A Construction for Clifford Algebras

This paper is dedicated to the memory of Zbigniew Oziewicz, to his generosity, intelligence and intensity in the search that is science and mathematics. The paper begins with a basic construction that produces Clifford algebras inductively, starting with a base algebra A that is associative and has an involution. This construction is an analog of the Cayley-Dickson Construction that produces the complex numbers, quaternions and octonions starting from the real numbers. Our basic construction always produces associative algebras and can be iterated an indefinite number of times. We generalize the basic construction to a group theoretic construction where a group G acts on the algebra A, and show how this group theoretic construction is related to matrix algebras. The paper then concentrates on applications of this algebra to the Dirac Equation and the Majorana-Dirac Equation.