John C. Baez's Blog, page 53

Geometric Quantization (Part 7)

 

I’ve been falling in love with algebraic geometry these days, as I realize how many of its basic concepts and theorems have nice interpretations in terms of geometric quantization. I had trouble getting excited about them before. I’m talking about things like the Segre embedding, the Veronese embedding, the Kodaira embedding theorem, Chow’s theorem, projective normality, ample line bundles, and so on. In the old days, all these things used to make me nod and go “that’s nice”, without great...

Applied Category Theory 2019 School

Dear scientists, mathematicians, linguists, philosophers, and hackers:

We are writing to let you know about a fantastic opportunity to learn about the emerging interdisciplinary field of applied category theory from some of its leading researchers at the ACT2019 School. It will begin February 18, 2019 and culminate in a meeting in Oxford, July 22–26. Applications are due January 30th; see below for details.

Applied category theory is a topic of interest for a growing community of researchers,...

Unsolved Mysteries of Fundamental Physics

In this century, progress in fundamental physics has been slow. The Large Hadron Collider hasn’t yet found any surprises, attempts to directly detect dark matter have been unsuccessful, string theory hasn’t made any successful predictions, and nobody really knows what to do about any of this. But there is no shortage of problems, and clues. Watch the talk I gave at the Cambridge University Physics Society for some ideas on this! Warning: this is for ordinary folks, not experts.

There are s...

Geometric Quantization (Part 6)

 

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.

To be specific, let’s consider the spin-3/2 particle. There’s nothing special about the number 3 here: everything I’ll say can be generalized. But the number 3 will give me a nice excuse to show you a picture of a curve called the ‘twisted cubic’.

We can build a spin-3/2 particle from three spin-1/2 particles, all having angular moment...

Interview

Happy New Year! People like to ponder grand themes each time the Earth completes another orbit around the Sun, so let’s give that a try.

Maria Mannone is a musician who studies the relation between mathematics, music and the visual arts. We met at a conference on The Philosophy and Physics of Noether’s Theorems. Later she decided to interview me for the blog Math is in the Air. There’s a version in English and one in Italian.

She let me reprint the interview here… so with no further ado, he...

Geometric Quantization (Part 5)

 

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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 classi...

Geometric Quantization (Part 4)

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 gi...

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. I’ve 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,...

Geometric Quantization (Part 2)

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 ge...

Applied Category Theory Seminar

We’re going to have a seminar on applied category theory here at U. C. Riverside! My students have been thinking hard about category theory for a few years, but they’ve decided it’s time to get deeper into applications. Christian Williams, in particular, seems to have caught my zeal for trying to develop new math to help save the planet.

We’ll try to videotape the talks to make it easier for you to follow along. I’ll also start discussions here and/or on the Azimuth Forum. It’ll work best if...