Split quaternions and semi-Euclidean projective spaces

Abstract

In this study, we give one-to-one correspondence between the elements of the unit split three-sphere S ( 3 , 2 ) with the complex hyperbolic special unitary matrices SU ( 2 , 1 ) . Thus, we express spherical concepts such as meridians of longitude and parallels of latitude on SU ( 2 , 1 ) by using the method given in Toth [Toth G. Glimpses of algebra and geometry. Springer-Verlag; 1998] for S 3 . The relation among the special orthogonal group SO ( R 3 ) , the quotient group of unit quaternions S 3 / { ± 1 } and the projective space RP 3 given as SO ( R 3 ) ≅ S 3 / { ± 1 } = RP 3 is known as the Euclidean projective spaces [Toth G. Glimpses of algebra and geometry. Springer-Verlag; 1998]. This relation was generalized to the semi-Euclidean projective space and then, the expression SO ( 3 , 1 ) ≅ S ( 3 , 2 ) / { ± 1 } = RP 2 3 was acquired. Thus, it was found that Hopf fibriation map of S ( 2 , 1 ) can be used for Twistors (in not-null state) in quantum mechanics applications. In addition, the octonions and the split-octonions can be obtained from the Cayley-Dickson construction by defining a multiplication on pairs of quaternions or split quaternions. The automorphism group of the octonions is an exceptional Lie group. The split-octonions are used in the description of physical law. For example, the Dirac equation in physics (the equation of motion of a free spin 1/2 particle, like e.g. an electron or a proton) can be represented by a native split-octonion arithmetic.