Damien Calaque has invited me to speak at FGSI 2019, a conference on the Foundations of Geometric Structures of Information. It will focus on scientific legacy of Cartan, Koszul and Souriau. Since Souriau helped invent geometric quantization, I decided to talk about this. That’s part of why I’ve been writing about it lately!

I’m looking forward to speaking to various people at this conference, including Mikhail Gromov, who has become interested in using category theory to understand biology and the brain.

Abstract. Edward Nelson famously claimed that quantization is a mystery, not a functor. In other words, starting from the phase space of a classical system (a symplectic manifold) there is no functorial way of constructing the correct Hilbert space for the corresponding quantum system. In geometric quantization one gets around this problem by equipping the classical phase space with extra structure: for example, a Kähler manifold equipped with a suitable line bundle. Then quantization becomes a functor. But there is also a functor going the other way, sending any Hilbert space to its projectivization. This makes quantum systems into specially well-behaved classical systems! In this talk we explore the interplay between classical mechanics and quantum mechanics revealed by these functors going both ways.

For more details, read these articles:

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.

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I was lucky enough to be able to audit this class, which was taught one time (so far). The class started off by asking us to read ‘Structures, Learning, and Ergosystems’ by Gromov, it was really fascinating, and I eventually ordered a printed copy of the paper. I just thought I would share a link to the class because I think it’s a really good reference list others might want to look at. I believe they are writing a book based on the class, which I am really hoping to get ahold of soon.

I can hardly follow your math proof but if you claim that the transition from classical to quantum and back, in first quantization, boils down to complex vector spaces and complex projective varieties, no need for Kahler spaces, Hilbert structure and symplectic structure, this is a tremendous simplification of the mathematics! Am I getting it right? A certain novel physical picture might emerge more easily. I wonder what physicists at the conference will have to say… Seize the day!

Yes, it’s a big simplification, and it’s very radical. There’s more to the story: we need Hilbert spaces for full-fledged quantum mechanics and we need symplectic manifolds for full-fledged classical mechanics. But I explain how that extra structure can be added. For the basic process of quantization and its adjoint (‘classicization’, or more mathematically speaking ‘projectivization’) we don’t need that extra structure!

The series is nowhere near done. But the talk goes further in some respects than the blog articles so far, because it describes quantization and projectivization as functors between full-fledged categories, not mere posets. These means that quantization sends symmetries of classical systems to symmetries of quantum ones, and projectivization sends symmetries of quantum systems to symmetries of classical ones! Since Noether’s theorem applies in both settings, this implies that both quantization and projectivization send observables to observables, with Poisson brackets getting sent to commutators and vice versa. This seems about as good as one can reasonably hope for.

Symmetries correspond in a one-to-one way with observables in the following sense. In quantum mechanics any self-adjoint operator generates a one-parameter unitary group; in classical mechanics any smooth function on a compact symplectic manifold generates a one-parameter group of symplectomorphisms. In my approach to geometric quantization my classical phase space is fundamentally a Kähler manifold embedded in projective space, and this changes the story, but I believe there’s still a version of this one-to-one correspondence.

Ah ok, I was thinking of symmetries as operators commuting with the Hamiltonian. But as your approach is purely kinematical there is no Hamiltonian, indeed.

I think I’m still unconvinced that one can construct a map from classical physics to quantum physics without butchering these concepts, but I’m looking forward to the next posts.

I wouldn’t say the approach is purely kinematical: you get to choose any one-parameter group of symmetries to describe time evolution, and this corresponds to a Hamiltonian.

The map from classical mechanics to quantum mechanics that I’m talking about is described in my talk slides, but I’ll probably also talk about it in future posts. Then I want to do a lot of other things!

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I was lucky enough to be able to audit this class, which was taught one time (so far). The class started off by asking us to read ‘Structures, Learning, and Ergosystems’ by Gromov, it was really fascinating, and I eventually ordered a printed copy of the paper. I just thought I would share a link to the class because I think it’s a really good reference list others might want to look at. I believe they are writing a book based on the class, which I am really hoping to get ahold of soon.

I can hardly follow your math proof but if you claim that the transition from classical to quantum and back, in first quantization, boils down to complex vector spaces and complex projective varieties, no need for Kahler spaces, Hilbert structure and symplectic structure, this is a tremendous simplification of the mathematics! Am I getting it right? A certain novel physical picture might emerge more easily. I wonder what physicists at the conference will have to say… Seize the day!

Yes, it’s a big simplification, and it’s very radical. There’s more to the story: we need Hilbert spaces for full-fledged quantum mechanics and we need symplectic manifolds for full-fledged classical mechanics. But I explain how that extra structure can be added. For the basic process of quantization and its adjoint (‘classicization’, or more mathematically speaking ‘projectivization’) we don’t need that extra structure!

As this seems to be the end of the series, I’m disappointed not to have seen answers to my objections here:

https://johncarlosbaez.wordpress.com/2018/12/27/geometric-quantization-part-3/#comment-139940

The series is nowhere near done. But the talk goes further in some respects than the blog articles so far, because it describes quantization and projectivization as functors between full-fledged categories, not mere posets. These means that quantization sends symmetries of classical systems to symmetries of quantum ones, and projectivization sends symmetries of quantum systems to symmetries of classical ones! Since Noether’s theorem applies in both settings, this implies that both quantization and projectivization send observables to observables, with Poisson brackets getting sent to commutators and vice versa. This seems about as good as one can reasonably hope for.

But how can a statement about symmetries yield information about observables? In general, most observables have nothing to do with symmetries.

Symmetries correspond in a one-to-one way with observables in the following sense. In quantum mechanics any self-adjoint operator generates a one-parameter unitary group; in classical mechanics any smooth function on a compact symplectic manifold generates a one-parameter group of symplectomorphisms. In my approach to geometric quantization my classical phase space is fundamentally a Kähler manifold embedded in projective space, and this changes the story, but I believe there’s still a version of this one-to-one correspondence.

Ah ok, I was thinking of symmetries as operators commuting with the Hamiltonian. But as your approach is purely kinematical there is no Hamiltonian, indeed.

I think I’m still unconvinced that one can construct a map from classical physics to quantum physics without butchering these concepts, but I’m looking forward to the next posts.

I wouldn’t say the approach is purely kinematical: you get to choose any one-parameter group of symmetries to describe time evolution, and this corresponds to a Hamiltonian.

The map from classical mechanics to quantum mechanics that I’m talking about is described in my talk slides, but I’ll probably also talk about it in future posts. Then I want to do a lot of other things!