Kontsevich quantization formula

In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary 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 ★ on the algebra of formal power series in ħ, A[[ħ]], subject to the following two axioms,

{\displaystyle {\begin{aligned}f*g&=fg+{\mathcal {O}}(\hbar )\\{}[f,g]&=f*g-g*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

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

where the Template:Mvar are linear bidifferential operators of degree at most Template:Mvar.

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 Template:Mvar are differential operators of order at most Template:Mvar. The corresponding induced ★-product, ★′, is then

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

For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" ★-product.

Kontsevich graphs

A Kontsevich graph is a simple directed graph without loops on 2 external vertices, labeled f and g; and Template:Mvar internal vertices, labeled Π. From each internal vertex originate two edges. All (equivalence classes of) graphs with Template:Mvar 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

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

Associated weight

For adding up these bidifferential operators there are the weights 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 Template:Mvar internal vertices is (n(n + 1))n. The sample graph above has the weight m(Γ) = 8. For this, it is helpful to enumerate the internal vertices from 1 to Template:Mvar.

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 ⊂ ℂ, endowed with a metric

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

and, for two points z, wH with zw, we measure the angle Template:Mvar between the geodesic from Template:Mvar to i and from Template:Mvar to Template:Mvar 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 Cn(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 Template:Mvar. The vertices f and g are at the fixed positions 0 and 1 in Template:Mvar.

The formula

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

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

Explicit formula up to second order

Enforcing associativity of the ★-product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in Template:Mvar, to just

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

References

1. M. Kontsevich (2003), Deformation Quantization of Poisson Manifolds, Letters of Mathematical Physics 66, pp. 157–216.
2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}