Conformal Group

Prerequisites: Poincare Group. Full path (9 pages)

Why is it interesting?
Intuitive
Concrete
Abstract
History

Why is it interesting?

The maximal spacetime symmetry group of massless particles is the conformal group.

It is an old idea in particle physics that, in some sense, at sufficiently high energies the masses of the elementary particles should become unimportant. In recent years this somewhat vague hope has acquired a more definite form in the theory of scale transformations, or dilatations.

Aspects of Symmetry: Selected Erice Lectures by Sidney Coleman

In particle physics, one longstanding hope has been that at high energies, particle masses can be neglected, so that the physics would become scale invariant. It turns out that in a local field theory, it is true, more or less in general, that scale invariance typically leads to conformal invariance. (This is because the violation of scale invariance and conformal invariance are both determined by the trace $T^\mu_\mu$ of the energy momentum tensor.)

page 621 Einstein Gravity in a Nutshell - A. Zee

"The simplest example of conformal matter is a perfect fluid of radiation. In the context of cosmology, this is extremely well motivated since the early Universe was, we believe, radiation dominated." https://arxiv.org/pdf/1612.02792.pdf

Intuitive

A conformal transformation is a map that may stretch or shrink lengths, by different amounts at different points, but keeps all angles fixed. A familiar example is the Mercator map of the Earth. Greenland looks far too large on it, yet the angle at which two roads cross is drawn correctly everywhere. Close to any single point, a conformal map looks like a rotation combined with a uniform zoom.

In spacetime, the conformal group collects all transformations of this kind. It contains the Poincaré transformations (translations, rotations and boosts), which keep all distances fixed. It also contains dilatations, which rescale everything by the same factor, and special conformal transformations, which rescale by a factor that changes from point to point. All of them map light cones to light cones, so they respect which events can be connected by a light signal.

Physicists care about this group because a theory without any length or mass scale cannot tell a system apart from a scaled-up copy of it. Free massless fields, such as the electromagnetic field in empty space in four dimensions, are of this type. At energies far above all particle masses a theory can come close to scale invariance, although quantum effects such as the running of coupling constants usually spoil it. Systems at the critical point of a continuous phase transition look the same at every length scale. In such situations the conformal group restricts the possible physics much more strongly than the Poincaré group alone.

"the word conformal means it preserves angles" http://fy.chalmers.se/~tfebn/YongsMScthesis.pdf

Concrete

The conformal group is $SO(4,2)$ or its double cover $SU(2,2)$ Source
For a nice discussion, see On Representations of the Conformal Group Which When Restricted to Its Poincare or Weyl Subgroups Remain Irreducible by J. Mickelsson
For the definition of the group and the algebra, see this chapter.
The conformal group is a subgroup of the diffeomorphism group. Under a conformal transformation, the metric changes as

$g_{\mu\nu}\to \Omega(x)g_{\mu\nu}$ or equivalently $d\tau \to \sqrt{\Omega(x)}\, d\tau ,$ where $\Omega(x) = \mathrm{e}^{2 \omega(x)}$ is a real, positive scalar factor.

"the conformal algebra is equivalent to SO(2, 4), the algebra of rotations and boosts in a six dimensional space with two time-like directions." http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy2001.pdf

$\begin{align} {\rm Translations}\quad (4\ {\rm param.})\quad & x^ i & \rightarrow\> \tilde{x}^ i =\, x^ i +a^ i \,,\tag{transl} \\ {\rm Lorentz\ transf.}\quad (6\ {\rm param.})\quad & x^ i & \rightarrow\>\tilde{x}^ i =\,\Lambda ^ i {}_ j \, x^ j \,,\tag{lor}\\ {\rm dila(ta)tion}\quad (1\ {\rm param.})\quad & x^ i & \rightarrow\>\tilde{x}^ i =\,\rho\, x^ i \,,\tag{dil}\\ {\rm prop.\ conf.\ transf.}\quad (4\ {\rm param.})\quad & x^ i & \rightarrow\> \tilde{x}^i =\,{\frac{x^i + \kappa ^i\,x^2}{1+ 2\kappa_j\,x^j +\kappa_j\kappa^j\,x^2}}.\tag{proper} \end{align}$ Here $a^i, \Lambda^i{}_j, \rho, \kappa^i$ are 15 constant parameters, and $x^2 := g_{ij}x^ix^j$ .

The Poincare subgroup (transl), (lor) leaves the spacetime interval $ds^2 = g_{ij}dx^idx^j$ invariant
Dilatations (dil) and proper conformal transformations (proper) change the spacetime interval by a scaling factor $ds^2 \rightarrow \rho^2 ds^2$ and $ds^2 \rightarrow \sigma^2 ds^2$ , respectively (with $\sigma^{-1} := 1+ 2\kappa_j\,x^j +\kappa_j\kappa^j\,x^2$ ).
The Weyl subgroup, is generated by transformations (transl)-(dil).

The light cone $ds^2 =0$ is left invariant by all these transformations.

A conformal field theory primer by Paul Fendley
Living Without Supersymmetry -- the Conformal Alternative and a Dynamical Higgs Boson by Philip D. Mannheim
http://physics.stackexchange.com/questions/63595/what-exactly-is-meant-by-the-conformal-group-of-minkowski-space
Very quick introduction to the conformal group and cft

Abstract

Properties

"the conformal group is non-compact and semisimple" Source
for the Casimir Operators see The conformal group, its casimir operators, and a four-position operator | Il Nuovo Cimento A (1971-1996) | Springer Nature Link
A detailed discussion of the geometrical interpretation of the conformal group can be found in Fulton et al."

Lie Algebra

Take note that $\mathfrak{su}(4)$ and $\mathfrak{su}(2,2)$ are both real forms of the same complex lie algebra $\mathfrak{sl}(4,\mathbb{C})$ . Source

See page 8 in http://dias.kb.dk/downloads/dias:67?locale=da for a good summary of the generators and the commutation relations.

Representations

In physical models, we need unitary representations to make sense of a probabilistic interpretation in a quantum field theory. The conformal group is non-compact and therefore all unitary representations are infinite dimensional (except for the trivial representation).

This can be understood, because the conformal algebra is equivalent to the $SO(2,4)$ , the algebra, which describes rotations and boosts in a six-dimensional space with two timelike directions. Therefore, as for the Poincare group it is always possible to boost the momentum label to higher and higher values.

A classification of the representations of the conformal group in a similar spirit as the classification of the representations of the Poincare group (= making use of the "little group" approach) can be found in section 3.4.1 in http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy2001.pdf.

As for the Poincare group, we focus on the maximal compact part of the group. For the conformal group this maximal compact part is $SO(2) \times SO(4) \subset SO(4,2).$ In addition, we have the isomorphisms $SO(4) \simeq SU(2) \times SU(2)$ and $SO(2) \simeq U(1)$ . Therefore, we can label the irreducible representation of the conformal group through "left" and "right" spins $(j_L,j_R)$ , which correspond to Casimir operators of the two $SU(2)$ factors, and one number, which labels the $U(1)$ representations.

"The non-compact generators act as raising and lowering operators, taking us between different states in a given representation." from p. 165 in http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy2001.pdf

All unitary irreducible representations of the conformal algebra were classified as follows

Name	$j_L,j_R$	$d$
identity	$j_L=j_R=0$	$d=0$
free chiral	$j_Lj_R=0$	$d=j_L+j_R+1$
chiral	$j_Lj_R=0$	$d>j_L+j_R+1$
free general	$j_Lj_R\neq 0$	$d=j_L+j_R+2$
general	$j_Lj_R\neq 0$	$d>j_L+j_R+2$

Source: Mack, Commun. Math. Phys. 55 (1977) 1.

The representation theory of the Poincare algebra augmented with the dilatation naturally leads to the notion of unparticles [39][40]. The theory of unparticles sometimes relies on a delicate difference between scale invariance and conformal invariance (see e.g. [41]).

https://arxiv.org/pdf/1302.0884.pdf

"there can be no infinite dimensional conformal group G for the Euclidean plane. What do physicists mean when they claim that the conformal group is infinite dimensional? The misunderstanding seems to be that physicists mostly think and calculate infinitesimally, while they write and talk globally. Many statements be- come clearer, if one replaces “group” with “Lie algebra” and “transformation” with “infinitesimal transformation” in the respective texts. […] For the Minkowski plane, there is really an infinite dimensional conformal group, as we will show in the next section. The associated complexified Lie algebra is again essentially the Witt algebra" Source

"This group doesn't act as symmetries of Minkowski spacetime, but under a (mathematically useful) completion, the "conformal compactification of Minkowski space". [The conformal group] is 15-dimensional and it's just the group $SO(2,4)$ , or if you prefer, the covering group $SU(2,2)$ !" http://math.ucr.edu/home/baez/symmetries.html

"Although the number of parameters is the same, $SO(n, 2)$ is a linear homogeneous transformation while the Poincar´e transformation is nonhomogeneous, and the special conformal transformation is non-linear and non-homogeneous, so how can they be contained in $SO(n, 2)$ ? The answer is, they are not contained in $SO(n, 2)$ , but some homogeneous linear transformations equivalent to them are." http://www.phys.nthu.edu.tw/~class/Group_theory/Chap%207.pdf

History

While the theory of representations of finite dimensional groups such as the ones Weyl studied in 1925-6 was a well-developed part of mathematics by the 1960s, little was known about the representations of infinite dimensional groups such as the group of conformal transformations in two dimensions. Without some restrictive condition on the groups and the representations to be considered, the general problem of understanding representations of infinite dimensional groups appears to be completely intractable. The mathematicians Victor Kac and Robert Moody in 1967 introduced some new algebraic structures that allowed the construction of a class of infinite dimensional groups, now known as Kac-Moody groups.

Not Even Wrong by P. Woit

Path to this page

Leads to: Conformal Field Theory.