nLab
space of states (in geometric quantization)

Idea

There are at least two formalizations of quantization, one of them is geometric quantization. In this context a quantum state is identified with a certain section (a polarized section) of a certain complex line bundle: the prequantum line bundle.

Here a section of the prequantum line bundle is what is physics ia called a wave function on the space of field configurations. A choice of polarization on this space is a choice of “canonical coordinates” and “canonical momenta”. Hence a polarized section, and hence a quantum state in the sense of geometric quantization, is, in physics language, a wave function of the canonical coordinates.

Related concepts

• space of states

  • wave function

• superposition, quantum interference

• entanglement

• quantum operator (in geometric quantization)

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation