查看“︁共形對稱”︁的源代码

在[[数学物理]]和[[共形場論]]中，[[时空]]的'''共形对称'''包括时空的[[龐加萊群]]。 

共形群有15個[[自由度 (物理学)|自由度]]：

* [[龐加萊群]]：10
* 特殊共形變換：4
* [[位似变换]]：1

== 共形群 ==
时空[[共形群]]有下面的[[群表示論|表示]]：<ref name="difrancesco">{{Cite book|last=Di Francesco|last2=Mathieu, Sénéchal|title=Conformal field theory|url=https://archive.org/details/conformalfieldth00sene|series=Graduate texts in contemporary physics|year=1997|publisher=Springer|isbn=978-0-387-94785-3|page=[https://archive.org/details/conformalfieldth00sene/page/n115 98]}}</ref>

: <math>\begin{align} & M_{\mu\nu} \equiv i(x_\mu\partial_\nu-x_\nu\partial_\mu) \,, \\
&P_\mu \equiv-i\partial_\mu \,, \\
&D \equiv-ix_\mu\partial^\mu \,, \\
&K_\mu \equiv i(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu) \,, \end{align}</math>

[[龐加萊群]]：<math>M_{\mu\nu}</math>是[[勞侖茲群]]的[[群的生成集合|生成集合]]、<math>P_\mu</math> 生成[[平移]]

<math>D</math> 生成[[位似变换]]

<math>K_\mu</math> 生成特[[殊形转换]]

== 交換子 ==
[[交換子]]是：<ref name="difrancesco">{{Cite book|last=Di Francesco|last2=Mathieu, Sénéchal|title=Conformal field theory|url=https://archive.org/details/conformalfieldth00sene|series=Graduate texts in contemporary physics|year=1997|publisher=Springer|isbn=978-0-387-94785-3|page=[https://archive.org/details/conformalfieldth00sene/page/n115 98]}}</ref>

: <math>\begin{align} &[D,K_\mu]= -iK_\mu \,, \\
&[D,P_\mu]= iP_\mu \,, \\
&[K_\mu,P_\nu]=2i (\eta_{\mu\nu}D-M_{\mu\nu}) \,, \\
&[K_\mu, M_{\nu\rho}] = i ( \eta_{\mu\nu} K_{\rho} - \eta_{\mu \rho} K_\nu ) \,, \\
&[P_\rho,M_{\mu\nu}] = i(\eta_{\rho\mu}P_\nu - \eta_{\rho\nu}P_\mu) \,, \\
&[M_{\mu\nu},M_{\rho\sigma}] = i (\eta_{\nu\rho}M_{\mu\sigma} + \eta_{\mu\sigma}M_{\nu\rho} - \eta_{\mu\rho}M_{\nu\sigma} - \eta_{\nu\sigma}M_{\mu\rho})\,, \end{align}</math>

舉例（特殊共形變換）：<ref>{{Cite book|last=Di Francesco|last2=Mathieu, Sénéchal|title=Conformal field theory|url=https://archive.org/details/conformalfieldth00sene|series=Graduate texts in contemporary physics|year=1997|publisher=Springer|isbn=978-0-387-94785-3|page=[https://archive.org/details/conformalfieldth00sene/page/n114 97]}}</ref>

: <math>
  x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2}
</math>

: <math>
\frac{{x}'^\mu}{{x'}^2}= \frac{x^\mu}{x^2} - a^\mu,
</math>

[[File:Conformal_grid_before_Möbius_transformation.svg|有框|平面[[格子]]]]
[[File:Conformal_grid_after_Möbius_transformation.svg|有框|特殊共形變換後]]

== 应用 ==

* [[共形場論]]<ref>{{Cite journal|title=Constraining conformal field theories with a higher spin symmetry|url=http://inspirehep.net/search?p=recid:1079967&of=hd|last=Juan Maldacena|last2=Alexander Zhiboedov|date=2013|journal=Journal of Physics A: Mathematical and Theoretical|issue=21|doi=10.1088/1751-8113/46/21/214011|volume=46|pages=214011|arxiv=1112.1016|bibcode=2013JPhA...46u4011M|access-date=2020-03-07|archive-date=2014-02-02|archive-url=https://web.archive.org/web/20140202092444/http://inspirehep.net/search?p=recid:1079967&of=hd|dead-url=no}}</ref>、[[场 (物理)]]
* [[普遍性 (物理学)|普遍性]]、[[相變]]
* 二维[[湍流]]、[[雷诺数]]
* [[粒子物理學]]、[[超對稱楊-米爾斯理論|N=4超对称杨-米尔斯的理论]]、[[世界面]]、[[弦理論|弦理论]]

== 相關條目 ==

* [[共形映射]]
* {{le|共形群|Conformal group}}（推广）
* [[重整化群]]

== 参考文献 ==
{{Reflist}}
[[Category:共形場論]]
[[Category:對稱]]