Conformal Symmetry - Generators and Commutation Relations

Generators and Commutation Relations

The conformal group has the following representation:

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

where 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:

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

other commutators vanish.

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

The special conformal transformations are given by

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

where is a parameter describing the transformation. This special conformal transformation can also be written as, 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.

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.

Read more about this topic: Conformal Symmetry

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