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Concept
Conformal symmetry
Summary
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group, known as the conformal 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. General relativity in two spacetime dimensions also enjoys conformal symmetry.
The Lie algebra of the conformal group has the following representation:
where are the Lorentz generators, generates translations, generates scaling transformations (also known as dilatations or dilations) and generates the special conformal transformations.
The commutation relations are as follows:
other commutators vanish. Here is the Minkowski metric tensor.
Additionally, is a scalar and is a covariant vector under the Lorentz transformations.
The special conformal transformations are given by
where is a parameter describing the transformation. This special conformal transformation can also be written as , where
which shows that it consists of an inversion, followed by a translation, followed by a second inversion.
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 field theory
In relativistic quantum field theories, the possibility of symmetries is strictly restricted by Coleman–Mandula theorem under physically reasonable assumptions.
Official source
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