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.
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 conformal group has the following representation:
where are the Lorentz generators, generates translations, generates scaling transformations and generates the special conformal transformations.

Commutation relations

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.

Applications

Conformal field theory

The largest possible global symmetry group of a non-supersymmetric interacting field theory is a direct product of the conformal group with an internal group. Such theories are known as conformal field theories.

Second-order phase transitions

One particular application is to critical phenomena in systems with local interactions. Fluctuations in such systems are conformally invariant at the critical point. That allows for classification of universality classes of phase transitions in terms of conformal field theories
Conformal invariance is also present in two-dimensional turbulence at high Reynolds number.

High-energy physics

Many theories studied in high-energy physics admit the conformal symmetry. A famous example is the N=4 supersymmetric Yang-Mills theory. Also, the worldsheet in string theory is described by a two-dimensional conformal field theory coupled to the two-dimensional gravity.