This room has been relatively silent recently. Just in case it gets frozen, I will remind that moderators can unfreeze chatrooms - so you can ask some moderator (from any site) whether they would be willing to do so. meta.stackexchange.com/tags/frozen-rooms/info How do I unfreeze a frozen chat room?

What does it mean to quantize a symplectic vector space?

Namely, if I have a lagrangian subspace of V, I have heard that there is a natural quantization associated to this data

But I am not sure what this means, and can't find much about it online

I also don't quite understand quantization, but here's what I think is supposed to happen:

(so if anyone catches me saying something misleading or wrong feel free to correct me!)

the idea is to take mathematical objects that represent some classical physical system and produce the mathematical objects which represent the corresponding quantum physics system.

One model for classical mechanics is a sympletic manifold M with a Hamiltonian function. Physically, M is phase space, and the functions on M are the observables. The symplectic structure induces a Poisson algebra structure on the observables.

One model for quantum mechanics is a Hilbert space of states together with a Lie algebra of observables. The Lie algebra is over C adjoin a formal parameter ħ, which has something to do with noncommutativity: for example, if Q denotes a position observable and P denotes a momentum observable, [P, Q] is ħ times something nonzero (which has something to do with the Heisenberg uncertainty principle).

Right

Given such a quantum system, you might want to take a "classical limit". In the real world, if you care about physics on orders much larger than ħ, classical mechanics is a very accurate approximation to quantum mechanics. So we hope that if we take the noncommutative algebra of quantum observables and mod out by ħ, we recover the commutative algebra of classical observables.

I think what quantization is doing is building this story all from the mathematical input data of the symplectic manifold and Hamiltonian function (and maybe also some additional data?)

I assume for symplectic vector spaces you'd just regard them as symplectic manifolds and do quantization that way, but honestly I'm not sure.

That is interesting, thank you. I will think about it, but I am not exactly sure where the Lagrangian subspace fits into the story.

Oh you begin with a Lagrangian subspace?

interesting. I have no idea how that fits into this story!

anyways, one reference I can suggest is John Baez's series of blog posts on this stuff: johncarlosbaez.wordpress.com/2018/12/01/…

Thanks! I will look into it. This quantization is very mystical to me, yet even more beautiful.

yeah, I've been learning a little about it recently. there's so much interesting structure here!

The nLab article on geometric quantization might also be useful (ncatlab.org/nlab/show/geometric+quantization).