Foundations of the Geometric Science of Information 2019

John Baez

February 6, 2019

From Classical to Quantum and Back

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

You can see the slides here. The first slide uses a picture by Abdelaziz Nait Merzouk, also shown above.

You can see more details here:

For a popularized account of these ideas, see:

© 2019 John Baez