Talk:PlanetPhysics/Deformation Quantization

Original TeX Content from PlanetPhysics Archive

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\begin{document}

 The term {\em 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' ''.

\begin{definition}

{\em Deformation quantization} is formally defined as the study of associative $*$--products of the form
$$f * g = fg \sum_{n >0}[{\hbar}^n C_n(f ,g)]$$, where $\hbar$ is a formal \htmladdnormallink{parameter}{http://planetphysics.us/encyclopedia/Parameter.html} (similar to Planck's constant, but allowed to vary), and $C_n$ are plane curves over $\mathcal{C}$ that are undergoing an Abelian quantization.
\end{definition}

The \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} is intensely studied by physical mathematicians, and has been intensively developed in recent years in the context of smooth Poisson \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}, perhaps because of its potential applications in \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html}. Thus, it would be natural to consider such deformations
`in the direction of Poisson brackets' by choosing
$$C_1( f ,g) = \{f, g\}$$
(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 $C_1$ brings no significant loss of generality for any \htmladdnormallink{quantization}{http://planetphysics.us/encyclopedia/MoyalDeformation.html} on a smooth (finite dimensional) manifold. Furthermore, the famous result of Hochschild, Kostant and Rosenberg implies that any $*$--product on a \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html}, commutative algebra is equivalent to an algebra defined as above with an antisymmetric $C_1$.

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.

\end{document}