Variational perturbation theory

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

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

into a convergent series in powers

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

↑ 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.
↑ 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.
↑ 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.
↑ 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.

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