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.^[1][2]

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,

${\displaystyle {\begin{aligned}f\star g&=fg+{\mathcal {O))(\hbar )\\{}[f,g]&=f\star g-g\star f=i\hbar \{f,g\}+{\mathcal {O))(\hbar ^{2})\end{aligned))}$

If one were given a Poisson manifold (M, {⋅, ⋅}), one could ask, in addition, that

${\displaystyle f\star g=fg+\sum _{k=1}^{\infty }\hbar ^{k}B_{k}(f\otimes g),}$

where the B_k 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,

${\displaystyle {\begin{cases}D:A[[\hbar ]]\to A[[\hbar ]]\\\sum _{k=0}^{\infty }\hbar ^{k}f_{k}\mapsto \sum _{k=0}^{\infty }\hbar ^{k}f_{k}+\sum _{n\geq 1,k\geq 0}D_{n}(f_{k})\hbar ^{n+k}\end{cases))}$

where D_n are differential operators of order at most n. The corresponding induced ${\displaystyle \star }$-product, ${\displaystyle \star '}$, is then

${\displaystyle f\,{\star }'\,g=D\left(\left(D^{-1}f\right)\star \left(D^{-1}g\right)\right).}$

For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" ${\displaystyle \star }$-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 G_n(2).

An example on two internal vertices is the following graph,

Associated bidifferential operator

Associated to each graph Γ, there is a bidifferential operator B_Γ( 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

${\displaystyle \Pi ^{i_{2}j_{2))\partial _{i_{2))\Pi ^{i_{1}j_{1))\partial _{i_{1))f\,\partial _{j_{1))\partial _{j_{2))g.}$

Associated weight

For adding up these bidifferential operators there are the weights w_Γ 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 m(Γ) = 8. 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 ⊂ ${\displaystyle \mathbb {C} }$, endowed with a metric

${\displaystyle ds^{2}={\frac {dx^{2}+dy^{2)){y^{2))};}$

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

${\displaystyle \phi (z,w)={\frac {1}{2i))\log {\frac {(z-w)(z-{\bar {w)))}{({\bar {z))-w)({\bar {z))-{\bar {w))))).}$

The integration domain is C_n(H) the space

${\displaystyle C_{n}(H):=\{(u_{1},\dots ,u_{n})\in H^{n}:u_{i}\neq u_{j}\forall i\neq j\}.}$

The formula amounts

${\displaystyle w_{\Gamma }:={\frac {m(\Gamma )}{(2\pi )^{2n}n!))\int _{C_{n}(H)}\bigwedge _{j=1}^{n}\mathrm {d} \phi (u_{j},u_{t1(j)})\wedge \mathrm {d} \phi (u_{j},u_{t2(j)})}$,

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

${\displaystyle f\star g=fg+\sum _{n=1}^{\infty }\left({\frac {i\hbar }{2))\right)^{n}\sum _{\Gamma \in G_{n}(2)}w_{\Gamma }B_{\Gamma }(f\otimes g).}$

Explicit formula up to second order

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

${\displaystyle {\begin{aligned}f\star g&=fg+{\tfrac {i\hbar }{2))\Pi ^{ij}\partial _{i}f\,\partial _{j}g-{\tfrac {\hbar ^{2)){8))\Pi ^{i_{1}j_{1))\Pi ^{i_{2}j_{2))\partial _{i_{1))\,\partial _{i_{2))f\partial _{j_{1))\,\partial _{j_{2))g\\&-{\tfrac {\hbar ^{2)){12))\Pi ^{i_{1}j_{1))\partial _{j_{1))\Pi ^{i_{2}j_{2))(\partial _{i_{1))\partial _{i_{2))f\,\partial _{j_{2))g-\partial _{i_{2))f\,\partial _{i_{1))\partial _{j_{2))g)+{\mathcal {O))(\hbar ^{3})\end{aligned))}$