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:
The homography group G(B) includes
which provides the action of a special conformal transformation.
A special conformal transformation can also be written
It is a composition of an inversion (xμ → xμ/x2), a translation (xμ → xμ − bμ), and an inversion:
Its infinitesimal generator is
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