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 $(A, {\cdot, \cdot})$ , 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,

\begin{align} f*g &=fg+\mathcal{O}(\hbar)\\ {}[f,g] &=f*g-g*f=i\hbar\{f,g\}+\mathcal{O}(\hbar^2) \end{align}

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

f*g=fg+\sum_{k=1}^\infty \hbar^kB_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,

\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\ge1, k\ge0} 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

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

\Pi^{i_2j_2}\partial_{i_2}\Pi^{i_1j_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 $H \subset ℂ$ , endowed with a metric

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

and, for two points $z, w \in H$ with $z \neq w$ , we measure the angle $φ$ between the geodesic from $z$ to $i \infty$ and from $z$ to $w$ counterclockwise. This is

\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

C_n(H):=\{(u_1,\dots,u_n)\in H^n: u_i\ne u_j\forall i\ne j\}.

The formula amounts

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

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

f*g = fg +\tfrac{i\hbar}{2}\Pi^{ij}\partial_i f\,\partial_j g -\tfrac{\hbar^2}{8}\Pi^{i_1j_1}\Pi^{i_2j_2}\partial_{i_1}\,\partial_{i_2}f \partial_{j_1}\,\partial_{j_2}g - \tfrac{\hbar^2}{12}\Pi^{i_1j_1}\partial_{j_1}\Pi^{i_2j_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

↑ M. Kontsevich (2003), Deformation Quantization of Poisson Manifolds, Letters of Mathematical Physics 66, pp. 157–216.
↑ Cattaneo, Alberto and Felder, Giovanni (2000). "A Path Integral Approach to the Kontsevich Quantization Formula". Communications in Mathematical Physics 212 (3): 591. doi:10.1007/s002200000229.