Conformal symmetry

In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes 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. In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes 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. The conformal group has the following representation: where M μ ν {displaystyle M_{mu u }} are the Lorentz generators, P μ {displaystyle P_{mu }} generates translations, D {displaystyle D} generates scaling transformations (also known as dilatations or dilations) and K μ {displaystyle K_{mu }} generates the special conformal transformations. The commutation relations are as follows: other commutators vanish. Here η μ ν {displaystyle eta _{mu u }} is the Minkowski metric tensor. Additionally, D {displaystyle D} is a scalar and K μ {displaystyle K_{mu }} is a covariant vector under the Lorentz transformations. The special conformal transformations are given by where a μ {displaystyle a^{mu }} is a parameter describing the transformation. This special conformal transformation can also be written as x μ → x ′ μ {displaystyle x^{mu } o x'^{mu }} , where