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 wellbehaved 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:

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 spin1/2 particle.
 Part 6: quantizing the spin3/2 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.
For a popularized account of these ideas, see:
© 2019 John Baez
baez@math.removethis.ucr.andthis.edu