Abstract
Corresponding to a finite dimensional Hilbert space H with dim H=n, we define a geometric algebra (H) with dim (H)=2ⁿ. The algebra (H) is a Hilbert space that contains H as a subspace. We interpret the unit vectors of H as states of individual fermions of the same type and (H) as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on (H) and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on H and (H) are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from H to (H) are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces.