"Everything is a Lagrangian submanifold..." Forget particles. Forget waves. Alan Weinstein quipped that the universe is built from something entirely different: Lagrangian submanifolds. What are those, and why should you care? To grasp Lagrangian submanifolds, you first need to know about phase space… (1/10) Usually people say phase space (incorrectly) when they actually mean “state space” so lets' define this phase space. It's an abstract space (not spacetime) where each point represents a particle's state is given by its position (q) and momentum (p). So, phase space is a space of (q, p) pairs. You can extend this to N particles, of course. (2/10)

The canonical quantization you're taught -- replace {f,g} with [f,g]/iℏ -- is, well, ill-defined. You can always add a classical 0 that becomes non-zero quantumly (thanks to non-commutation). But there's another way: geometric quantization. (1/7) It all starts w/ symplectic geometry. Think of a classical phase space, but forget coordinates for the moment. What you require is something called a symplectic form -- a closed, non-degenerate 2-form, ω... (2/7)