Geometry of spin 1/2 particles

Abstract

The geometric algebras of space and spacetime are derived by sucessively extending the real number system to include new mutually anticommuting square roots of $\pm 1$. The quantum mechanics of spin 1/2 particles are then expressed in these geometric algebras. Classical 2 and 4 component spinors are represented by geometric numbers which have parity, providing new insight into the familiar bra-ket formalism of Dirac. The classical Dirac Equation is shown to be equivalent to the Dirac-Hestenes equation, so long as the issue of parity is not taken into consideration, the latter quantity being constructed in such a way that it is parity invarient.