variational perturbation theory

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

:$s=\backslash sum\_\{n=0\}^\backslash infty\; a\_n\; g^n$,

into a convergent series in powers

:$s=\backslash sum\_\{n=0\}^\backslash infty\; b\_n\; /(g^\backslash omega)^n$,

where $\backslash omega$ 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.

{{cite journal

|last1=Kleinert |first1=H. |author-link=Hagen Kleinert

|year=1995

|title=Systematic Corrections to Variational Calculation of Effective Classical Potential

|url=http://users.physik.fu-berlin.de/~kleinert/213/213.pdf

|journal=Physics Letters A

|volume=173 |issue=4–5 |pages=332–342

|bibcode=1993PhLA..173..332K

|doi=10.1016/0375-9601(93)90246-V

}}

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.

{{cite journal

|last1=Kleinert |first1=H. |author-link=Hagen Kleinert

|last2=Janke |first2=W.

|year=1993

|title=Convergence Behavior of Variational Perturbation Expansion - A Method for Locating Bender-Wu Singularities

|url=http://users.physik.fu-berlin.de/~kleinert/235/235.pdf

|journal=Physics Letters A

|volume=206 |issue=5–6 |pages=283–289

|arxiv=quant-ph/9509005

|bibcode=1995PhLA..206..283K

|doi=10.1016/0375-9601(95)00521-4

}}

{{cite journal

|last1=Guida |first1=R.

|last2=Konishi |first2=K.

|last3=Suzuki |first3=H.

|year=1996

|title=Systematic Corrections to Variational Calculation of Effective Classical Potential

|journal=Annals of Physics

|volume=249 |issue=1 |pages=109–145

|arxiv=hep-th/9505084

|bibcode=1996AnPhy.249..109G

|doi=10.1006/aphy.1996.0066

}}

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.

{{cite journal

|last1=Kleinert |first1=H. |author-link=Hagen Kleinert

|year=1998

|title=Strong-coupling behavior of φ^4 theories and critical exponents

|url=http://users.physik.fu-berlin.de/~kleinert/257/257.pdf

|journal=Physical Review D

|volume=57 |issue=4 |page=2264

|bibcode=1998PhRvD..57.2264K

|doi=10.1103/PhysRevD.57.2264

}} Applications focus on the theory of critical phenomena. It has led to the most accurate predictions of critical exponents.

More details can be read [http://users.physik.fu-berlin.de/~kleinert/b8/crit.htm here].