Conformal symmetry

The conformal group has the following representation:[2]

${\displaystyle {\begin{aligned}&M_{\mu \nu }\equiv i(x_{\mu }\partial _{\nu }-x_{\nu }\partial _{\mu })\,,\\&P_{\mu }\equiv -i\partial _{\mu }\,,\\&D\equiv -ix_{\mu }\partial ^{\mu }\,,\\&K_{\mu }\equiv i(x^{2}\partial _{\mu }-2x_{\mu }x_{\nu }\partial ^{\nu })\,,\end{aligned}}}$

where ${\displaystyle M_{\mu \nu }}$ are the Lorentz generators, ${\displaystyle P_{\mu }}$ generates translations, ${\displaystyle D}$ generates scaling transformations (also known as dilatations or dilations) and ${\displaystyle K_{\mu }}$ generates the special conformal transformations.

The commutation relations are as follows:[2]

${\displaystyle {\begin{aligned}&[D,K_{\mu }]=-iK_{\mu }\,,\\&[D,P_{\mu }]=iP_{\mu }\,,\\&[K_{\mu },P_{\nu }]=2i(\eta _{\mu \nu }D-M_{\mu \nu })\,,\\&[K_{\mu },M_{\nu \rho }]=i(\eta _{\mu \nu }K_{\rho }-\eta _{\mu \rho }K_{\nu })\,,\\&[P_{\rho },M_{\mu \nu }]=i(\eta _{\rho \mu }P_{\nu }-\eta _{\rho \nu }P_{\mu })\,,\\&[M_{\mu \nu },M_{\rho \sigma }]=i(\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\nu \sigma }M_{\mu \rho })\,,\end{aligned}}}$

other commutators vanish. Here ${\displaystyle \eta _{\mu \nu }}$ is the Minkowski metric tensor.

Additionally, ${\displaystyle D}$ is a scalar and ${\displaystyle K_{\mu }}$ is a covariant vector under the Lorentz transformations.

The special conformal transformations are given by[3]

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

where ${\displaystyle a^{\mu }}$ is a parameter describing the transformation. This special conformal transformation can also be written as ${\displaystyle x^{\mu }\to x'^{\mu }}$, where

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

which shows that it consists of an inversion, followed by a translation, followed by a second inversion.

A coordinate grid prior to a special conformal transformation

The same grid after a special conformal transformation

In two dimensional spacetime, the transformations of the conformal group are the conformal transformations. There are infinitely many of them.

In more than two dimensions, Euclidean conformal transformations map circles to circles, and hyperspheres to hyperspheres with a straight line considered a degenerate circle and a hyperplane a degenerate hypercircle.

In more than two Lorentzian dimensions, conformal transformations map null rays to null rays and light cones to light cones with a null hyperplane being a degenerate light cone.

• Conformal map
• Conformal group
• Coleman–Mandula theorem
• Renormalization group
• Scale invariance
• Superconformal algebra
• Conformal Killing equation