Why is it interesting?

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

Layman

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

Student

\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.

• 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

Researcher

• "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."

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

Examples

Example1

Example2:

FAQ

History