  In this paper we study a Clifford algebra generalization of the quaternions
and its relationship with braid group representations related to Majorana
fermions. The Fibonacci model for topological quantum computing is based on the
fusion rules for a Majorana fermion. Majorana fermions can be seen not only in
the structure of collectivies of electrons, as in the quantum Hall effect, but
also in the structure of single electrons both by experiments with electrons in
nanowires and also by the decomposition of the operator algebra for a fermion
into a Clifford algebra generated by two Majorana operators. The purpose of
this paper is to discuss these braiding representations, important for
relationships among physics, quantum information and topology. A new result in
this paper is the Clifford Braiding Theorem. This theorem shows that the
Majorana operators give rise to a particularly robust representation of the
braid group that is then further represented to find the phases of the fermions
under their exchanges in a plane space. The more robust representation in our
braiding theorem will be the subject of further work.

Strongly Correlated Electrons;Geometric Topology