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. In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations.
Vector presentation A special conformal transformation can be written
x ′ μ = x μ − b μ x 2 1 − 2 b ⋅ x + b 2 x 2 = x 2 | x − b x 2 | 2 ( x μ − b μ x 2 ) . {\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 (xμ → xμ/x2 = yμ), a translation (yμ → yμ − bμ = zμ), and another inversion (zμ → zμ/z2 = x′μ)
x ′ μ x ′ 2 = x μ x 2 − b μ . {\displaystyle {\frac {x'^{\mu }}{x'^{2}}}={\frac {x^{\mu }}{x^{2}}}-b^{\mu }\,.}
Its infinitesimal generator is
K μ = − i ( 2 x μ x ν ∂ ν − x 2 ∂ μ ) . {\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.
Projective presentation The inversion can also 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 ) ( 1 0 t 1 ) = U ( q + t : 1 ) . {\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);
… excerpt ends here. Continue reading the full article.
