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 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 ⋆ {\displaystyle \star } {\displaystyle \star } 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 ) {\displaystyle {\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}}} {\displaystyle {\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, {⋅, ⋅}), one could ask, in addition, that

f ⋆ g = f g + ∑ k = 1 ∞ ℏ k B k ( f ⊗ g ) , {\displaystyle f\star g=fg+\sum _{k=1}^{\infty }\hbar ^{k}B_{k}(f\otimes g),} {\displaystyle f\star g=fg+\sum _{k=1}^{\infty }\hbar ^{k}B_{k}(f\otimes g),}

where the Bk 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 {\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}}} {\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 Dn are differential operators of order at most n. The corresponding induced ⋆ {\displaystyle \star } {\displaystyle \star }-product, ⋆ ′ {\displaystyle \star '} {\displaystyle \star '}, is then

f ⋆ ′ g = D ( ( D − 1 f ) ⋆ ( D − 1 g ) ) . {\displaystyle f\,{\star }'\,g=D\left(\left(D^{-1}f\right)\star \left(D^{-1}g\right)\right).} {\displaystyle 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" ⋆ {\displaystyle \star } {\displaystyle \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 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

Π i 2 j 2 ∂ i 2 Π i 1 j 1 ∂ i 1 f ∂ j 1 ∂ j 2 g . {\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.} {\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 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 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 {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }, endowed with the Poincaré metric

d s 2 = d x 2 + d y 2 y 2 ; {\displaystyle ds^{2}={\frac {dx^{2}+dy^{2}}{y^{2}}};} {\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

ϕ ( z , w ) = 1 2 i log ⁡ ( z − w ) ( z − w ¯ ) ( z ¯ − w ) ( z ¯ − w ¯ ) . {\displaystyle \phi (z,w)={\frac {1}{2i}}\log {\frac {(z-w)(z-{\bar {w}})}{({\bar {z}}-w)({\bar {z}}-{\bar {w}})}}.} {\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 Cn(H) the space

C n ( H ) := { ( u 1 , … , u n ) ∈ H n : u i ≠ u j ∀ i ≠ j } . {\displaystyle C_{n}(H):=\{(u_{1},\dots ,u_{n})\in H^{n}:u_{i}\neq u_{j}\forall i\neq j\}.} {\displaystyle C_{n}(H):=\{(u_{1},\dots ,u_{n})\in H^{n}:u_{i}\neq u_{j}\forall i\neq 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 ) ) {\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)})} {\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

f ⋆ g = f g + ∑ n = 1 ∞ ( i ℏ 2 ) n ∑ Γ ∈ G n ( 2 ) w Γ B Γ ( f ⊗ g ) . {\displaystyle 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).} {\displaystyle 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 ⋆ {\displaystyle \star } {\displaystyle \star }-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 ) {\displaystyle {\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}}} {\displaystyle {\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

  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.