Conformal symmetry

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:^[1]

$\begin{align} & M_{\mu\nu} \equiv i(x_\mu\partial_\nu-x_\nu\partial_\mu) \,, \\ &P_\mu \equiv-i\partial_\mu \,, \\ &D \equiv-ix_\mu\partial^\mu \,, \\ &K_\mu \equiv i(x^2\partial_\mu-2x_\mu x_\nu\partial^\nu) \,, \end{align}$

where M_μν are the Lorentz generators, P_μ generates translations, D generates scaling transformations (also known as dilatations or dilations) and K_μ generates the special conformal transformations.

The commutation relations are as follows:^[1]

$\begin{align} &[D,K_\mu]=-iK_\mu \,, \\ &[D,P_\mu]=iP_\mu \,, \\ &[K_\mu,P_\nu]=2i\eta_{\mu\nu}D-2iM_{\mu\nu} \,, \\ &[K_\mu, M_{\nu\rho}] = i ( \eta_{\mu\nu} K_{\rho} - \eta_{\mu \rho} K_\nu ) \,, \\ &[P_\rho,M_{\mu\nu}] = i(\eta_{\rho\mu}P_\nu - \eta_{\rho\nu}P_\mu) \,, \\ &[M_{\mu\nu},M_{\rho\sigma}] = i (\eta_{\nu\rho}M_{\mu\sigma} + \eta_{\mu\sigma}M_{\nu\rho} - \eta_{\mu\rho}M_{\nu\sigma} - \eta_{\nu\sigma}M_{\mu\rho})\,, \end{align}$

other commutators vanish.

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

The special conformal transformations are given by^[2]

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

where a^μ is a parameter describing the transformation. This special conformal transformation can also be written as $x^\mu \to x'^\mu$, where

$\frac{{X}'^\mu}{{X'}^2}= \frac{x^\mu}{x^2} - a^\mu,$

which shows that it consists of an inversion, followed by a translation, followed by a second inversion.

A coordinate grid prior to a special conformal transformation

The same grid after a special 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.

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:

• N = 4 supersymmetric Yang-Mills.
• The theory over the worldsheet in string theory.

See also

• Coleman–Mandula theorem
• Renormalization group
• Scale invariance
• Superconformal algebra
• Harry Bateman
• Ebenezer Cunningham

References

1. ^ ^{a b} Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. p. 98. ISBN 9780387947853. 
2. ^ Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. p. 97. ISBN 9780387947853.