geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
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 is called a wave function or probability amplitude 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.
So the space of states is the subspace of the space of sections of the prequantum line bundle on those that are polarized.
For details see at geometric quantization – Space of quantum states.
duality between algebra and geometry in physics:
holographic principle in quantum field theory
bulk field theory | boundary field theory |
---|---|
dimension $n+1$ | dimension $n$ |
field | source |
wave function | correlation function |
space of quantum states | conformal blocks |