Quantum Theory for Mathematicians pp 483-526 | Cite as

# Geometric Quantization on Manifolds

## Abstract

Geometric quantization is a type of *quantization*, which is a general term for a procedure that associates a quantum system with a given classical system. In practical terms, if one is trying to deduce what sort of quantum system should model a given physical phenomenon, one often begins by observing the classical limit of the system. Electromagnetic radiation, for example, is describable on a macroscopic scale by Maxwell’s equations. On a finer scale, quantum effects (photons) become important. How should one determine the correct *quantum* theory of electromagnetism? It seems that the only reasonable way to proceed is to “quantize” Maxwell’s equations—and then to compare the resulting quantum system to experiment.

## Keywords

Line Bundle Symplectic Manifold Cotangent Bundle Geometric Quantization Vertical Polarization## References

- [4].R.J. Blattner, Quantization and representation theory. In
*Harmonic Analysis on Homogeneous Spaces*(Proceedings of Symposia in Pure Mathematics, vol. XXVI, Williams College, Williamstown, MA, 1972). (American Mathematical Society, Providence, RI, 1973), pp. 147–165Google Scholar - [20].B.C. Hall, Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type. Comm. Math. Phys.
**226**, 233–268 (2002)CrossRefMATHMathSciNetGoogle Scholar - [29].J. Lee,
*Introduction to Smooth Manifolds*, 2nd edn. (Springer, London, 2006)Google Scholar - [45].N. Woodhouse,
*Geometric Quantization*, 2nd edn. (Oxford University Press, Oxford, 1992)MATHGoogle Scholar