在数学物理和共形场论中，时空的共形对称包括时空的庞加莱群。 共形群有15个自由度： 庞加莱群：10 特殊共形变换：4 位似变换：1 时空共形群有下面的表示：[1] M μ ν ≡ i ( x μ ∂ ν − x ν ∂ μ ) , P μ ≡ − i ∂ μ , D ≡ − i x μ ∂ μ , K μ ≡ i ( x 2 ∂ μ − 2 x μ x ν ∂ ν ) , {\displaystyle {\begin{aligned}&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{aligned}}} 庞加莱群： M μ ν {\displaystyle M_{\mu \nu }} 是劳仑兹群的生成集合、 P μ {\displaystyle P_{\mu }} 生成平移 D {\displaystyle D} 生成位似变换 K μ {\displaystyle K_{\mu }} 生成特殊形转换 交换子是：[1] [ D , K μ ] = − i K μ , [ D , P μ ] = i P μ , [ K μ , P ν ] = 2 i ( η μ ν D − M μ ν ) , [ K μ , M ν ρ ] = i ( η μ ν K ρ − η μ ρ K ν ) , [ P ρ , M μ ν ] = i ( η ρ μ P ν − η ρ ν P μ ) , [ M μ ν , M ρ σ ] = i ( η ν ρ M μ σ + η μ σ M ν ρ − η μ ρ M ν σ − η ν σ M μ ρ ) , {\displaystyle {\begin{aligned}&[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{aligned}}} 举例（特殊共形变换）：[2] x μ → x μ − a μ x 2 1 − 2 a ⋅ x + a 2 x 2 {\displaystyle x^{\mu }\to {\frac {x^{\mu }-a^{\mu }x^{2}}{1-2a\cdot x+a^{2}x^{2}}}} x ′ μ x ′ 2 = x μ x 2 − a μ , {\displaystyle {\frac {{x}'^{\mu }}{{x'}^{2}}}={\frac {x^{\mu }}{x^{2}}}-a^{\mu },} 平面格子 特殊共形变换后 共形场论[3]、场 (物理) 普遍性、相变 二维湍流、雷诺数 粒子物理学、N=4超对称杨-米尔斯的理论、世界面、弦理论 共形映射 共形群（英语：Conformal group）（推广） 重整化群 [1]Di Francesco; Mathieu, Sénéchal. Conformal field theory. Graduate texts in contemporary physics. Springer. 1997: 98. ISBN 978-0-387-94785-3. [2]Di Francesco; Mathieu, Sénéchal. Conformal field theory. Graduate texts in contemporary physics. Springer. 1997: 97. ISBN 978-0-387-94785-3. [3]Juan Maldacena; Alexander Zhiboedov. Constraining conformal field theories with a higher spin symmetry. Journal of Physics A: Mathematical and Theoretical. 2013, 46 (21): 214011 [2020-03-07]. Bibcode:2013JPhA...46u4011M. arXiv:1112.1016 . doi:10.1088/1751-8113/46/21/214011. （原始内容存档于2014-02-02）. Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for EdgeWikiwand for Firefox
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是劳仑兹群的生成集合、
生成平移
生成位似变换
生成特殊形转换
![{\displaystyle {\begin{aligned}&[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{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5c1cebad87b5e53e8458678669adeec3e825296d)
平面格子
特殊共形变换后
![{\displaystyle {\begin{aligned}&[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{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5c1cebad87b5e53e8458678669adeec3e825296d)
