Perturbation theory

Perturbation theory refers to any situation in which a solution to an equation is analyzed by using an existing nearby solution (possibly solving a nearby equation rather than the original equation) as a reference. In many cases the reference solution is trivial (the zero solution). In order for perturbation theory to work, one or more of the following should be true:

• The desired initial data should be close to the reference initial data.
• The desired equation should be close to the reference equation.
• The time interval on which the analysis is performed should be small.

More informally, perturbation theory requires the usage of the $\epsilon$ symbol somewhere in the analysis.

Perturbation theory relies heavily on iteration schemes, bootstrap arguments, and the continuity method.

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 understand 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

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

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.

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.

Finally, the success of this method relies on the possibility to obtain a proper analytical solution to the leading order equation.

Nonlinear PDE and Perturbation Methods

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 FraA2006.

This can be seen by considering a NLKG equation

$\Box \phi -\lambda V(\phi )=0.$

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.

In order to see this we apply the computation given in the previous section by rescaling time as $t\rightarrow {\sqrt {\lambda }}t$ and take a solution series with a large parameter

$\phi =\sum _{n=0}^{\infty }{\frac {1}{\lambda ^{n}}}\phi _{n}.$

The non trivial set of equations is so obtained

$\partial _{\tau }^{2}\phi _{0}+V(\phi _{0})=0$

$\partial _{\tau }^{2}\phi _{1}+V'(\phi _{0})\phi _{1}=\Delta ^{2}\phi _{0}$

$\vdots$

where $\tau =\lambda t$. Indeed, this is a gradient expansion.

An interesting problem that applies to a given PDE is

$\Box \phi -\lambda V(\phi )=-j$

where $j$ is a driving term. When a small parameter series has to be computed we obtain that at the leading order one has generally to solve

$\Box \phi _{0}=-j$

that is very easy to solve by the Green function method

$\phi _{0}=\phi _{H}+\int d^{D}x'G(x-x')j(x')$

where

$\Box \phi _{H}=0$

and

$\Box G=-\delta ^{D}$

being $\delta ^{D}$ a Dirac distribution of the given dimensionality $D$.

A similar result holds also for the large parameter series FraA2007,FraB2007. We note that the leading order of the gradient expansion is now

$\partial _{\tau }^{2}\phi _{0}+V(\phi _{0})=j.$

A small time series holds that has the form

$\phi _{0}=\sum _{n=0}^{\infty }a_{n}\int d\tau 'G(\tau -\tau ')(\tau -\tau ')^{n}j(\tau ')$

being

$\partial _{\tau }^{2}G+V(G)=\delta$

and the coefficients $a_{n}$ are computed by deriving the equation we started from and with the initial conditions and will generally depend on the values of the source and its derivatives at the intial time. The success of the method relies on the ability to obtain analitically the Green function.

An interesting case is given by $V(\phi )=\phi ^{3}$ FraB2006. In this case one has (undoing any normalization)

$G(t)=\theta (t)\left({\frac {2}{\lambda }}\right)^{\frac {1}{4}}sn\left[\left({\frac {\lambda }{2}}\right)^{\frac {1}{4}}\Lambda t,i\right]$I

where $\theta$ is the step function, $sn$ is a Jacobi elliptic function and $\Lambda$ is a scale parameter being the theory scale invariant. This gives immediately the mass spectrum of the quantum theory as FraB2006

$\mu _{n}=(2n+1){\frac {\pi }{2K(i)}}\left({\frac {\lambda }{2}}\right)^{\frac {1}{4}}\Lambda$

where $n$ is an integer and $K(i)=\int _{0}^{\frac {\pi }{2}}{\frac {d\theta }{\sqrt {1+\sin ^{2}\theta }}}$.