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 $\omega$ 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.

References

1. Hagen, 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. Hagen, H.; Janke, W. (1993). "Convergence Behavior of Variational Perturbation Expansion - A Method for Locating Bender-Wu Singularities" (PDF). Physics Letters A. 206: 283–289. arXiv:quant-ph/9509005. Bibcode:1995PhLA..206..283K. doi:10.1016/0375-9601(95)00521-4.
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. Hagen, 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.

• 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". Physical Review A. 34 (6): 5080–5084. Bibcode:1986PhRvA..34.5080F. doi:10.1103/PhysRevA.34.5080. PMID 9897894.