Geometric Algebras and Fermion Quantum Field Theory

Corresponding to a finite dimensional Hilbert space $H$ with $\dim H=n$, we define a geometric algebra $\text{gscript}(H)$ with $\dim \text{sqbrac}\text{gscript}(H)=2^n$. The algebra $\text{gscript}(H)$ 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 $\text{gscript}(H)$ as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on $\text{gscript}(H)$ and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on $H$ and $\text{gscript}(H)$ are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from $H$ to $\text{gscript}(H)$ are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces.