# 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.

## References

1. ^ Kleinert, H. (1995). "Systematic Corrections to Variational Calculation of Effective Classical Potential" (PDF). Physics Letters A. 173 (4–5): 332–342. Bibcode:1993PhLA..173..332K. doi:10.1016/0375-9601(93)90246-V.
2. ^
3. ^ Guida, R.; Konishi, K.; Suzuki, H. (1996). "Systematic Corrections to Variational Calculation of Effective Classical Potential". Annals of Physics. 249 (1): 109–145. arXiv:hep-th/9505084. Bibcode:1996AnPhy.249..109G. doi:10.1006/aphy.1996.0066.
4. ^ Kleinert, H. (1998). "Strong-coupling behavior of φ^4 theories and critical exponents" (PDF). Physical Review D. 57 (4): 2264. Bibcode:1998PhRvD..57.2264K. doi:10.1103/PhysRevD.57.2264.