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^[1] 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.

Vector presentation

A special conformal transformation can also be written^[2]

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

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

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

References

1. 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
2. Di Francesco; Mathieu, Sénéchal (1997). Conformal field theory. Graduate texts in contemporary physics. Springer. pp. 97–98. ISBN 978-0-387-94785-3.