On an Alternative Approach to the Relation between Bosons and Fermions: Employing Clifford Space

We further explore the idea that physics takes place in Clifford space which should be considered as a generalization of spacetime. Following the old observation that spinors can be represented as members of left ideals of Clifford algebra, we point out that the transformations which mix bosons and fermions could be represented by means of operators acting on Clifford algebra-valued (polyvector) fields. A generic polyvector field can be expanded either in terms of bosonic or in terms of fermionic fields. In particular, a scalar field can transform into a mixture of bosonic and/or fermionic fields. The idea that Clifford algebra might provide a clue to the unification of fundamental interactions has been explored by a number of researchers [1, 2, 3], [4]–[6]. Crawford considered local transformations which involve all 2 generators of Clifford algebra acting on column spinors. The compensating gauge fields behaved as Yang-Mills fields. Instead of column spinors one can represent spinors geometrically, as memebers of left or right ideals of Clifford algebra [7, 11]. Chisholm used geometric spinors, but he took only spinor of one left ideal, and left out those belonging to other left ideals. The first who took advantage of the full Clifford algebra in describing wave functions was Pezzaglia. He proposed that wave functions be Clifford algebra valued objects, called polyvectors, satisfying the generalized Klein-Gordon or the Dirac equation. In his particular model the wave function depended