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

{\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 $(M, {\cdot, \cdot})$ , one could ask, in addition, that

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,

{\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 $\star$ -product, $\star '$ , is then

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" $\star$ -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

\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 $\mathbb {C}$ , 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}\neq u_{j}\forall i\neq 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\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 $\star$ -product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in $ħ$ , to just

{\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}}

References

M. Kontsevich (2003), Deformation Quantization of Poisson Manifolds, Letters of Mathematical Physics 66, pp. 157–216.
Cattaneo, Alberto; Felder, Giovanni (2000). "A Path Integral Approach to the Kontsevich Quantization Formula". Communications in Mathematical Physics. 212 (3): 591–611. arXiv:math/9902090. Bibcode:2000CMaPh.212..591C. doi:10.1007/s002200000229. S2CID 8510811.

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[1] M. Kontsevich (2003), Deformation Quantization of Poisson Manifolds, Letters of Mathematical Physics 66, pp. 157–216.

[2] Cattaneo, Alberto; Felder, Giovanni (2000). "A Path Integral Approach to the Kontsevich Quantization Formula". Communications in Mathematical Physics. 212 (3): 591–611. arXiv:math/9902090. Bibcode:2000CMaPh.212..591C. doi:10.1007/s002200000229. S2CID 8510811.