Special conformal transformation

In projective geometry, a special conformal transformation is a linear fractional transformation that is not an affine transformation. Thus the generation of a special conformal transformation involves use of multiplicative inversion, which is the generator of linear fractional transformations that is not affine.

A coordinate grid prior to a special conformal transformation

The same grid after a special conformal transformation

In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations.

Vector presentation

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) such that:

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

Its infinitesimal generator is

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

Alternative presentation

The inversion can also be taken[2] 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 conjugates of translation by inversion:

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

The matrix describes the action of a special conformal transformation.

References

1. Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. pp. 97–98. ISBN 978-0-387-94785-3.
2. Arthur Conway (1911) "On the application of quaternions to some recent developments of electrical theory", Proceedings of the Royal Irish Academy 29:1–9, particularly page 9