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advanced_tools:group_theory:conformal_group

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

- 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 $$ d\tau \to \Omega(x) d\tau , $$ where $\Omega(x) = \mathrm{e}^{i \omega(x)}$ is a 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{eqnarray} {\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{eqnarray} 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.

**Properties**

- "the conformal group is non-compact and semisimple" Source
- for the Casimir Operators see http://link.springer.com/article/10.1007%2FBF02727449
- 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 spacelike 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$ |

chiral | $j_Lj_R\neq 0 $ | $d>j_L+j_R+2$ |

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

See also:Irreducible unitary representations of $SU(2, 2)$ by A.W Knapp, B Speh and section 4.3 in http://dias.kb.dk/downloads/dias:67?locale=da

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]).

"*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

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 asKac-Moody groups.Not Even Wrong by P. Woit

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