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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Quanta
2025-07-01
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| Online Access: | https://dankogeorgiev.com/ojs/index.php/quanta/article/view/100 |
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| author | Stan Gudder |
| author_facet | Stan Gudder |
| author_sort | Stan Gudder |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-91e2edf058eb4ef7a75bef1f52abd337 |
| institution | DOAJ |
| issn | 1314-7374 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Quanta |
| record_format | Article |
| series | Quanta |
| spelling | doaj-art-91e2edf058eb4ef7a75bef1f52abd3372025-08-20T03:14:33ZengQuantaQuanta1314-73742025-07-0114486510.12743/quanta.92100Geometric Algebras and Fermion Quantum Field TheoryStan Gudder0University of DenverCorresponding 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.https://dankogeorgiev.com/ojs/index.php/quanta/article/view/100 |
| spellingShingle | Stan Gudder Geometric Algebras and Fermion Quantum Field Theory Quanta |
| title | Geometric Algebras and Fermion Quantum Field Theory |
| title_full | Geometric Algebras and Fermion Quantum Field Theory |
| title_fullStr | Geometric Algebras and Fermion Quantum Field Theory |
| title_full_unstemmed | Geometric Algebras and Fermion Quantum Field Theory |
| title_short | Geometric Algebras and Fermion Quantum Field Theory |
| title_sort | geometric algebras and fermion quantum field theory |
| url | https://dankogeorgiev.com/ojs/index.php/quanta/article/view/100 |
| work_keys_str_mv | AT stangudder geometricalgebrasandfermionquantumfieldtheory |