>The Paravector Model of Spacetime

Abstract

The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a natural complex structure on a four-dimensional space. If paravectors, which are formed sums of scalars and vectors, are taken as the “real” elements of the space, then the space can be shown to have a Minkowski spacetime metric, and the paravectors may be identified with spacetime vectors. Physical Lorentz transformations of spacetime vectors are described by spin transformations of the paravectors. The transformation elements are unimodular elements of the algebra, and they form the six-parameter group SL(2,C), the two-fold covering group of restricted Lorentz transformations, SO+(l,3). Its elements are also reducible spinors, whose elements carry a reducible spin representation of SL(2,C). The spin representation is reduced by splitting elements into complementary minimal left ideals. The complex spin space that results has a symplectic structure and its elements belong to Sp(2).KeywordsRest FrameLorentz TransformationGeometric AlgebraSpacetime DiagramObject FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.