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

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.

Researcher

Examples

Example1

Example2:

FAQ

History