Geometric quantization is often presented as a way to take a symplectic manifold and construct a Hilbert space, but in fact that’s a better description of ‘prequantization’, which is just the first step in geometric quantization. Even that’s not completely accurate: we need to equip our symplectic manifold with a bit of extra structure just to prequantize it. But more importantly, when we do this, the resulting ‘prequantum Hilbert space’ is too big: we need to chop it down significantly to get a Hilbert space suitable for the quantum description of our physical system. And to chop it down, we need to equip our symplectic manifold with a lot more extra structure. So much extra structure, in fact, that the whole idea of ‘quantizing a symplectic manifold’ starts sounding more like the wistful, nostalgic description of a naive hope than an honest account of what we’re really doing.
So, it’s probably better to admit that ‘quantizing a classical system’ is not a real thing. More precisely—because people will rightly object to that over-bold statement—there’s no systematic procedure that takes us all the way from the mathematical structures used to describe a large class of classical systems to those used to describe quantum systems, with no further inputs required. Sure, we can make quantization more systematic by limiting the class of systems it’s supposed to handle. We can also make it more systematic by having it do less: for example, just prequantization instead of full-fledged quantization. Both those options have been explored for many decades. Let’s instead try something new.
For starters, let’s accept the fact that the world was not created classically by God on the first day and quantized on the second.
The world is quantum; under some conditions it looks approximately classical. So instead of treating the lack of a purely functorial quantization procedure for symplectic manifolds as a failure, let’s accept it and try some new ways of thinking about the meaning of geometric quantization.
To make things specific, let’s consider Kähler quantization. Here we start not with a raw symplectic manifold, but much more. For starters, we take a Kähler manifold, which is a smooth manifold equipped with a beautiful trio of structures:
1) a symplectic structure
2) a complex structure
3) a Riemannian metric
fitting together in a way that obeys this equation:
whenever we have two tangent vectors at any point of This equation says how any two of are enough to determine the third.
Each tangent space of a Kähler manifold is a complex Hilbert space—which is downright suspicious, given that in quantization we’re aiming to get a complex Hilbert space. Here’s how it works:
1) the symplectic structure gives the imaginary part of the inner product on each tangent space of
2) the complex structure gives the operation of multiplication by on each tangent space, and
3) the Riemannian metric gives the real part of the inner product on each tangent space.
says that the imaginary part of the inner product of and equals the real part of the inner product of and
In particular, each finite-dimensional Hilbert space is a Kähler manifold. Similarly, an infinite-dimensional Hilbert space is a kind of infinite-dimensional Kähler manifold.
So, you can think of a complex Hilbert space as a special kind of Kähler manifold: a flat and simply-connected one. Or—less precisely but quite evocatively—you can think of a Kähler manifold as a curved generalization of a complex Hilbert space!
If Kähler quantization were a systematic procedure for extracting a Hilbert space from a Kähler manifold, these facts would lead us to a clear view of what this procedure is doing. Namely, it’s taking a general Kähler manifold and ‘flattening it out’ in a very clever way, producing a Hilbert space. Perhaps a better word than ‘flattening’ would be ‘linearizing’, since this emphasizes the all-important linearity built into quantum mechanics.
Something like this is true—but it can’t be the whole story, because Kähler quantization requires more input than a mere Kähler manifold to produce a Hilbert space! We also need a complex line bundle over our manifold Vectors in our Hilbert space will not be just complex-valued functions on rather, they will be (nice) sections of this line bundle.
It would be nice to understand this in a deeply physical way. For example, we might try to insist that the ‘true’ space of classical states is not the manifold but something with one extra dimension built from this line bundle This thing is not a symplectic manifold: it’s called a contact manifold. Fans of contact geometry will argue, quite convincingly, that the ‘phases’ so important in quantum mechanics are already lurking in classical mechanics, invisible when we use symplectic manifolds, but evident when we use contact manifolds! If we take this seriously, maybe we shouldn’t be trying to geometrically quantize symplectic manifolds in the first place: maybe we should be using contact manifolds.
However, even taking our Kähler manifold and this complex line bundle over it as input is not enough to systematically construct the long-sought Hilbert space. We need more!
1) We need to equip with a connection whose curvature is This is our way of making compatible with the symplectic structure on We need this even for prequantization: it lets us turn smooth functions on (‘classical observables’) into operators on the space of sections of (‘quantum observables’).
2) We need to equip with the structure of a holomorphic line bundle. This is our way of making compatible with the complex structure on We need this to cut down the big vector space obtained in prequantization to a smaller one: the space of holomorphic sections of
3) We need to equip with the structure of a hermitian line bundle. I would like to say this is our way of making compatible with the Riemannian structure on since that would make everything very symmetrical—but it doesn’t seem correct! Instead, we need this to put an inner product on the space of holomorphic sections of so we can get a Hilbert space.
All this must sound like a real slag-heap of mathematics, if you’re not already in love with it! It’s beautiful when you get to know it, especially when you look at examples. But I find it difficult to motivate everything on physical grounds, and I also find it a bit difficult to keep track of all these interacting structures.
So, next time I’ll present a simplified version of geometric quantization, where we throw out most of these structures and keep only enough to get a complex vector space, not a complex Hilbert space. In fact I’ll throw out the symplectic and Riemannian structures on our manifold and only keep the complex structure! This is pretty heretical from the viewpoint of physics, but in fact it’s quite standard in mathematics. Indeed, I don’t want to get your hopes up too much: if you know some algebraic geometry, you should find most of what I say quite familiar. But I think this is a good way to get started on setting up a pair of adjoint functors: ‘quantization’ and the reverse process I mentioned last time: ‘projectivization’.
Since I’ve got your ear, let me ask a question. Suppose we have a holomorphic hermitian line bundle on a Kähler manifold . I can think of three ways in which a connection on can be compatible with all this structure:
1) We can demand that the curvature of the connection equal
2) We can demand that the connection be compatible with the holomorphic structure of Let me spell this out a little. Since is a holomorphic line bundle there’s a operator that we can apply to any section of and get an -valued 1-form. This is an -valued (0,1)-form, meaning that written out in holomorphic complex coordinates it has terms but no terms. On the other had, we can use the connection to take the covariant derivative of which is a -valued 1-form and take its (0,1) part, denoted Then, we can demand that these agree:
3) We can demand that the connection be hermitian. This means that the directional derivative of the inner product of two sections of can be computed using a kind of product rule where we differentiate each section using Namely:
for any vector field on and any sections of
I’m confused by when we can find a connection obeying all three of these conditions. Here are some things I know.
First, any holomorphic hermitian line bundle has a unique connection obeying 2) and 3). This is called the Chern connection and its construction is actually given here:
• Wikipedia, Hermitian metrics on a holomorphic vector bundle.
Does this connection also obey 1)? I see no reason why, in general: condition 1) involves the symplectic structure on while conditions 2) and 3) don’t mention the symplectic or Riemannian structure on just its complex structure.
Second, we can find a connection obeying 1) iff represents, in de Rham cohomology, the Chern class of the line bundle If is an integral 2-form we can always find some line bundle for which this is true.
Can we find a holomorphic line bundle for which this is true? Can we then choose a holomorphic connection on this line bundle obeying 1)? If I understood Picard groups better, I might know.
But as you can see, my understanding of how conditions 1)-3) interact is rather weak. I feel I should have seen this spelled out carefully in a book on geometric quantization—but I don’t think I have.
• 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.