Special conformal transformation

In mathematical physics, a special conformal transformation is a type of spherical wave transformation and an expression of conformal symmetry.

Special conformal transformations arise from translation of spacetime and inversion. The inversion can be taken to be multiplicative inversion of biquaternions B. The complex algebra B can be extended to P(B) through the projective line over a ring. Homographies on P(B) include translations:

$U(q,1){\begin{pmatrix}1&0\\t&1\end{pmatrix}}=U(q+t,1).$

The homography group G(B) includes

${\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}1&0\\t&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}1&t\\0&1\end{pmatrix}},$

which provides the action of a special conformal transformation.

A coordinate grid prior to a special conformal transformation

The same grid after a special conformal transformation

Vector presentation

A special conformal transformation can also be written

$x'^{\mu }={\frac {x^{\mu }-b^{\mu }x^{2}}{1-2b\cdot x+b^{2}x^{2}}}\,.$

It is a composition of an inversion (x^μ → x^μ/x²), a translation (x^μ → x^μ − b^μ), and an inversion:

${\frac {x'^{\mu }}{x'^{2}}}={\frac {x^{\mu }}{x^{2}}}-b^{\mu }\,.$

Its infinitesimal generator is

$K_{\mu }=-i(2x_{\mu }x^{\nu }\partial _{\nu }-x^{2}\partial _{\mu })\,.$