✎ Conformal Group [The Physics Travel Guide]

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====== Conformal Group ======

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox 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 [[advanced_tools:group_theory:poincare_group|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. 
 
<tabbox Researcher>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

--> Example1#

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--> Example2:#

<--

</tabbox>

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