Geometric quantization

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

One of the earliest attempts at a natural quantization was Weyl quantization, done by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. Here, the position and momentum are reinterpreted as the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1949, J.E. Moyal considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to define the Moyal product of a pair of functions. More generally, this technique leads to deformation quantization, where the Moyal product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold.

ee also

* Half-form
* Lagrangian foliation
* Kirillov orbit method

External links

* [http://arxiv.org/abs/math-ph/0208008 William Ritter's review of Geometric Quantization] presents a general framework for all problems in physics and fits geometric quantization into this framework

* [http://math.ucr.edu/home/baez/quantization.html John Baez's review of Geometric Quantization] , by John Baez is short and pedagogical

* [http://www.unine.ch/phys/string/lecturesGQ.ps.gz Matthias Blau's primer on Geometric Quantization] , one of the very few good primers (ps format only)