Conformal symmetry

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In theoretical physics, conformal symmetry is a symmetry under dilatation (scale invariance) and under the special conformal transformations. Together with the Poincaré group these generate the conformal symmetry group.

The conformal group has the following representation in spacetime:

M_{\mu\nu}\equiv-i(x_\mu\partial_\nu-x_\nu\partial_\mu),
P_\mu\equiv-i\partial_\mu,
D\equiv-x_\mu\partial^\mu,
K_\mu\equiv-{i\over2}(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu),

where Mμν are the Lorentz generators, Pμ generates translations, D generates scaling transformations (also known as dilatations or dilations) and Kμ generates the special conformal transformations.

The commutation relations, in addition to those of the Poincaré group, are as follows:

[D,D] = 0 , [D,Kμ] = − Kμ,
[D,Pμ] = Pμ , [Kμ,Kν] = 0,
[Kμ,Pν] = ημνD − iMμν,

Additionally, D is a scalar and Kμ is a covariant vector under the Lorentz transformations.

Proper conformal transformations are given by

x^\mu \to \frac{x^\mu+a^\mu x^2}{1+ 2a\cdot x + a^2 x^2}

where aμ is a parameter describing the transformation.

A coordinate grid prior to a proper conformal transformation
A coordinate grid prior to a proper conformal transformation
The same grid after a proper conformal transformation
The same grid after a proper conformal transformation

In two dimensional spacetime, the transformations of the conformal group are the conformal transformations.

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.

[edit] Uses

The largest possible 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.

One particular application is to critical phenomena (phase transitions of the second order) in systems with local interactions. The 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 discovered in two-dimensional turbulence at high Reynolds number.

Several spaces and theories in high-energy physics admit the conformal symmetry:

[edit] See also

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