Kontsevich quantization formula

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 ★ 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 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 ★-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 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 ⊂ 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

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

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 ħ, 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. Cattaneo, Alberto and Felder, Giovanni (2000). "A Path Integral Approach to the Kontsevich Quantization Formula". Communications in Mathematical Physics. 212 (3): 591. arXiv:math/9902090. Bibcode:2000CMaPh.212..591C. doi:10.1007/s002200000229.CS1 maint: uses authors parameter (link)