# Variational perturbation theory

In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say

${\displaystyle s=\sum _{n=0}^{\infty }a_{n}g^{n}}$,

into a convergent series in powers

${\displaystyle s=\sum _{n=0}^{\infty }b_{n}/(g^{\omega })^{n}}$,

where ${\displaystyle \omega }$ is a critical exponent (the so-called index of "approach to scaling" introduced by Franz Wegner). This is possible with the help of variational parameters, which are determined by optimization order by order in ${\displaystyle g}$. The partial sums are converted to convergent partial sums by a method developed in 1992.[1]

Most perturbation expansions in quantum mechanics are divergent for any small coupling strength ${\displaystyle g}$. They can be made convergent by VPT (for details see the first textbook cited below). The convergence is exponentially fast.[2][3]

After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in quantum field theory with its anomalous dimensions.[4] Applications focus on the theory of critical phenomena. It has led to the most accurate predictions of critical exponents. More details can be read here.