Kontsevich quantization formula - Alchetron, the free social encyclopedia

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.

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,

f ∗ g = f g + O ( ℏ ) [ f , g ] = f ∗ g − g ∗ f = i ℏ { f , g } + O ( ℏ 2 )

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

f ∗ g = f g + ∑ k = 1 ∞ ℏ k B k ( f ⊗ 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,

{ D : A [ [ ℏ ] ] → A [ [ ℏ ] ] ∑ k = 0 ∞ ℏ k f k ↦ ∑ k = 0 ∞ ℏ k f k + ∑ n ≥ 1 , k ≥ 0 D n ( f k ) ℏ n + k

where D_n are differential operators of order at most n. The corresponding induced ★-product, ★′, is then

f ∗ ′ g = D ( ( D − 1 f ) ∗ ( D − 1 g ) ) .

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

Π i 2 j 2 ∂ i 2 Π i 1 j 1 ∂ i 1 f ∂ j 1 ∂ 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

d s 2 = d x 2 + d y 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

ϕ ( z , w ) = 1 2 i log ⁡ ( z − w ) ( z − w ¯ ) ( z ¯ − w ) ( z ¯ − w ¯ ) .

The integration domain is C_n(H) the space

C n ( H ) := { ( u 1 , … , u n ) ∈ H n : u i ≠ u j ∀ i ≠ j } .

The formula amounts

w Γ := m ( Γ ) ( 2 π ) 2 n n ! ∫ C n ( H ) ⋀ j = 1 n d ϕ ( u j , u t 1 ( j ) ) ∧ d ϕ ( u j , u t 2 ( 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 = f g + ∑ n = 1 ∞ ( i ℏ 2 ) n ∑ Γ ∈ G n ( 2 ) w Γ B Γ ( f ⊗ 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 = f g + i ℏ 2 Π i j ∂ i f ∂ j g − ℏ 2 8 Π i 1 j 1 Π i 2 j 2 ∂ i 1 ∂ i 2 f ∂ j 1 ∂ j 2 g − ℏ 2 12 Π i 1 j 1 ∂ j 1 Π i 2 j 2 ( ∂ i 1 ∂ i 2 f ∂ j 2 g − ∂ i 2 f ∂ i 1 ∂ j 2 g ) + O ( ℏ 3 )

References

Kontsevich quantization formula Wikipedia

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