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| | == Testing MathJax == |
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| | | $$ |
| == Duality in perturbation theory ==
| | \int_a^b f'(s) ds = f(b) - f(a). |
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| 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.
| | It works! |
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| A typical perturbation problem can be formulated with the equation
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| <math>
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| \partial_t u = L(u) + \lambda V(u)
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| </math>
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| being <math>\lambda</math> an arbitrary ordering parameter. A solution series with a small parameter <math>\lambda\rightarrow 0</math> can be computed taking
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| <math>
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| u = \sum_{n=0}^{\infty} \lambda^n u_n
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| </math>
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| giving the following equations to be solved
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| <math>
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| \partial_t u_0 = L(u_0)
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| </math>
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| <math>
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| \partial_t u_1 = L'(u_0)u_1 + V(u_0)
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| </math>
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| <math>
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| \vdots
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| </math>
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| 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
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| <math>
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| \partial_t v_0 = V(v_0)
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| </math>
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| <math>
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| \partial_t v_1 = V'(v_0)v_1 + L(v_0)
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| </math>
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| <math>
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| \vdots
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| </math>
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| 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
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| <math>
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| \lambda\partial_{\tau} u = L(u) + \lambda V(u)
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| </math>
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| 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
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| <math>
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| \epsilon\partial_t v_0 = V(v_0)
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| </math>
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| that is an equation for adiabatic evolution that in the proper limit <math>\epsilon\rightarrow 0</math> will give the static solution <math>V(v_0)=0</math>. So, the dual series
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| <math>
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| v = \sum_{n=0}^{\infty}\frac{1}{\lambda^n}v_n
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| </math>
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| 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.
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| The main mathematical problem of this kind of methods is the existence of the solution series. For the most interesting cases this series are not converging and represent asymptotic approximations to the true solution.
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| Finally, the success of this method relies on the possibility to obtain a proper analytical solution to the leading order equation. In order to see this we apply the computation given in the previous section by rescaling time as <math>t\rigtharrow \lambda t</math> and take a solution series with a large parameter
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| \phi = \sum_{n=0}^{\infty}\frac{1}{\lambda^n}\phi_n.
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| The non trivial set of equations is so obtained
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| <math>\partial_t^2\phi_0+\lambda V(\phi)=0</math>
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| <math>\partial_t^2\phi_1+V'(\phi_0)\phi=\grad^2\phi_0</math>
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| <math>\vdots</math>
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| == Nonlinear PDE and Perturbation Methods ==
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| The application of the perturbation methods described above to PDE gives an interesting result, i.e. the dual series to the small parameter solution series is a gradient expansion [[Fra2006]].
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| This can be seen by considering a NLKG equation
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| <math>\Box\phi+\lambda V(\phi)=0.</math>
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| The choice of the perturbation term to compute a small or a large parameter series depends also on the way the derivatives of the field are managed.
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