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== Duality in perturbation theory ==
== Duality in perturbation theory ==


In this section we will show how a duality principle holds in perturbation theory
Duality in perturbation theory has been introduced in [[Fra1998]]. It can be formulated by saying that a  solution series with a large parameter is dual to a solution series with a small parameter as it can be obtained by interchanging the choice of the perturbation term in the given equation.
showing how to derive a strong coupling expansion with the leading order ruled by
 
an adiabatic dynamics in order to study the evolution of a physical system.  
A typical perturbation problem can be formulated with the equation
We consider the following perturbation problem


<math>
<math>
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</math>
</math>


being <math>\lambda</math> an arbitrary ordering parameter: As is well known
being <math>\lambda</math> an arbitrary ordering parameter. A solution series with a small parameter <math>\lambda\rightarrow 0</math> can be computed taking
an expansion parameter is obtained by the computation of the series itself. The standard
approach assume the limit <math>\lambda\rightarrow 0</math> and putting


<math>
<math>
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</math>
</math>


one gets the equations for the series
giving the following equations to be solved


<math>
<math>
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</math>
</math>


where a derivative with respect to the ordering parameter is indicated by a prime. We recognize here a conventional small perturbation theory as it should be. But the ordering parameter is just a conventional matter and so one may ask what does it mean to consider <math>L(u)</math> as a perturbation instead with respect to the same parameter. Indeed one formally could write the set of equations
where a derivative with respect to the ordering parameter is indicated by a prime. The choice of the ordering parameter is just a conventional matter and one can choice to consider <math>L(u)</math> as a perturbation instead with respect to the same parameter. Indeed one formally could write the set of equations


<math>
<math>
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</math>
</math>


where we have interchanged <math>L(u)</math> and <math>V(u)</math> and renamed the solution as <math>v</math>. The question to be answered is what is the expansion parameter now and what derivative the prime means. To answer this question we rescale the time variable as <math>\tau = \lambda t</math> into eq.(\ref{eq:eq1}) obtaining the equation
where <math>L(u)</math> and <math>V(u)</math> are interchanged with the new solution <math>v</math>. In order to undertsand the expansion parameter we rescale the time variable as <math>\tau = \lambda t</math> into the equation to be solved obtaining


<math>
<math>
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</math>
</math>


and let us introduce the small parameter <math>\epsilon=\frac{1}{\lambda}</math>. It easy to see that applying again the small perturbation theory to the parameter <math>\epsilon\rightarrow 0</math> we get the set of equations (\ref{eq:set}) but now the time is scaled as <math>t/\epsilon</math>, that is, at the leading order the development parameter of the series will enter into the scale of the time evolution producing a proper slowing down ruled by the equation
and we introduce the small parameter <math>\epsilon=\frac{1}{\lambda}</math>. One sees that applying again the small perturbation theory to the parameter <math>\epsilon\rightarrow 0</math> we get the required set of equations but now the time is scaled as <math>t/\epsilon</math>, that is, at the leading order the development parameter of the series will enter into the scale of the time evolution producing a proper slowing down ruled by the equation


<math>
<math>
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</math>
</math>


that we can recognize as an equation for adiabatic evolution that in the proper limit <math>\epsilon\rightarrow 0</math> will give the static solution <math>V(u_0)=0</math>. We never assume this latter solution but rather we will study the evolution of eq.(\ref{eq:lead}). Finally, the proof is complete as we have obtained a dual series
that is an equation for adiabatic evolution that in the proper limit <math>\epsilon\rightarrow 0</math> will give the static solution <math>V(u_0)=0</math>. So, the dual series


<math>
<math>
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</math>
</math>


by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit <math>\lambda\rightarrow\infty</math> dual to the small perturbation theory <math>\lambda\rightarrow 0</math> we started with and having an adiabatic equation at the leading order.
is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit <math>\lambda\rightarrow\infty</math> dual to the small perturbation theory <math>\lambda\rightarrow 0</math> we started with and having an adiabatic equation at the leading order.
 
It is interesting to note that, for a partial differential equation,
we can be forced into a homogeneous equation because, generally, if we require
also a scaling into space variables we gain no knowledge at all on the evolution of a
physical system. On the other side, requiring a scaling on the space variables and not on
the time variable will wash away any evolution of the system. So, on most physical systems
a strong perturbation means also a homogeneous solution but this is not a general rule. As
an example one should consider fluid dynamics where two regimes dual each other can be found
depending if it is the Eulerian or the Navier-Stokes term to prevail. In general relativity
things stay in a way to get a homogeneous equation at the leading order. The reason for this
is that products of derivatives or second order derivatives in space coordinates are
the only elements forming the Einstein tensor beside time dependence.

Revision as of 08:15, 14 June 2007

Welcome to the sandbox! Please feel free to edit this page as you please by clicking on the "edit" tab at the top of this page. Terry 14:58, 30 July 2006 (EDT)

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Duality in perturbation theory

Duality in perturbation theory has been introduced in Fra1998. It can be formulated by saying that a solution series with a large parameter is dual to a solution series with a small parameter as it can be obtained by interchanging the choice of the perturbation term in the given equation.

A typical perturbation problem can be formulated with the equation

being an arbitrary ordering parameter. A solution series with a small parameter can be computed taking

giving the following equations to be solved

where a derivative with respect to the ordering parameter is indicated by a prime. The choice of the ordering parameter is just a conventional matter and one can choice to consider as a perturbation instead with respect to the same parameter. Indeed one formally could write the set of equations

where and are interchanged with the new solution . In order to undertsand the expansion parameter we rescale the time variable as into the equation to be solved obtaining

and we introduce the small parameter . One sees that applying again the small perturbation theory to the parameter we get the required set of equations but now the time is scaled as , that is, at the leading order the development parameter of the series will enter into the scale of the time evolution producing a proper slowing down ruled by the equation

that is an equation for adiabatic evolution that in the proper limit will give the static solution . So, the dual series

is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit dual to the small perturbation theory we started with and having an adiabatic equation at the leading order.