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

External links

• Kleinert H., Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3. Auflage, World Scientific (Singapore, 2004) (readable online here) (see Chapter 5)
• Kleinert H. and Verena Schulte-Frohlinde, Critical Properties of φ⁴-Theories, World Scientific (Singapur, 2001); Paperback ISBN 981-02-4658-7 (readable online here) (see Chapter 19)
• Feynman, R. P.; Kleinert, H. (1986). "Effective classical partition functions" (PDF). Physical Review A. 34 (6): 5080–5084. Bibcode:1986PhRvA..34.5080F. doi:10.1103/PhysRevA.34.5080. PMID 9897894.