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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

$\partial _{t}u=L(u)+\lambda V(u)$

being $\lambda$ an arbitrary ordering parameter. A solution series with a small parameter $\lambda \rightarrow 0$ can be computed taking

$u=\sum _{n=0}^{\infty }\lambda ^{n}u_{n}$

giving the following equations to be solved

$\partial _{t}u_{0}=L(u_{0})$

$\partial _{t}u_{1}=L'(u_{0})u_{1}+V(u_{0})$

$\vdots$

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 $L(u)$ as a perturbation instead with respect to the same parameter. Indeed one formally could write the set of equations

$\partial _{t}v_{0}=V(v_{0})$

$\partial _{t}v_{1}=V'(v_{0})v_{1}+L(v_{0})$

$\vdots$

where $L(u)$ and $V(u)$ are interchanged with the new solution $v$. In order to undertsand the expansion parameter we rescale the time variable as $\tau =\lambda t$ into the equation to be solved obtaining

$\lambda \partial _{\tau }u=L(u)+\lambda V(u)$

and we introduce the small parameter $\epsilon ={\frac {1}{\lambda }}$. One sees that applying again the small perturbation theory to the parameter $\epsilon \rightarrow 0$ we get the required set of equations but now the time is scaled as $t/\epsilon$, 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

$\epsilon \partial _{t}v_{0}=V(v_{0})$

that is an equation for adiabatic evolution that in the proper limit $\epsilon \rightarrow 0$ will give the static solution $V(v_{0})=0$. So, the dual series

$v=\sum _{n=0}^{\infty }{\frac {1}{\lambda }}v_{n}$

is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit $\lambda \rightarrow \infty$ dual to the small perturbation theory $\lambda \rightarrow 0$ we started with and having an adiabatic equation at the leading order.