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In mathematics, the '''Kontsevich quantization formula''' describes how to construct a generalized [[Moyal product|<small>★</small>-product]] operator algebra from a given arbitrary [[Poisson manifold]].  This operator algebra amounts to the [[Weyl quantization|deformation quantization]] of the corresponding Poisson algebra. It is due to [[Maxim Kontsevich]].<ref>M. Kontsevich (2003), [http://arxiv.org/abs/q-alg/9709040 ''Deformation Quantization of Poisson Manifolds''], ''Letters of Mathematical Physics'' '''66''', pp. 157&ndash;216.</ref><ref>{{cite journal
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| authors= Cattaneo,  Alberto and [[Giovanni Felder|Felder, Giovanni]]
| year = 2000
| title = A Path Integral Approach to the Kontsevich Quantization Formula
| journal = Communications in Mathematical Physics
| volume = 212
| issue = 3
| pages = 591
| publisher =
| jstor =
| doi = 10.1007/s002200000229
}}</ref>
 
==Deformation quantization of a Poisson algebra==
Given a [[Poisson algebra]] (''A'',&nbsp;{&middot;,&middot;}), a [[Deformation theory|deformation]] quantization is an '''associative''' unital product <small>★</small> on the algebra of formal power series in ''ħ'', ''A''[&nbsp;[&nbsp;''ħ''&nbsp;]&nbsp;], subject to the following two axioms,
 
:<math>f*g=fg+\mathcal{O}(\hbar),</math>
 
and
 
:<math>[f,g]:=f*g-g*f=i\hbar\{f,g\}+\mathcal{O}(\hbar^2).</math>
 
If one were given a Poisson manifold (''M'',&nbsp;{.,.}), one could ask, in addition, that
 
:<math>f*g=fg+\sum_{k=1}^\infty \hbar^kB_k(f\otimes g),</math>
 
where the ''B<sub>k</sub>'' are linear bi[[differential operator]]s of degree at most&nbsp;''k''.
 
Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
:<math>D\colon A[[\hbar]]\mapsto A[[\hbar]]: \sum_{k=0}^\infty \hbar^k f_k \mapsto \sum_{k=0}^\infty \hbar^k f_k +\sum_{n\ge1, k\ge0} D_n(f_k)\hbar^{n+k} ~,</math>
where ''D<sub>n</sub>'' are differential operators of order at most ''n''.
The corresponding induced <small>★</small>-product,  <small>★</small>', is then
:<math>f\,{*}'\,g = D((D^{-1}f)*(D^{-1}g)).</math>
 
For the archetypal example, one may well consider [[Hilbrand J. Groenewold|Groenewold]]'s original [[Weyl quantization|"Moyal&ndash;Weyl" <small>★</small>-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 {{mvar|Π}}.  From each internal vertex originate two edges.  All (equivalence classes of) graphs with ''n'' internal vertices are accumulated in the set {{math|''G''<sub>''n''</sub>(2)}}.
 
An example on two internal vertices is the following graph,
:[[Image:KontsevichGraph2.svg|||Kontsevich graph for n=2]]
 
===Associated bidifferential operator===
Associated to each graph ''Γ'', there is a bidifferential operator {{math|''B''<sub>''Γ''</sub>(''f'',&nbsp;''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 {{mvar|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 {{mvar|Π}} is the [[Poisson bivector]] of the Poisson manifold.
 
The term for the example graph is <math>\Pi^{i_2j_2}\partial_{i_2}\Pi^{i_1j_1}\partial_{i_1}f\,\partial_{j_1}\partial_{j_2}g</math>.
 
===Associated weight===
For adding up these bidifferential operators there are the weights {{math|''w<sub>Γ</sub>''}} of the graph {{mvar|Γ}}.  First of all, to each graph there is a multiplicity {{math|''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''&nbsp;+&nbsp;1))<sup>''n''</sup>.  The sample graph above has the weight {{math|''m''(Γ)&nbsp;{{=}}&nbsp;8}}. For this, it is helpful to enumerate the internal vertices from 1 to&nbsp;''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 {{math|''H'' ⊂ ℂ}}, endowed with a [[metric tensor|metric]] <math>ds^2=(dx^2+dy^2)/y^2</math>;  and, for two points  ''z'',&nbsp;''w''&nbsp;∈&nbsp;''H'' with ''z''&nbsp;≠&nbsp;''w'', we measure the angle {{mvar|φ}} between the geodesic from {{mvar|z}} to ''i∞'' and from  {{mvar|z}}  to {{mvar|w}} counterclockwise. This is
 
:<math>\phi(z,w)=\frac{1}{2i}\log\frac{(z-w)(z-\bar{w})}{(\bar{z}-w)(\bar{z}-\bar{w})}.</math>
 
The integration domain is ''C''<sub>''n''</sub>(''H'') the space
:<math>C_n(H):=\{(u_1,\dots,u_n)\in H^n: u_i\ne u_j\forall i\ne j\}.</math>
 
The formula amounts
:<math>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)})</math>,
where ''t''1(''j'') and ''t''2(''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&nbsp;''H''.
 
==The formula==
Given the above three definitions, the Kontsevich formula for a star product is now
:<math>f*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).</math>
 
===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 {{mvar|ħ}}, to just
 
:<math>\begin{align} f*g & = fg +\tfrac{i\hbar}{2}\Pi^{ij}\partial_i f\,\partial_j g -\tfrac{\hbar^2}{8}\Pi^{i_1j_1}\Pi^{i_2j_2}\partial_{i_1}\,\partial_{i_2}f \partial_{j_1}\,\partial_{j_2}g \\
& {} - \tfrac{\hbar^2}{12}\Pi^{i_1j_1}\partial_{j_1}\Pi^{i_2j_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{align}</math>
 
==References==
{{Reflist}}
{{DEFAULTSORT:Kontsevich Quantization Formula}}
[[Category:Mathematical quantization]]

Latest revision as of 01:59, 21 May 2014

Your writing should be well presented and fulfills all academic requirements. The second thing to do before writing the paper is to make an outline. When you have any kind of questions regarding exactly where and also tips on how to use writing my essay, you can email us on our web site. The scholarship essay that you write is supposed to present you well and thus should be detailed and giving reasons as to why you should be awarded the scholarship. Most of the time it is very difficult to find the topic and due to this most of the students feel very much depress and then they type write my essay or write my college essay from any of the best custom writing service then that company facilitate them according to their need. After all, they aren't cheating by using an academic writing service.

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If students opt for taking help from service provider companies then they will have plenty of options to choose. Why not starts your medical essay wriitng with your personal experiences starts exhibited in the form of anecdote. Narrative Essay writing: It is the simplest form of essay writing where the writer directly puts forth his personal experience & views as well. Leaving any of them can create confusion among your readers which will compel them to turn it down. Introduction part should be able to capture the attention of readers.

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