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 con ...
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 ...
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 ...
(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 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. 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 ...
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 I ...
, 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