共形對稱

共形群有15個自由度：

共形群

时空共形群有下面的表示：^[1]

{\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}}

龐加萊群： $M_{\mu \nu }$ 是勞侖茲群的生成集合、 $P_{\mu }$ 生成平移

$K_{\mu }$ 生成特殊形转换

{\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}}

舉例（特殊共形變換）：^[2]

x^{\mu }\to {\frac {x^{\mu }-a^{\mu }x^{2}}{1-2a\cdot x+a^{2}x^{2}}}

{\frac {{x}'^{\mu }}{{x'}^{2}}}={\frac {x^{\mu }}{x^{2}}}-a^{\mu },

[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）.