Kontsevich quantization formula
Encyclopedia
In mathematics, the Kontsevich quantization formula describes how to construct an generalized ∗-product
operator algebra from a given Poisson manifold
. This operator algebra amounts to the deformation quantization
of the Poisson algebra. It is due to Maxim Kontsevich
.
(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,
and
If one were given a Poisson manifold (M, {.,.}), one could ask, in addition, that
where the Bk are linear bidifferential operator
s of degree at most k.
Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
where Dn are differential operators of order at most n.
The corresponding induced ∗-product, ∗', is
For the archetypal example, one may well consider Groenewold
's original "Moyal–Weyl" ∗-product
.
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,
The term for the example graph is
.
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, 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
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 j. The vertices f and g are at the fixed positions 0 and 1 in H.
Moyal product
In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket .This particular star product is also sometimes called...
operator algebra from a given Poisson manifold
Poisson manifold
In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra C^\infty\, of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra...
. This operator algebra amounts to the deformation quantization
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
of the Poisson algebra. It is due to Maxim Kontsevich
Maxim Kontsevich
Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...
.
Deformation quantization of a Poisson algebra
Given a Poisson algebraPoisson algebra
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups...
(A, {.,.}), a deformation
Deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
quantization is an associative unital product ∗ on the algebra of formal power series in ħ, A[ [ ħ ] ], subject to the following two axioms,
and
If one were given a Poisson manifold (M, {.,.}), one could ask, in addition, that
where the Bk are linear bidifferential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
s of degree at most k.
Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
where Dn are differential operators of order at most n.
The corresponding induced ∗-product, ∗', is
For the archetypal example, one may well consider Groenewold
Hilbrand J. Groenewold
Hilbrand Johannes Groenewold was a Dutch theoretical physicist who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization....
's original "Moyal–Weyl" ∗-product
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
.
Kontsevich graphs
A Kontsevich graph is a simple directed graphDirected graph
A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...
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,
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 n 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 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
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
; 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 isThe 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 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 nowFormula up to second order
Enforcing associativity of the ∗-product, it is easy to check directly that the Kontsevich formula must reduce to second order in ħ to just