Split quaternions and semi-Euclidean projective spaces

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; 19981 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)) congruent to 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) congruent to 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. (C) 2008 Elsevier Ltd. All rights reserved.