Editing Conformal symmetry (section)

==Commutation relations==

The [[Commutator|commutation]] relations are as follows:{{sfn|Di Francesco|Mathieu|Sénéchal|1997|p=98}}

: <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>
other commutators vanish. Here <math>\eta_{\mu\nu}</math> is the [[Minkowski metric]] tensor.

Additionally, <math>D</math> is a scalar and <math>K_\mu</math> is a covariant vector under the [[Lorentz transformation]]s.

The special  conformal transformations are given by{{sfn|Di Francesco|Mathieu|Sénéchal|1997|p=97}}
:<math>
   x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2}
</math>
where <math>a^{\mu}</math> is a parameter describing the transformation.  This special conformal transformation can also be written as <math> x^\mu  \to x'^\mu </math>, where
:<math>
\frac{{x}'^\mu}{{x'}^2}= \frac{x^\mu}{x^2} - a^\mu,
</math> 
which shows that it consists of an inversion, followed by a translation, followed by a second  inversion.

[[Image:Conformal grid before Möbius transformation.svg|frame|A coordinate grid prior to a special conformal transformation]]
[[Image:Conformal grid after Möbius transformation.svg|frame|The same grid after a special conformal transformation]]

In two-dimensional [[spacetime]], the transformations of the conformal group are the [[conformal geometry|conformal transformations]]. There are [[Conformal field theory#Two dimensions|infinitely many]] of them.

In more than two dimensions, [[Euclidean space|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 [[Minkowski space|Lorentzian dimension]]s, conformal transformations map null rays to null rays and [[Light cone|light cones]] to light cones, with a null [[hyperplane]] being a degenerate light cone.