4. Spherical Wave Transformations and the Group of Conformal Transformations of a Space of Four Dimensions.
The group of spherical wave transformations may be reduced to a known group by putting t = is. The quadratic form which remains invariant is then of the type
and so the transformation is a conformal one.
The group of conformal transformations in a space of four dimensions has been studied by Sophus Lie,[1] who has shown that it is composed of reflexions, translations, rotations, magnifications, and inversions.[2] The transformations which are of importance in the present case are the imaginary ones, and it should be noticed that by a combination of two imaginary inversions we can obtain a transformation of the type
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which is quite different from an inversion or simple displacement. This corresponds to the real spherical wave transformation[3]
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An imaginary rotation in the four-dimensional space may be specified, in a particular case, by the equations
Putting
- ↑ Göttinger Nachrichten (1871), Transformationgruppen, Bd. 3, p. 351.
- ↑ Every transformation belonging to the group is a birational transformation.
- ↑ See a paper by the author "The Conformal Transformations of a Space of Four Dimensions and their Applications to Geometrical Optics," Proc. London Math. Soc., Ser. 2, Vol. 7, p. 70 (1909).