查看“︁維度正規化”︁的源代码

在[[量子場論]]中，'''維度正規化'''是一種[[正規化]]辦法。Giambiagi、Bollini、<ref>Bollini 1972, p. 20.</ref><ref name="physicstoday">{{Cite journal|title=Revolutionary physics in reactionary Argentina|url=https://archive.org/details/sim_physics-today_2014-02_67_2/page/38|last=Bietenholz|first=Wolfgang|last2=Prado|first2=Lilian|date=2014-02-01|journal=Physics Today|issue=2|doi=10.1063/PT.3.2277|volume=67|pages=38–43|bibcode=2014PhT....67b..38B|issn=0031-9228}}</ref> [[杰拉德·特·胡夫特]]和[[马丁纽斯·韦尔特曼]]<ref>{{Citation|last=Hooft|first=G. 't|last2=Veltman|first2=M.|title=Regularization and renormalization of gauge fields|doi=10.1016/0550-3213(72)90279-9|year=1972|journal=Nuclear Physics B|issn=0550-3213|volume=44|number=1|pages=189–213|bibcode=1972NuPhB..44..189T}}</ref>都提出了這個辦法。物理學家使用維度正規化來計算[[费曼图]]的積分。积分的值是d的[[亚纯函数|亞純函數]]；d是時空的維度。

== 其他應用 ==
{{en-link|Wilson–Fisher固定点|Wilson–Fisher fixed point}}、[[分形维数]]<ref>{{Cite journal|title=Accurate critical exponents for Ising-like systems in non-integer dimensions|url=https://hal.archives-ouvertes.fr/jpa-00210418/document|last=Le Guillo|first=J.C.|last2=Zinn-Justin|first2=J.|journal=Journal de Physique|year=1987|volume=48|access-date=2020-03-07|archive-date=2020-07-10|archive-url=https://web.archive.org/web/20200710093101/https://hal.archives-ouvertes.fr/jpa-00210418/document|dead-url=no}}</ref>

== 舉例 ==

: <math>\int\frac{d^dp}{(2\pi)^d}\frac{1}{\left(p^2+m^2\right)^2},</math>

若''d''&nbsp;=&nbsp;4&nbsp;&#x2212;&nbsp;''ε，''上文的積分是

: <math>\int_0^\infty \frac{dp}{(2\pi)^{4-\varepsilon}} \frac{2\pi^{(4-\varepsilon)/2}}{\Gamma\left(\frac{4-\varepsilon}{2}\right)} \frac{p^{3-\varepsilon}}{\left(p^2+m^2\right)^2}=\frac{2^{\varepsilon -4}\pi^{\frac{\varepsilon}{2}-1}}{\sin \left(\frac{\pi\varepsilon}{2}\right) \Gamma \left(1-\frac{\varepsilon}{2}\right)}m^{-\varepsilon}=\frac{1}{8\pi^2\varepsilon}-\frac{1}{16\pi^2}\left(\ln \frac{m^2}{4\pi}+\gamma\right)+ \mathcal{O}(\varepsilon).</math>

有人認為[[Ζ函數正規化]]和維度正規化是等效的等同因為[[解析延拓]]。<ref> A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini, ''Analytic Aspects of Quantum Field'', World Scientific Publishing, 2003, {{ISBN|981-238-364-6}}</ref>

== 參考文獻 ==
{{Reflist}}

== 閱讀 ==
* {{Citation|last=Bollini|first=Carlos|last2=Giambiagi|first2=Juan Jose|title=Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter.|journal=Il Nuovo Cimento B (1971-1996)|publisher=Il Nuovo Cimento B|volume=12|number=1|pages=20–26|year=1972|url=https://link.springer.com/article/10.1007/BF02895558|doi=10.1007/BF02895558|doi_brokendate=2020-01-22|accessdate=2020-03-07|archive-date=2017-10-16|archive-url=https://web.archive.org/web/20171016230059/https://link.springer.com/article/10.1007/BF02895558|dead-url=no}}
* {{Citation|last=Etingof|first=Pavel|title=Quantum fields and strings: a course for mathematicians, Vol. 1,(Princeton, NJ, 1996/1997)|chapter-url=http://www.math.ias.edu/QFT/fall/|publisher=Amer. Math. Soc.|place=Providence, R.I.|isbn=978-0-8218-2012-4|mr=1701608|year=1999|chapter=Note on dimensional regularization|pages=597–607|accessdate=2020-03-07|archive-date=2019-05-29|archive-url=https://web.archive.org/web/20190529231754/https://www.math.ias.edu/QFT/fall/|dead-url=no}}
* {{Citation|last=Hooft|first=G. 't|last2=Veltman|first2=M.|title=Regularization and renormalization of gauge fields|doi=10.1016/0550-3213(72)90279-9|year=1972|journal=Nuclear Physics B|issn=0550-3213|volume=44|number=1|pages=189–213|bibcode=1972NuPhB..44..189T}}

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