Hyperbolic phase transformation group and its applicaton to four-dimensional relativistic spacetime

Abstract

The hyperbolic complex space RH defined by the Clifford algebra and the hyperbolic phase transformation group U4(H) acting on RH are endowed with definite physical meaning in this paper. The hyperbolic complex space RH is isomorphic to the 4-dimensional(4D) Minkowski spacetime, and the hyperbolic phase transformation group U4(H) in RH is just Lorentz transformation group on 4D relativistic spacetime. Furthermore, the general expressions of Lorentz transformation and the velocity transformation on 4D Minkowski spacetime are naturally derived using the composite transformations of the group U4(H). Hence, the well-known special Lorentz transformation in the special relativity(SR) is contained as a special case in our discussions.