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 let
be the poset of linearly normal varieties in
, ordered by inclusion. Let
be the poset of linear subspaces of
also ordered by inclusion. Define the order-preserving map
as follows: for any let
be the smallest linear subspace of
such that
is contained in the projectivization of that subspace. And define the order-preserving map
as follows: for any let
be the projectivization of
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 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 I want to get the vector space of all sections of the dual of the tautological line bundle on
The functors
only do this job if and only if
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 for each different
We can get around this as follows.
For each we have an inclusion
sending each vector to the same list of numbers with a zero tacked on at the end. This gives an inclusion of posets
So, we can take the colimit of the diagram
or in more lowbrow terms, the union of all the posets each included in the next, and get a big poset I’ll call
(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
so we can take the colimit of
and get a poset I’ll call
This poset seems familiar to me: it’s the poset of all finite-dimensional subspaces of a certain vector space called
the colimit of all the spaces
The poset
is less familiar to me. We can intuitively think of it as the poset of all linearly normal subvarieties of
that sit inside
for some finite
. However, while I’m familiar with
as a topological space, the colimit of all the manifolds
I have never heard people talk about ‘subvarieties’ of
This could easily be a hole in my education. (Is
just a non-noetherian scheme?)
Now, the nice thing is that our maps of posets
get along with the inclusions
forming two commutative squares that I’m too lazy to draw here. So, we get well-defined maps of posets
Last time we saw that the maps and
were adjoint for each
That implies the same result for
and
Don’t be scared; this just means that
for all 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 is a reflective subcategory of
That implies the same result for
and
This just means that
for all And again, this is easy to check directly.
So, we are back where we started—but now we can stop worrying about which our classical state spaces are subvarieties of, or which
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 and
we can take their tensor product and get
So, we get a multiplication on
And this multiplication will be associative if we identify
with
in the right way—namely, using the lexicographic ordering on the set of basis vectors
of
So, let’s do this. then becomes a monoidal poset—that is, a monoid that’s also a poset, where the multiplication gets along with the ordering as follows:
(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 is a subvariety of
and
is a subvariety of
their product
is naturally a subvariety of
using a nice trick called the Segre embedding
I’m hoping that if is linearly normal in
and
is linearly normal in
their product is linearly normal in
If this is true, we get a product on
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!
So, if we think of and
as monoidal categories, then
and
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.
• 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.
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.
trialogueblog.wordpress.com/category/linear-algebra-course-notes/
Thanks, How about equation arrays?
What about them?
Thanks for your response. Sorry for incomplete question, I wanted to know how does one format an equation array, say a matrix equation in a WP post.
I like to do equation arrays using the \begin{array} and \end{array} commands; the main trick is to read the warning I wrote just above the box where you enter your comment! So, this:
gives this:
The exact spacing in what I wrote is irrelevant except that you need to start with $latex with at least one space after it!
That is nice, thanks.
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.
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.