Variational perturbation theory explained

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

s = \sum ^{inf ty} _{n =0} a _{n} g ^{n} ,into a convergent series in powerss = \sum ^{inf ty} _{n =0} b _{n} /(g ^{\omega)} ^{n} ,where \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 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

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.

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 (readable online here) (see Chapter 19)
• Feynman. R. P.. Richard P. Feynman . Kleinert . H. . Hagen Kleinert . 1986 . Effective classical partition functions . . 34 . 6 . 5080–5084 . 1986PhRvA..34.5080F . 10.1103/PhysRevA.34.5080 . 9897894.

Notes and References

1. Kleinert . H. . Hagen Kleinert . 1995 . Systematic Corrections to Variational Calculation of Effective Classical Potential . . 173 . 4–5 . 332–342 . 1993PhLA..173..332K . 10.1016/0375-9601(93)90246-V.
2. Kleinert . H. . Hagen Kleinert . Janke . W. . 1993 . Convergence Behavior of Variational Perturbation Expansion - A Method for Locating Bender-Wu Singularities . . 206 . 283–289 . quant-ph/9509005 . 1995PhLA..206..283K . 10.1016/0375-9601(95)00521-4.
3. Guida . R. . Konishi . K. . Suzuki . H. . 1996 . Systematic Corrections to Variational Calculation of Effective Classical Potential . . 249 . 1 . 109–145 . hep-th/9505084 . 1996AnPhy.249..109G . 10.1006/aphy.1996.0066.
4. Kleinert . H. . Hagen Kleinert . 1998 . Strong-coupling behavior of φ^4 theories and critical exponents . . 57 . 4 . 2264 . 1998PhRvD..57.2264K . 10.1103/PhysRevD.57.2264.