In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
in a small expansion parameter, say
:
,
into a
convergent series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted
:S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k.
The th partial sum ...
in powers
:
,
where
is a
critical exponent
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
*Critical Software, a company specializing in ...
(the so-called index of "approach to scaling" introduced by
Franz Wegner
Franz Joachim Wegner (born 15 June 1940) is emeritus professor for theoretical physics at the University of Heidelberg.
Education
Franz Wegner attained a doctorate in 1968 with thesis advisor Wilhelm Brenig at the Technical University Munich with ...
). 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 a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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 classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
with its
anomalous dimensions.
[
] Applications focus on the theory of
critical phenomena
In physics, critical phenomena is the collective name associated with the
physics of critical points. Most of them stem from the divergence of the
correlation length, but also the dynamics slows down. Critical phenomena include scaling relations ...
. It has led to the most accurate predictions of
critical exponent
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
*Critical Software, a company specializing in ...
s.
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
, last=Feynman, first=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 Ins ...
, 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