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. I recently discovered through conversations on Twitter how few physicists have heard of the classical spin-j particle. They all know that the quantum spin-j particle has a Hilbert space , an irreducible representation of But the corresponding classical system whose quantization gives this Hilbert space seems remarkably little-known, especially given how simple it is. So, I’ll describe it and its geometric quantization slowly and carefully, before feeding it into our functor
which does this quantization quickly and automatically.
The space of states of the a classical spin-j particle is the sphere, The radius of the sphere is the total angular momentum of the particle, namely j. A point on this sphere represents a possible angular momentum vector of the particle: a vector with length j. There are 3 important observables in this system: the components of the angular momentum, and
To do classical mechanics with this system we want a symplectic structure on the sphere. A symplectic structure determines Poisson brackets of observables, and we want to make sure that
There’s a unique symplectic structure that does the job. A symplectic structure on the sphere is just a nowhere vanishing 2-form (since such a 2-form is automatically closed and nondegenerate), and you can almost guess just from the fact that it had better be rotation-invariant, so it has to be some multiple of the usual area 2-form on the unit sphere. If you figure out which multiple gives the above Poisson brackets, you see that must be times the area 2-form for the unit sphere. With painful explicitness:
This is a bit peculiar: you might expect here instead of since the area of a sphere of radius in Euclidean space is proportional to But that’s not what the calculation gives! If you want some more justification for the Poisson brackets we’ve chosen, which force this symplectic structure, you can think of the sphere as a coadjoint orbit of the group and use the fact that any such coadjoint orbit gets a Poisson structure, which happens to give the above Poisson bracket formula in this example. Or, you can use dimensional analysis to see must be proportional to since they both have units of action. But let’s not get into that here.
To geometrically quantize our symplectic manifold using Kähler quantization, we need to equip it with lots of extra structure, as explained in Part 2. For starters, we need to give it a complex structure. There’s a unique one invariant under rotations, namely the one that makes our sphere into a copy of the Riemann sphere There is then a unique Kähler structure on whose imaginary part is our symplectic structure The real part is a Riemannian metric: the usual metric on a round sphere of radius in Euclidean space. (Again, that square root looks peculiar, but that’s what we get.)
Next we need to choose a hermitian line bundle over the sphere, equipped with a hermitian connection whose curvature is By the miracle of algebraic topology, this exists precisely when the integral of over the sphere is times an integer. Since this area is this forces So, while we can work with the classical spin- particle for any real we can only quantize it when takes the usual integer or half-integer values!
Let’s focus on the spin-1/2 particle for a while. In this case we can easily describe the relevant line bundle over our sphere. It’s just the bundle I’ve been calling the dual of the tautological line bundle on The holomorphic sections of this bundle are just linear functions So, the space of sections is or just if we identify this space with its dual using the usual inner product.
That’s good! We got the right answer! But as you can see, the process seems rather long and tortuous.
If we use our functor
everything goes a lot faster, mainly because all the choices we had to make are built into the definition of An object in this category, you’ll remember, is a linearly normal subvariety for some arbitrary When we quantize it we get where is the smallest linear subspace with
In other words, quantization ‘flattens out’ or ‘linearizes’ replacing this possibly quite interesting projective variety by the smallest vector space whose projectivization contains this variety.
For the spin-1/2 particle, we take And we get since this is the smallest subspace of whose projectivization contains Voilà!
That was pretty trivial. But it was trivial for an interesting reason. Remember from Part 4 that we have an adjoint functor going back from the quantum to the classical:
This sends any subspace to its projectivization Moreover, we have
In words: if we quantize the projectivization of a quantum system we get that quantum system back.
And that’s what we’re doing in the case of the spin-1/2 particle! The space of states of the classical spin-1/2 particle, was the projectivization of So when we quantize it, we just get back.
That’s not how it will work for the spin-1 particle, or any higher-spin particle. The spin-j particle still has as its classical state space, but when we quantize it we get That’s what I’ll talk about next time.
Now, experts may be yawning at this point, because they already know how to handle such a higher-spin particle! We simply choose a different Kähler structure on and a different line bundle over it. Namely, we rescale the Kähler structure we’ve already been talking about to make the total area of the sphere be If j = k/2, this means we have to multiply our earlier Kähler structure by k. And to get a line bundle with a connection whose curvature is the imaginary part of this rescale Kähler structure, we just take the kth tensor power of our line bundle
The holomorphic sections of this new line bundle are homogeneous polynomials of degree k on The space of all these has dimension and this is the right space of states for quantum spin-j particle.
All this is great, and it’s an illustration of a theme I’ll eventually want to talk about much more: when you’re doing the Kähler quantization of some physical system, you can always multiply your Kähler structure by a natural number k, and take the kth tensor power of your line bundle, and get a new system!
What does this mean physically? I’ll give away part of the answer now: it corresponds to dividing Planck’s constant by k! You see, angular momentum is really measured in units of Planck’s constant, so using this procedure to go from the spin-1/2 particle to the spin-5/2 particle (for example) is the same as dividing Planck’s constant by 5.
However, in the setup described last time, we had no explicit choice over what Kähler structure or line bundle to use on our subvariety It inherited those structures from So to change those structures, we need to embed in a different way, perhaps into a different projective space.
This may seem a bit silly, and in a way it is—but not completely.
As I mentioned last time, we are describing the geometry of varieties extrinsically, through their embedding in projective space. This is a bit old-fashioned compared to the more modern intrinsic viewpoint. At some point I’d like to switch to an intrinsic approach. However, the interplay between the intrinsic and extrinsic approaches is a time-honored theme in algebraic geometry, and we should exploit it in geometric quantization!
In particular, even if we start with an abstract variety not embedded in any projective space, once we choose a line bundle over it we can often embed in a projective space built from By definition, we can do this whenever is a ‘very ample’ line bundle. Sometimes isn’t very ample but is whenever k is large enough; then we call ‘ample’. The study of ample line bundles is a big deal in algebraic geometry.
So, it’s not so silly to start our research on geometric quantization by taking our varieties to come equipped with an embedding into a projective space, and then later think about how we get such an embedding if we don’t already have one.
So, next time I’ll geometrically quantize the spin-j particle by taking the space of states of a classical spin-j particle, and embedding it, not in itself, but in some higher-dimensional projective space. The math involved is well-known; the interesting part to me is its physical interpretation.
• 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.