In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.
Deformation quantization of a Poisson algebra Given a Poisson algebra (A, {⋅, ⋅}), a deformation quantization is an associative unital product ⋆ {\displaystyle \star } on the algebra of formal power series in ħ, A[[ħ]], subject to the following two axioms,
f ⋆ g = f g + O ( ℏ )
[ f , g ] = f ⋆ g − g ⋆ f = i ℏ { f , g } + O ( ℏ 2 ) {\displaystyle {\begin{aligned}f\star g&=fg+{\mathcal {O}}(\hbar )\\{}[f,g]&=f\star g-g\star f=i\hbar \{f,g\}+{\mathcal {O}}(\hbar ^{2})\end{aligned}}}
If one were given a Poisson manifold (M, {⋅, ⋅}), one could ask, in addition, that
f ⋆ g = f g + ∑ k = 1 ∞ ℏ k B k ( f ⊗ g ) , {\displaystyle f\star g=fg+\sum _{k=1}^{\infty }\hbar ^{k}B_{k}(f\otimes g),}
where the Bk are linear bidifferential operators of degree at most k. Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
{ D : A [ [ ℏ ] ] → A [ [ ℏ ] ] ∑ k = 0 ∞ ℏ k f k ↦ ∑ k = 0 ∞ ℏ k f k + ∑ n ≥ 1 , k ≥ 0 D n ( f k ) ℏ n + k {\displaystyle {\begin{cases}D:A[[\hbar ]]\to A[[\hbar ]]\\\sum _{k=0}^{\infty }\hbar ^{k}f_{k}\mapsto \sum _{k=0}^{\infty }\hbar ^{k}f_{k}+\sum _{n\geq 1,k\geq 0}D_{n}(f_{k})\hbar ^{n+k}\end{cases}}}
where Dn are differential operators of order at most n. The corresponding induced ⋆ {\displaystyle \star } -product, ⋆ ′ {\displaystyle \star '} , is then
f ⋆ ′ g = D ( ( D − 1 f ) ⋆ ( D − 1 g ) ) . {\displaystyle f\,{\star }'\,g=D\left(\left(D^{-1}f\right)\star \left(D^{-1}g\right)\right).}
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