Geometric Algebras and Fermion Quantum Field Theory
Corresponding to a finite dimensional Hilbert space $H$ with $\dim H=n$, we define a geometric algebra $\mathcal{G}(H)$ with $\dim\left[\mathcal{G}(H)\right]=2^n$. The algebra $\mathcal{G}(H)$ is a Hilbert space that contains $H$ as a subspace. We interpret the unit vectors of $H$ as states of indiv...
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| Format: | Article |
| Language: | English |
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Quanta
2025-07-01
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| Series: | Quanta |
| Online Access: | https://dankogeorgiev.com/ojs/index.php/quanta/article/view/100 |
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| Summary: | Corresponding to a finite dimensional Hilbert space $H$ with $\dim H=n$, we define a geometric algebra $\mathcal{G}(H)$ with $\dim\left[\mathcal{G}(H)\right]=2^n$. The algebra $\mathcal{G}(H)$ 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 $\mathcal{G}(H)$ as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on $\mathcal{G}(H)$ and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on $H$ and $\mathcal{G}(H)$ are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from $H$ to $\mathcal{G}(H)$ are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces.
Quanta 2025; 14: 48–65. |
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| ISSN: | 1314-7374 |