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== 2D Euclidean quantum field theory ==
In [[quantum field theory]], the '''operator product expansion''' ('''OPE''') is a [[Laurent series]] expansion of two operators. A [[Laurent series]] is a generalization of the [[Taylor series]] in that finitely many powers of the inverse of the expansion variable(s) are added to the Taylor series: pole(s) of finite order(s) are added to the series.
 
Heuristically, in [[quantum field theory]] one is interested in the result of physical observables represented by [[Operator (mathematics)|operators]]. If one wants to know the result of making two physical observations at two points <math>z</math> and <math>w</math>, one can time order these operators in increasing time.
 
If one maps coordinates in a conformal manner, one is often interested in radial ordering. This is the analogue of time ordering where increasing time has been mapped to some increasing radius on the complex plane.  One is also interested in [[normal order]]ing of creation operators.
 
A radial-ordered '''OPE''' can be written as a normal-ordered '''OPE''' minus the non-normal-ordered terms.  The non-normal-ordered terms can often be written as a [[commutator]], and these have useful simplifying identities. The radial ordering supplies the convergence of the expansion.
 
The result is a convergent expansion of the product of two operators in terms of some terms that have poles in the complex plane (the Laurent terms) and terms that are finite.  This result represents the expansion of two operators at two different points as an expansion around just one point, where the poles represent where the two different points are the same point e.g.
 
:<math>1/(z-w)</math>.
 
Related to this is that an [[Operator (mathematics)|operator]] on the complex plane is in general written as a function of <math>z</math> and <math>\bar{z}</math>. These are referred to as the [[holomorphic]] and [[antiholomorphic function|anti-holomorphic]] parts respectively, as they are continuous and differentiable except at the (finite number of) singularities. One should really call them  [[meromorphic]] but [[holomorphic]] is common parlance. In general, the operator product expansion may not separate into holormorphic and anti-holomorphic parts, especially if there are <math>\log z</math> terms in the expansion. However, derivatives of the '''OPE''' can often separate the expansion into holomorphic and anti-holomorphic expansions. This expression is also an '''OPE'''  and in general is more useful.
 
== General ==
 
In [[quantum field theory]], the '''operator product expansion''' ('''OPE''') is a [[convergent expansion]] of the product of two [[field (physics)|fields]] at different points as a sum (possibly infinite) of local fields.
 
More precisely, if x and y are two different points, and A and B are [[operator-valued field]]s, then there is an [[open neighborhood]] of y, O such that for all x in O/{y}
 
:<math>A(x)B(y)=\sum_{i}c_i(x-y)^i C_i(y)</math>
 
where the sum is over finitely or countably many terms, C<sup>i</sup> are operator-valued fields, c<sub>i</sub> are [[analytic function]]s over O/{y} and the sum is convergent in the [[operator topology]] within O/{y}.
 
OPEs are most often used in [[conformal field theory]].
 
The notation <math>F(x,y)\sim G(x,y)</math> is often used to denote that the difference G(x,y)-F(x,y) remains analytic at the points x=y. This is an [[equivalence relation]].
 
==See also==
* [[QCD sum rules]]
 
==External links==
* [http://www.scholarpedia.org/article/Operator_product_expansion The OPE at Scholarpedia]
 
[[Category:Quantum field theory]]
[[Category:Conformal field theory]]
[[Category:String theory]]

Latest revision as of 20:35, 12 May 2014

But recently, researchers all over the world have been studying a more severe side effect. The Obama administration has been trying to garner a peace agreement between Israel and the Palestinian Authority. Almonds, pistachios, musli (safed musli), dates, raisins are used in aphrodisiac preparations of Unani Medicines. Get emergency medical assistance if you experience priapism - a condition that involves painful and prolonged erections. Low-priced genanerics help men obtain, solid as a rock long lasting erection for hours.

Not only that, the process of cell repair can also be done well ( the regenerate ) , that old cells replaced with better ones. This environment thus promotes the growth of the anaerobic bacteria. A large number of thousand older men are suffering from this term. Tadalafil is the particular generic name in this product. Because of it they have to feel shame somehow but they have nothing to worry now as they can Buy Viagra and get relief from their worries.

Men who are impotent from diabetes must seek treatment every month and each month, the attending doctor must draw a blood test called HBA1C, or another blood test called fructosamine. The tight jeans increase the temperature in your groin. For many leaders this can be a bewildering, confidence-sapping time - for them, this doesn't feel like a transition from adolescence to maturity. They might also incorporate a shallower central dip that is more suited to the female pelvic structure. mes, liquids and gels and want a fast, esthetically pleasing solution, this is the perfect product.

Another herbal remedy that has been studied for its effects on erectile dysfunction is Korean red ginseng. So, I would advise anyone suffering from low testosterone symptoms to get out there and do something about this condition in order to get their lives back. These symptoms include lack of energy and motivation, fatigue, sleeping problems, weight gain and increase in fat deposits in the abdomen, impotence, lack of interest in sex and other side effects. Often cause for pity, this problem can usually be handled if the actual problem is discovered. The Sizes can also be considerably larger or smaller than inconspicuous tumors.

Mutations in the HFE, SLC40A1, HAMP, TFR2, and HFE2 genes cause this genetic disorder. Some of these habits include excessive drinking and smoking. Not all studies point to such benefits, however, and many conventional doctors remain skeptical. Impotence is a common sexual condition that involves a partial, total or an inconsistent inability to achieve or retain an erection that is satisfactory for the purpose of sexual intercourse. Do you know what foods keep plaque out of your arteries.

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