’ll call \texttt{Class}. (Sorry, this notation conflicts with the one I used last time. I should have put subscripts everywhere last time.)

Similarly, we have inclusions of posets

\texttt{Quant}_{n-1} \hookrightarrow \texttt{Quant}_n

so we can take the colimit of

\cdots \hookrightarrow \texttt{Quant}_{n-1} \hookrightarrow \texttt{Quant}_n \hookrightarrow \texttt{Quant}_{n+1} \hookrightarrow \cdots

and get a poset I’ll call \texttt{Quant}.

This poset \texttt{Quant} seems familiar to me: it’s the poset of all finite-dimensional subspaces of a certain vector space called \mathbb{C}^\infty, the colimit of all the spaces \mathbb{C}^n. The poset \texttt{Class} is less familiar to me. We can intuitively think of it as the poset of all linearly normal subvarieties of \mathbb{C}\mathrm{P}^\infty that sit inside \mathbb{C}\mathrm{P}^n for some finite n. However, while I’m familiar with \mathbb{C}\mathrm{P}^\infty as a topological space, the colimit of all the manifolds \mathbb{C}\mathrm{P}^n, I have never heard people talk about ‘subvarieties’ of \mathbb{C}\mathrm{P}^\infty. This could easily be a hole in my education. (Is \mathbb{C}\mathrm{P}^\infty just a non-noetherian scheme?)

Now, the nice thing is that our maps of posets

\texttt{Q}_n \colon \texttt{Class}_n \to \texttt{Quant}_n,   \qquad \texttt{P}_n \colon \texttt{Quant}_n \to \texttt{Class}_n

get along with the inclusions

\texttt{Class}_n \hookrightarrow \texttt{Class}_{n+1} , \qquad   \texttt{Quant}_n \hookrightarrow \texttt{Quant}_{n+1}

forming two commutative squares that I’m too lazy to draw here. So, we get well-defined maps of posets

\texttt{Q} \colon \texttt{Class} \to \texttt{Quant},    \qquad \texttt{P} \colon \texttt{Quant} \to \texttt{Class}

Last time we saw that the maps \texttt{Q}_n and \texttt{P}_n were adjoint for each n. That implies the same result for \texttt{Q} and \texttt{P}. Don’t be scared; this just means that

\texttt{Q}(M) \subseteq V \quad \iff \quad M \subseteq \texttt{P}(V)

for all M \in \texttt{Class}, V \in \texttt{Quant}, and this is easy to check directly. (Or, if you’re a category theorist, dream up an abstract nonsense proof.)

Last time we also saw that \texttt{Quant}_n is a reflective subcategory of \texttt{Class}_n. That implies the same result for \texttt{Quant} and \texttt{Class}. This just means that

\texttt{Q}( \texttt{P} (V)) = V

for all V \in \texttt{Quant}. And again, this is easy to check directly.

So, we are back where we started—but now we can stop worrying about which \mathbb{C}\mathrm{P}^{n-1} our classical state spaces are subvarieties of, or which \mathbb{C}^n our quantum state spaces are subspaces of.

This lets us do some new things!

It’s fun to combine quantum systems using the tensor product of their vector spaces, and now we can do it. If we have linear subspaces V \subseteq \mathbb{C}^m and W \subseteq \mathbb{C}^n, we can take their tensor product and get V \otimes W \subseteq \mathbb{C}^m \otimes \mathbb{C}^n \cong \mathbb{C}^{mn}. So, we get a multiplication on \texttt{Quant}. And this multiplication will be associative if we identify \mathbb{C}^m \otimes \mathbb{C}^n with \mathbb{C}^{mn} in the right way—namely, using the lexicographic ordering on the set of basis vectors e_i \otimes e_j of \mathbb{C}^m \otimes \mathbb{C}^n.

So, let’s do this. \texttt{Quant} then becomes a monoidal poset—that is, a monoid that’s also a poset, where the multiplication gets along with the ordering as follows:

V \subseteq V', \; W \subseteq W' \quad \implies \quad V \otimes W \subseteq W \otimes W'

(If you want to show off, you can equivalently say a monoidal poset is just a monoidal category that’s a poset.)

We can also combine classical systems, using the cartesian product. IF M is a subvariety of \mathbb{C}\mathrm{P}^{m-1} and N is a subvariety of \mathbb{C}\mathrm{P}^{n-1}, their product M \times N is naturally a subvariety of \mathbb{C}\mathrm{P}^{mn-1}, using a nice trick called the Segre embedding

\mathbb{C}\mathrm{P}^{m-1} \times \mathbb{C}\mathrm{P}^{n-1} \hookrightarrow \mathbb{C}\mathrm{P}^{mn-1}

I’m hoping that if M is linearly normal in \mathbb{C}\mathrm{P}^{m-1} and N is linearly normal in \mathbb{C}\mathrm{P}^{n-1}, their product is linearly normal in \mathbb{C}\mathrm{P}^{mn-1}. If this is true, we get a product on \texttt{Class}, which again will be associative if we carefully use the lexicographic ordering when defining the Segre embedding.

And now for the punchline: quantization and projectivization get along with these ways of combining classical and quantum systems!

\texttt{Q}(M \times N) = \texttt{Q}(M) \otimes \texttt{Q}(N)
\texttt{P}(V \otimes W) = \texttt{P}(V) \times \texttt{P}(W)

So, if we think of \texttt{Class} and \texttt{Quant} as monoidal categories, then \texttt{Q} and \texttt{P} are monoidal functors!

It would also be interesting to combine classical systems by disjoint union, and show that this corresponds to combining quantum ones by direct sum. But I think I should stop here.

Enough formalism! Let’s look at examples! Next time we will.

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.

8 Responses to Geometric Quantization (Part 4)

  1. strangeset says:

    Wow, very useful for a new learner like me. Thanks for posting. I am also trying to write-up my linear Algebra book on WP, do you have any tips on how to? I have latex versions and for the time being I have just posted the pdf links.

  2. strangeset says:

    Thanks, How about equation arrays?

  3. Now let’s start looking at some examples of the adjoint functors introduced in Part 4: quantization and projectivization. It’s really the examples that bring the subject to life. They give new insights into hoary old topics in physics, and also raise some puzzles about the relation between classical and quantum mechanics.

    I’ll start with the classical spin-j particle and its quantization.

  4. Now let’s do some more interesting examples of geometric quantization using the functor described in Part 4. Let’s look at the spin-j particle with j > 1/2.

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