Unification of Spins and Charges in Grassmann Space Enables Unification of All Interactions

Abstract

In a space of $d $ Grassmann coordinates two types of generators of Lorentz transformations can be defined, one of spinorial and the other of vectorial character. Both kinds of operators appear as linear operators in Grassmann space, definig the fundamental and the adjoint representations of the group $ SO(1,d$$-$$1) $, respectively. The eigenvalues of commuting operators belonging to the subgroup $(SO(1,4)) $ can be identified with spins of either fermionic or bosonic fields, while the operators belonging to subgroups of $ SO(d$$-$$5) $ ${\supset SU(3)}$ $ {\times SU(2)}$ $ {\times U(1)} $, determine the Yang-Mills charges. The theory offers unification of all the internal degrees of freedom of fermionic and bosonic fields - spins and all Yang-Mills charges. When accordingly all interactions are unified, Yang-Mills fields appear as part of the gravitational field. The theory suggests that elementary particles are either in the fundamental representations with respect to the groups determining the spin and the charges, or they are in the adjoint representations with respect to the groups, which determine the spin and the charges.