Geometric Algebras and Fermion Quantum Field Theory

TL;DR

This paper introduces a geometric algebra framework for fermion quantum field theory, connecting finite and infinite dimensional Hilbert spaces, and explores creation operators, evolution, and boson-fermion fields.

Contribution

It develops a geometric algebra approach to fermion quantum fields, including operator extensions and a generalization to infinite dimensions.

Findings

Defined a geometric algebra for finite-dimensional Hilbert spaces.

Constructed creation operators and their matrix representations.

Extended the framework to infinite-dimensional separable Hilbert spaces.

Abstract

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