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 \mathbb{C}\mathrm{P}^n that’s picked out by homogeneous polynomial equations.

In fact this is not as different as it sounds: if we give \mathbb{C}\mathrm{P}^n its Fubini–Study metric, any smooth projective variety M \subseteq \mathbb{C}\mathrm{P}^n becomes a Kähler manifold! We just restrict the Fubini–Study metric to M.

Furthermore, M \subseteq \mathbb{C}\mathrm{P}^n has a god-given line bundle over it. You see, any point in \mathbb{C}\mathrm{P}^n is really a line through the origin in \mathbb{C}^{n+1}, and these lines define a line bundle over \mathbb{C}\mathrm{P}^n. 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 \mathbb{C}^{n+1} gives one, and that’s how they all arise! Let’s call this dual bundle L_{\mathbb{C}\mathrm{P}^n}. If we restrict L_{\mathbb{C}\mathrm{P}^n} to M, we get a holomorphic line bundle with enough sections to be useful in geometric quantization. Let’s call this bundle L_M.

Even better, M and this line bundle L_M have the whole laundry list of extra structure and properties that I was bemoaning last time. As I already said, M is Kähler. Furthermore, L_M is holomorphic. Better yet, there’s a god-given hermitian metric on L_M, coming from the standard inner product on \mathbb{C}^{n+1} (and thus its dual). So, as I explained near the end last time, we get a connection on L_M called the Chern connection. And if I’m not horribly mistaken, the curvature of this connection gives the symplectic structure on M, at least up to those factors of 2\pi and i 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, \texttt{Class} and \texttt{Quant}, and make geometric quantization into a functor

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

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 n and let \texttt{Class} be the poset of smooth linearly normal subvarieties of \mathbb{C}\mathrm{P}^n, ordered by inclusion. Don’t worry—I’ll tell you what ‘linearly normal’ means in a minute. I’ll let \texttt{Quant} be the poset of linear subspaces of \mathbb{C}^{n+1}, also ordered by inclusion.

You can think of any object of \texttt{Class} 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 \texttt{Quant} as a ‘space of quantum states’, since as a subspace of \mathbb{C}^{n+1} it’s a Hilbert space.

Here’s how the functor \texttt{Q} works on objects. As we’ve seen, any smooth projective variety M \subseteq \mathbb{C}\mathrm{P}^n comes with a line bundle L_M \to M, namely the restriction of the dual of the tautological line bundle to M. To quantize M, we form the space of holomorphic sections of L_M.

Now, this doesn’t instantly look like a subspace of \mathbb{C}^{n+1}. But we can identify it with one as follows. Every holomorphic section of L_{\mathbb{C}\mathrm{P}^n} restricts to a holomorphic section of L_M. And when M is linearly normal, every holomorphic section of L_M arises this way! This is just the definition of ‘linearly normal’.

In this case, the space of holomorphic sections of L_M is a quotient of the space of holomorphic sections of L_{\mathbb{C}\mathrm{P}^n}. But I’ve already mentioned that the latter space can be identified with {\mathbb{C}^{n+1}}^\ast. So, we’ve got a quotient space of {\mathbb{C}^{n+1}}^\ast. But by the magic of linear algebra, this corresponds to a subspace of \mathbb{C}^{n+1}. And that’s \texttt{Q}(M).

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 M \subseteq \mathbb{C}\mathrm{P}^n there’s a smallest subspace V \subseteq \mathbb{C}^{n+1} such that M sits inside the projective space \mathrm{P}V \subseteq \mathbb{C}\mathrm{P}^n. We have

\texttt{Q}(M) = V

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

M \subseteq M' \quad \implies \quad \texttt{Q}(M) \subseteq \texttt{Q}(M')

So, \texttt{Q} 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

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

It works like this. An object V \in \texttt{Quant} is a linear subspace of \mathbb{C}^{n+1}. This gives a projective variety \mathrm{P}V \subseteq \mathbb{C}\mathrm{P}^n. Since this variety is linearly normal, we can define

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

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

V \subseteq V' \quad \implies \quad \texttt{P}(V) \subseteq \texttt{P}(V')

so \texttt{P} is a functor.

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

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

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

(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:

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

for all V \in \texttt{Quant}. Again this is obvious: the smallest linear subspace whose projectivization contains the projectivization of V is V.

Thanks to this extra property we say \texttt{Quant} is a reflective subcategory of \texttt{Class}. A more familiar example of this situation is how \textrm{AbGp} is a reflective subcategory of \textrm{Gp}. The forgetful functor \textrm{AbGp} \to \textrm{Gp} 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 \texttt{Quant} and \texttt{Class} 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 \mathbb{C}\mathrm{P}^n, 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

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

did not use the inner product on \mathbb{C}^{n+1} at all! That’s why I was talking about the dual space {\mathbb{C}^{n+1}}^\ast. 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 \mathbb{C}^{n+1} with an inner product: any inner product, for example its standard one.


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.

Part 7: the Veronese embedding as a method of ‘cloning’ a classical system, and taking the symmetric tensor powers of a Hilbert space as the corresponding method of cloning a quantum system.

Part 8: cloning a system as changing the value of Planck’s constant.

18 Responses to Geometric Quantization (Part 3)

  1. You’re horribly mistaken. Let M \subset \mathbb{CP}^n be a discrete set of k>n+1 points. Then the space of sections of L_M is k-dimensional, so the restriction map can’t be onto.

    I doubt that this messes anything else up for you. There’s still a restriction map, it’s just not always onto. The adjective to look for is “projectively normal”.

    For a more expensive example, consider the map \mathbb C^2 \to Sym^n(\mathbb C^2) = \mathbb C^{n+1}, \vec v \mapsto \vec v \otimes \cdots \otimes \vec v. This projectivizes to the nth “Veronese embedding” of \mathbb{CP}^1, into \mathbb{CP}^n, with image a “rational normal curve”, and it does have the onto-ness property you ask for. When you pull back \mathcal O(1) from \mathbb{CP}^n, you get \mathcal O(n) on \mathbb{CP}^1. The rational normal curve is a “degree n curve”.

    But! now take a generic linear projection \mathbb C^{n+1} \to \mathbb C^4, restrict to the image of the \mathbb C^2 above, and projectivize; you get a well-defined degree n embedding of \mathbb{CP}^1 into \mathbb{CP}^3. Once again, the line bundle on the ambient \mathbb{CP}^3 only has a 4-dim space of sections, whereas its restriction to the degree n curves has a (n+1)-dim space.

    • John Baez says:

      Thanks! But damn you! I’ll have to change my theorem—the easy way will be to restrict to projectively normal varieties, but there could also be some other way, involving more substantial changes.

      For those who are just half-following along, this remark of mine was wrong:

      To quantize M \subseteq \mathbb{C}\mathrm{P}^n, we form the space of holomorphic sections of L_M.

      Now, this doesn’t instantly look like a subspace of \mathbb{C}^{n+1}. But we can identify it with one […]

      In fact we can do this in a reasonable way only when M is projectively normal.

      There’s always a restriction map taking sections of the tautological bundle over \mathbb{C}\mathrm{P}^n to sections of L_M over M. But this is not always onto, as Allen’s examples show. If this restriction map is onto we say M is projectively normal. In this case, everything I said works out.

      So, the cheap fix is to adjust the definition of \texttt{Class} so that it’s the poset of projectively normal smooth subvarieties of \mathbb{C}\mathrm{P}^n, ordered by inclusion.

      I think I’ll take the liberty of correcting my post this way.

      • allenknutson says:

        This restriction is rarely 1:1. That’s the question of whether the subvariety linearly spans the projective space. Rather, you were asserting it should be onto, which is what I was giving counterexamples to.

        • John Baez says:

          In my last comment, “one-to-one” was just a stupid typo for “onto”. Honest. Everything is working fine now, at least in the deep recesses of my own brain. I need the restriction to be onto. I’ll go back and change my comment, to spread less confusion.

    • John Baez says:

      By the way, I’d appreciate it if someone could comment on another thing I’m not sure about! I took a smooth subvariety M of \mathbb{C}\mathrm{P}^n and restricted the dual of the tautological line bundle L_{\mathbb{C}\mathrm{P}^n} to M, getting a bundle L_M. Then I said:

      As I already said, M is Kähler. Furthermore, L_M is holomorphic. Better yet, there’s a god-given hermitian metric on L_M, coming from the standard inner product on \mathbb{C}^{n+1} (and thus its dual). So, as I explained near the end last time, we get a connection on L_M called the Chern connection. And if I’m not horribly mistaken, the curvature of this connection gives the symplectic structure on M, at least up to those factors of 2\pi and i that I keep losing track of.

      In section 4 of this article:

      • Mirko Mauri, Kodaira embedding theorem.

      Mauri notes that the curvature of the Chern connection on L_{\mathbb{C}\mathrm{P}^n} is the usual symplectic structure on \mathbb{C}\mathrm{P}^n (the imaginary part of the Fubini–Study metric). So my claim is true in this case.

      I think the general case follows if the Chern connection on L_M is the restriction to L_M \to M of the Chern connection on L_{\mathbb{C}\mathrm{P}^n} \to \mathbb{C}\mathrm{P}^n. I’m hoping this follows from a general fact about Chern connections.

      It’s a bit ridiculous to be doing this research before learning all the usual stuff about algebraic geometry, but right now I feel that the best way to learn algebraic geometry is to try to understand Kähler quantization: everything that seemed dry and boring before is now something I desperately need to know!

      • Rogier Brussee says:

        It is indeed true that the the curvature of the restriction of the connection on the tautological bundle is the symplectic form. That is because connections and hermitian metrics and \bar \partial operator restrict and it is obvious that the restriction of the connection is is a hermitian connection.

        • John Baez says:

          Thanks! So things seem to be working!

          I really appreciate all the help you’ve been giving me here. I’m busy learning about certain aspects of of complex algebraic geometry and Kähler geometry, now that I finally have a good reason to want to learn them!

  2. Daniele Corradetti says:

    That’s really really interesting! Hope to have some time during the holiday to put my head on it! Do you also have some concrete example to show how the things work?

    • John Baez says:

      I have tons of concrete examples, and I’m dying to talk about them because they lead to lots of puzzles (and clues) about the physical meaning of geometric quantization. I’m not sure whether I should discuss these examples before developing more of the formalism, or afterward. The adjoint functors I described in this post are supposed to be a preliminary for some better ones. But maybe you’re persuading me to do examples before developing more abstract nonsense.

  3. Comparison says:

    Small correction : An object V \in \texttt{Class} is a linear subspace

    should be An object V \in \texttt{Quantum} is a linear subspace

  4. It’s a nice construction. But I think this works because you’re only considering the state space of the quantum system. The state space in itself contains virtually no information about the quantum system (all separable Hilbert spaces are isomorphic, and the only invariant of a finite dimensional Hilbert space is its dimension…). The information lies in the algebra of observables.

    If there is any adjunction in the spirit of what you describe, it should go in the other way. Information is lost when going from a quantum system to its classical counterpart. (It is really a limit, taking \hbar to zero.) But I’d bet there is no such thing, because some quantum systems have no classical limit, as we discussed here:
    https://golem.ph.utexas.edu/category/2017/12/an_m5brane_model.html

    • John Baez says:

      Of course I plan to bring observables into the game too. I also plan to talk about classical limits. But since I’m talking about geometric quantization, I’ll only be talking about quantum systems that do have classical limits.

  5. Ammar Husain says:

    Is there something about product of posets \texttt{Class}_n and \texttt{Class}_m into \texttt{Class}_{nm+n+m} and its relation to the P_{n,m,nm+n+m}? Future post?

  6. Avi Levy says:

    If I am not mistaken, \texttt{Q} is the convex hull functor restricted to whichever class of projective varieties John considers “nice enough” for his purposes :)

    In this case, there is an analogy between pure and mixed ergodic states, with the “classical” space corresponding to the pure states and the “quantum” space corresponding to the mixed states.

    At a glance, one way coming to mind to attempt to extend \texttt{Q} from inclusion maps to arbitrary morphisms f \colon V\to W of projective varieties is to pick a generic maximal finite linearly independent subset S of V (I believe genericity implies f(S) has the same property) and let \texttt{Q}(f) be the linear interpolation of the restriction of f to S – although it is not clear to me whether this is independent of the choice of S, as it must be.

    • John Baez says:

      Avi Levy wrote:

      If I am not mistaken, \texttt{Q} is the convex hull functor restricted to whichever class of projective varieties John considers “nice enough” for his purposes :)

      Do you know a reference where people have studied the ‘convex hull’ of a subset of \mathbb{C}\mathrm{P}^n? Anyway, yes, this is probably that, though I’d prefer to call it the ‘projective hull’: it’s the smallest projective space PV \subseteq \mathbb{C}\mathrm{P}^n containing the given subset.

      Here’s what it means for a projective variety M \subseteq \mathbb{C}\mathrm{P}^n to be ‘nice enough’ for my purposes: the restriction map sending sections of the dual of the tautological bundle on \mathbb{C}\mathrm{P}^n to the sections of the restriction of that bundle to M is onto.

      (I’m getting really sick of saying ‘dual of the tautological bundle’ so I may start calling it \mathcal{O}(-1) the way the big boys do, but so far I’ve been calling it L.)

      At some point when I try to do things more slickly I’ll try to switch to an ‘intrinsic’ approach where I don’t treat varieties as coming with a fixed embedding in \mathbb{C}\mathrm{P}^n.

  7. John Baez says:

    I screwed up in this post: I said ‘projectively normal’ when I meant ‘linearly normal’. I fixed it! Now it says this:

    Now, this doesn’t instantly look like a subspace of \mathbb{C}^{n+1}. But we can identify it with one as follows. Every holomorphic section of L_{\mathbb{C}\mathrm{P}^n} restricts to a holomorphic section of L_M. And when M is linearly normal, every holomorphic section of L_M arises this way! This is just the definition of ‘linearly normal’.

    ‘Projectively normal’ is a stronger condition which is explained in the Wikipedia link. I wish I understood their explanation of linear normality a bit better:

    The variety V in its projective embedding is “projectively normal” if its homogeneous coordinate ring R is integrally closed. This condition implies that V is a normal variety, but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a rational quartic curve in three dimensions. Another equivalent condition is in terms of the linear system of divisors on V cut out by the tautological line bundle L on projective space, and its dth powers for d = 1, 2, 3, …; when V is non-singular, it is projectively normal if and only if each such linear system is a complete linear system.

    I have no intuition for what it means for the homogenous coordinate ring to be integrally closed.

    I also don’t have any intuition for what this “complete linear system” condition means.

    This would be okay if I were sure I understood this:

    In a more geometric way one can think of L as the Serre twist sheaf O(1) on projective space, and use it to twist the structure sheaf O_V k times, for any k. Then V is called k-normal if the global sections of O(k) map surjectively to those of O_V(k) for a given k.

    I find this confusing. Isn’t O(1) the sheaf of sections of the dual of the tautological line bundle? Yet here Wikipedia seems to be claiming it’s the sheaf of sections of L, which in the previous passage it claimed was the tautological line bundle!

    I think this is just a mistake on their part. I think they should say sections of L^* give the Serre twist sheaf O(1). Should I correct this? I don’t understand the previous passage well enough to know if they wanted the tautological line bundle or its dual back there.

    Anyway, going on:

    If V is 1-normal it is called linearly normal, and projective normality is the condition that V is k-normal for all k ≥ 1.

    Is this way of stating projective normality only true for normal varieties, or for all varieties?

    Linear normality may be said geometrically: V as projective variety cannot be obtained by an isomorphic linear projection from a projective space of higher dimension, except in the trivial way of lying in a proper linear subspace.

    This sounds nice and geometrical and very relevant to what I”m
    doing, but I don’t quite understand it. What’s a “linear projection from a projective space of a higher dimension”? A linear projection from a vector space of higher dimension to one of lower dimension has a nontrivial kernel so it doesn’t give a regular map between their projective spaces—I’ve been meaning to ask what sort of map we call this partially defined map. (A rational map I guess?) But I guess sometimes it maps a subvariety of the higher-dimensional projective space isomorphically to a subvariety of the lower-dimensional one? Is that the idea?

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