In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, variational perturbation theory (VPT) is a mathematical method to convert divergent
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
in a small expansion parameter, say
:
,
into a
convergent series in powers
:
,
where
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
. The partial sums are converted to convergent partial sums by a method developed in 1992.
Most
perturbation expansions in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
are divergent for any small coupling strength
. They can be made convergent by VPT (for details see the first textbook cited below). The convergence is exponentially fast.
After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
with its
anomalous dimensions.
[
] Applications focus on the theory of
critical phenomena. It has led to the most accurate predictions of
critical exponents.
More details can be rea
here
References
External links
*
Kleinert H., ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 3. Auflage
World Scientific (Singapore, 2004)(readable onlin
here (see Chapter 5)
*
Kleinert H. and Verena Schulte-Frohlinde, ''Critical Properties of φ
4-Theories''
World Scientific (Singapur, 2001) Paperback (readable onlin
here (see Chapter 19)
* {{cite journal
, last1=Feynman, first1=R. P., author1-link=Richard P. Feynman
, last2=Kleinert , first2=H. , author2-link=Hagen Kleinert
, year=1986
, title=Effective classical partition functions
, journal=
Physical Review A
''Physical Review A'' (also known as PRA) is a monthly peer-reviewed scientific journal published by the American Physical Society covering atomic, molecular, and optical physics and quantum information. the editor was Jan M. Rost ( Max Planck ...
, volume=34 , issue=6 , pages=5080–5084
, bibcode=1986PhRvA..34.5080F
, doi=10.1103/PhysRevA.34.5080
, pmid=9897894, url=https://authors.library.caltech.edu/3553/1/FEYpra86.pdf
Asymptotic analysis
Perturbation theory