Conformal symmetry is a property of spacetime that ensures angles remain unchanged even when distances are altered. If you stretch, compress, or otherwise distort spacetime, the local angular relationships between lines or curves stay the same. This idea extends the familiar Poincaré group —which accounts for rotations, translations, and boosts—into the more comprehensive conformal group.
Conformal symmetry encompasses special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation.
Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations. They called a generic expression of conformal symmetry a spherical wave transformation. General relativity in two spacetime dimensions also enjoys conformal symmetry.
Generators
The Lie algebra of the conformal group has the following representation:
: <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>
where <math>M_{\mu\nu}</math> are the Lorentz generators, <math>P_\mu</math> generates translations, <math>D</math> generates scaling transformations (also known as dilatations or dilations) and <math>K_\mu</math> generates the special conformal transformations.
Commutation relations
The commutation relations are as follows:
: <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 transformations.
The special conformal transformations are given by
:<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