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deformation quantization
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(Definition)
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The term deformation quantization was introduced by Moshe Flato, who suggested that “any nontrivial associative deformation of an algebra of functions should be interpreted as a kind of `quantization' ”.
Definition 0.1
Deformation quantization is formally defined as the study of associative  –products of the form
, where  is a formal parameter (similar to Planck's constant, but allowed to vary), and  are plane curves over
 that are undergoing an Abelian quantization.
The concept is intensely studied by physical mathematicians, and has been intensively developed in recent years in the context of smooth Poisson manifolds, perhaps because of its potential applications in theoretical physics. Thus, it would be natural to consider such deformations `in the direction of Poisson brackets' by choosing
(cf. Drinfel'd), which is naturally antisymmetric. However, independently of the symplectic structure, one can consider more general deformations than that defined above, because the antisymmetry of  brings no significant loss of generality for any quantization on a smooth (finite dimensional) manifold. Furthermore, the famous result of Hochschild, Kostant and Rosenberg implies that any  –product on a regular, commutative algebra is equivalent to an algebra defined as above with an antisymmetric  .
An especially interesting study is concerned with the deformation quantization on varieties with singularities. The cohomological implications of such singularities should lead to some very interesting mathematical properties and also to novel mathematical results.
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"deformation quantization" is owned by bci1.
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See Also: quantization, Poisson ring
| Other names: |
Abelian quantization |
| Also defines: |
deformation, Abelian quantization, varieties with singularities,  parameter, associative  --products, cohomological properties |
| Keywords: |
Quantization, Deformation, Harrison Cohomology, Singular Curves |
Cross-references: regular, quantization, theoretical physics, manifolds, concept, parameter
There are 7 references to this object.
This is version 10 of deformation quantization, born on 2009-06-20, modified 2009-06-22.
Object id is 812, canonical name is DeformationQuantization.
Accessed 1921 times total.
Classification:
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Pending Errata and Addenda
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![$\displaystyle f * g = fg \sum_{n >0}[{\hbar}^n C_n(f ,g)]$](http://images.physicslibrary.org/cache/objects/812/l2h/img2.png "$\displaystyle f * g = fg \sum_{n >0}[{\hbar}^n C_n(f ,g)]$")
