Kontsevich quantization formula

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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,

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

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,

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

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,

Kontsevich graph for n=2

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

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

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

The integration domain is Cn(H) the space

The formula amounts

,

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

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

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 }}