A Binary Index Notation for Clifford Algebras

Abstract

Hagmark and Lounesto’s binary labeling of the generators of a Clifford algebra can be extended by ordering the basis elements in either ascending order, denoted by a post-scripted binary index, or descending order, denoted by a pre-scripted binary index. Reversion is then represented by swapping prescripts and postscripts, grade involution by changing the sign of the index, and conjugation by both swapping and changing the sign. Bit inversion of the binary index is a duality operation that does not suffer the handedness problems of the Clifford dual or of the Hodge dual. Generators whose binary indices are bit inverses of each other commute (anti-commute) if the product of their grades is even (odd). If the number of generators is even (odd), the commutators (anti-commutators) of basis vectors labeled with binary indices and of covectors labeled by bit inversion of binary indices yield generalizations of the Heisenberg commutation relations.