deformation quantization
DeformationQuantization
2009-06-20 16:27:04
2009-06-20 16:37:54
Definition
deformation
Abelian quantization
$\hbar$ parameter
associative $*$--products
Quantization
Deformation
Harrison Cohomology
Singular Curves
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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 parameter (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 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.