Rotation Groups in Higher Dimensions

Symmetries play two different roles in the determination of physical systems. The more obvious role proceeds by identifying the invariance of a system's mechanics under specific transformations, mainly under rotations or reflections of coordinates. Such transformations form the invariance group of the system. Less direct procedures disregard initially certain aspects of the system's mechanics, utilizing broader symmetries of a preliminary formulation in further analysis. Transformations preserving such symmetries form non-invariance groups. Extending the r -transformations of physical space to higher dimensions may proceed by adding one dimension at a time. The first step of this procedure deals with the Coulomb–Kepler system of two-body motion, with interparticle potential proportional to 1/ r and its invariance complements space coordinates with an eccentricity parameter, forming a four-dimensional space with an invariance group. Further extensions of rotational symmetry occur with the use of imaginary angles that extend the symmetry of physical space coordinates to include the time variable of special relativity.