- Geometric quantization
In
mathematical physics , geometric quantization is a mathematical approach to defining a quantum theory corresponding to a givenclassical theory . It attempts to carry outquantization , for which there isin 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 theHeisenberg picture ofquantum mechanics and theHamilton equation in classical physics should be built in.One of the earliest attempts at a natural quantization was
Weyl quantization , done byHermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (aself-adjoint operator on aHilbert space ) with a real-valued function on classicalphase space . Here, the position and momentum are reinterpreted as the generators of theHeisenberg group , and the Hilbert space appears as agroup 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 theMoyal product of a pair of functions. More generally, this technique leads todeformation quantization , where the Moyal product is taken to be a deformation of the algebra of functions on asymplectic manifold orPoisson 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)
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