The Complex Algebra of Physical Space: A Framework for Relativity

A complex and, equivalently, hyperbolic extension of the algebra of physical space (APS) is discussed that allows one to distinguish space-time vectors from paravectors of APS, while preserving the natural origin of the Minkowski space-time metric. The CAPS formalism is Lorentz covariant and gives expression to persistent vectors in physical space as time-like planes in space-time. Commuting projectors ${P_{\pm} = \frac{1}{2} (1 \pm h)}$ project CAPS onto two-sided ideals, one of which is APS. CAPS has the same dimension as the space-time algebra (STA) if both are considered real algebras, and it distinguishes covariant roles of elements, as does STA. Its structure, however, is closer to APS, with a volume element that belongs to the center of the algebra and a simple relation between space-times of opposite signature. Furthermore, CAPS, unlike STA, distinguishes point-like space-time inversion of a Dirac spinor from a physical rotation. To illustrate its use, CAPS is applied to the Dirac equation and to the fundamental symmetry transformations of the equation and Dirac spinors. The physical interpretations of both the equation and the spinor are clarified, and it is seen that the space-time frame ${\{\gamma_{\mu}\}}$ arises fully from relative vectors and does not imply the existence of an absolute space-time frame.