en>Mz7: Reverted edits by 132.3.65.82 (talk) to last revision by Deville (HG)

2014-03-01T00:18:01Z

Reverted edits by 132.3.65.82 (talk) to last revision by Deville (HG)

en>Hgilbert: rev. vandalism

2013-10-30T01:22:56Z

rev. vandalism

← Older revision		Revision as of 03:22, 30 October 2013
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	~~Hi there~~, ~~I am Andrew Berryhill. Ohio~~ is ~~exactly~~ where ~~her home~~ is. ~~Invoicing~~ is ~~my occupation~~. ~~One~~ of the ~~issues she loves most~~ is ~~canoeing and she's been doing it~~ for ~~quite a whilst~~.<br><br>~~Feel free~~ to ~~visit my web~~ page ~~psychics~~ online - [http://~~mydatefinder~~.eu/~~groups~~/~~excellent~~-~~advice~~-~~for~~-~~selecting~~-~~the~~-~~ideal~~-~~hobby/?ac~~=~~21677~~/ ~~mydatefinder~~.~~eu linked web-site~~],		In [[mathematics]], '''variational perturbation theory''' (VPT) is a mathematical method to convert divergent [[power series]] in a small expansion parameter, say
			:<math>s=\sum_{n=0}^\infty a_n g^n</math>,
			into a [[convergent series]] in powers
			:<math>s=\sum_{n=0}^\infty b_n /(g^\omega)^n</math>,
			where <math>\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 method (quantum mechanics)\|variational parameters]], which are determined by optimization order by order in <math>g</math>. The partial sums are converted to convergent partial sums by a method developed in 1992.<ref>
			{{cite journal
			\|last1=Hagen \|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 \|page=332–342
			\|bibcode=1993PhLA..173..332K
			\|doi=10.1016/0375-9601(93)90246-V
			}}</ref>

			Most perturbation expansions in [[quantum mechanics]] are divergent for any small coupling strength <math>g</math>. They can be made convergent by VPT (for details see the first textbook cited below). The convergence is exponentially fast.<ref>
			{{cite journal
			\|last1=Hagen \|first1=H. \|author-link=Hagen Kleinert
			\|last2=Janke \|first2=W.
			\|year=1993
			\|title=Convergence Behavior of Variatioal 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= \|page=283–289
			\|arxiv=quant-ph/9509005
			\|bibcode=1995PhLA..206..283K
			\|doi=10.1016/0375-9601(95)00521-4
			}}</ref><ref>
			{{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 \|page=109–145
			\|arxiv=hep-th/9505084
			\|bibcode=1996AnPhy.249..109G
			\|doi=10.1006/aphy.1996.0066
			}}</ref>

			After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in quantum field theory with its [[Anomalous scaling dimension\|anomalous dimensions]].<ref>
			{{cite journal
			\|last1=Hagen \|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
			}}</ref> Applications focus on the theory of [[critical phenomena]]. It has led to the most accurate predictions of [[critical exponent]]s.
			More details can be read [http://users.physik.fu-berlin.de/~kleinert/b8/crit.htm here].

			== References ==
			{{Reflist}}

			== References ==
			* [[Hagen Kleinert\|Kleinert H.]], ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 3. Auflage, [http://www.worldscibooks.com/physics/5057.html World Scientific (Singapore, 2004)] (readable online [http://www.physik.fu-berlin.de/~kleinert/b5 here]) (see Chapter 15)
			* [[Hagen Kleinert\|Kleinert H.]] and Verena Schulte-Frohlinde, ''Critical Properties of φ<sup>4</sup>-Theories'', [http://www.worldscibooks.com/physics/4733.html World Scientific (Singapur, 2001)]; Paperback ISBN 981-02-4658-7 (readable online [http://www.physik.fu-berlin.de/~kleinert/b8 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]]
			\|volume=34 \|issue=6 \|pages=5080–5084
			\|bibcode=1986PhRvA..34.5080F
			\|doi=10.1103/PhysRevA.34.5080
			\|pmid=9897894
			}}

			[[Category:Asymptotic analysis]]
			[[Category:Perturbation theory]]

en>Mgiganteus1: /* Technical Drawing */ lowercase

2012-07-28T21:05:18Z

Technical Drawing: lowercase

New page

Hi there, I am Andrew Berryhill. Ohio is exactly where her home is. Invoicing is my occupation. One of the issues she loves most is canoeing and she's been doing it for quite a whilst.<br><br>Feel free to visit my web page psychics online - [http://mydatefinder.eu/groups/excellent-advice-for-selecting-the-ideal-hobby/?ac=21677/ mydatefinder.eu linked web-site],

Oval - Revision history

en>Mz7: Reverted edits by 132.3.65.82 (talk) to last revision by Deville (HG)

en>Hgilbert: rev. vandalism

en>Mgiganteus1: /* Technical Drawing */ lowercase