Geometric Algebras and Fermion Quantum Field Theory

• URL: http://arxiv.org/abs/2507.20394v1
• Date: Sun, 27 Jul 2025 19:25:01 GMT
• Title: Geometric Algebras and Fermion Quantum Field Theory
• Authors: Stan Gudder,
• Abstract summary: We define a geometric algebra $gscript (H)$ with $dimsqbracgscript (H)=2n$.<n>The algebra $gscript (H)$ is a Hilbert space that contains $H$ as a subspace.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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