Variational perturbation theory
In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say
- [math]\displaystyle{ s=\sum_{n=0}^\infty a_n g^n }[/math],
into a convergent series in powers
- [math]\displaystyle{ s=\sum_{n=0}^\infty b_n /(g^\omega)^n }[/math],
where [math]\displaystyle{ \omega }[/math] 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 [math]\displaystyle{ g }[/math]. 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 [math]\displaystyle{ g }[/math]. 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
- ↑ "Systematic Corrections to Variational Calculation of Effective Classical Potential". Physics Letters A 173 (4–5): 332–342. 1995. doi:10.1016/0375-9601(93)90246-V. Bibcode: 1993PhLA..173..332K. http://users.physik.fu-berlin.de/~kleinert/213/213.pdf.
- ↑ "Convergence Behavior of Variational Perturbation Expansion - A Method for Locating Bender-Wu Singularities". Physics Letters A 206: 283–289. 1993. doi:10.1016/0375-9601(95)00521-4. Bibcode: 1995PhLA..206..283K. http://users.physik.fu-berlin.de/~kleinert/235/235.pdf.
- ↑ Guida, R.; Konishi, K.; Suzuki, H. (1996). "Systematic Corrections to Variational Calculation of Effective Classical Potential". Annals of Physics 249 (1): 109–145. doi:10.1006/aphy.1996.0066. Bibcode: 1996AnPhy.249..109G.
- ↑ "Strong-coupling behavior of φ^4 theories and critical exponents". Physical Review D 57 (4): 2264. 1998. doi:10.1103/PhysRevD.57.2264. Bibcode: 1998PhRvD..57.2264K. http://users.physik.fu-berlin.de/~kleinert/257/257.pdf.
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)
- "Effective classical partition functions". Physical Review A 34 (6): 5080–5084. 1986. doi:10.1103/PhysRevA.34.5080. PMID 9897894. Bibcode: 1986PhRvA..34.5080F. https://authors.library.caltech.edu/3553/1/FEYpra86.pdf.
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