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 \star on the algebra of formal power series in ħ, A ħ , subject to the following two axioms, If one were given a Poisson manifold (M, {⋅, ⋅}) , one could ask, in addition, that 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, where Dn are differential operators of order at most n. The corresponding induced \star-product, \star', is then For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" \star-product.

Kontsevich graphs

A Kontsevich graph is a simple directed graph without loops on 2 external vertices, labeled f and g; and n internal vertices, labeled Π . From each internal vertex originate two edges. All (equivalence classes of) graphs with n internal vertices are accumulated in the set Gn(2) . An example on two internal vertices is the following graph,

Associated bidifferential operator

Associated to each graph Γ , there is a bidifferential operator BΓ( f, g) defined as follows. For each edge there is a partial derivative on the symbol of the target vertex. It is contracted with the corresponding index from the source symbol. The term for the graph Γ is the product of all its symbols together with their partial derivatives. Here f and g stand for smooth functions on the manifold, and Π is the Poisson bivector of the Poisson manifold. The term for the example graph is

Associated weight

For adding up these bidifferential operators there are the weights wΓ of the graph Γ . First of all, to each graph there is a multiplicity m(Γ) which counts how many equivalent configurations there are for one graph. The rule is that the sum of the multiplicities for all graphs with n internal vertices is (n(n + 1))n . The sample graph above has the multiplicity . For this, it is helpful to enumerate the internal vertices from 1 to n. In order to compute the weight we have to integrate products of the angle in the upper half-plane, H, as follows. The upper half-plane is H ⊂ \mathbb{C} , endowed with the Poincaré metric and, for two points z, w ∈ H with z ≠ w , we measure the angle φ between the geodesic from z to i∞ and from z to w counterclockwise. This is The integration domain is Cn(H) the space The formula amounts where t1(j) and t2(j) are the first and second target vertex of the internal vertex j. The vertices f and g are at the fixed positions 0 and 1 in H.

The formula

Given the above three definitions, the Kontsevich formula for a star product is now

Explicit formula up to second order

Enforcing associativity of the \star-product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in ħ, to just