Special conformal transformation

A special conformal transformation can be written^[1]

${\displaystyle x'^{\mu }={\frac {x^{\mu }-b^{\mu }x^{2}}{1-2b\cdot x+b^{2}x^{2}}}={\frac {x^{2}}{|x-bx^{2}|^{2}}}(x^{\mu }-b^{\mu }x^{2})\,.}$

It is a composition of an inversion (x^μ → x^μ/x² = y^μ), a translation (y^μ → y^μ − b^μ = z^μ), and another inversion (z^μ → z^μ/z² = x′^μ)

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

Its infinitesimal generator is

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

Special conformal transformations have been used to study the force field of an electric charge in hyperbolic motion.^[2]

The inversion can also be taken^[3] 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:

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

The homography group G(B) includes of translations at infinity with respect to the embedding q → U(q:1);

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

The matrix describes the action of a special conformal transformation.^[4]