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  "body": "\n## Summary\n\nGeometric quantization is a method to build a quantum system from a classical one, using the geometry of a [[symplectic manifold|CAN490EPRT]].\nIt first attaches to the manifold a complex line bundle with a connection whose curvature is the symplectic form.\nThen it cuts down the space of sections of this bundle with a choice of polarization.\nThe method was developed by Kostant and Souriau in the 1960s and 1970s, and it is closely tied to the orbit method in representation theory.\n\n## Explanations\n\n### L1 human\n\nA classical system lives on a phase space, and its observables are functions there; a quantum system lives in a Hilbert space, and its observables are operators.\nGeometric quantization tries to pass from the first to the second in a way that depends only on the geometry, not on chosen coordinates.\nWave functions become sections of a line bundle over phase space.\nA polarization says that they should depend on only half of the variables, such as positions and not momenta.\nFor the sphere with total area $2\\pi\\hbar k$, the method produces a space of dimension $k + 1$, which is the state space of a quantum spin.\n",
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