Quadratic Hamiltonians in Fermionic Fock Spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915947863867392 |
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| author | Bru, Jean-Bernard Metraud, Nathan |
| author_facet | Bru, Jean-Bernard Metraud, Nathan |
| contents | Quadratic Hamiltonians are important in quantum field theory and quantum statistical mechanics. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case studied here. Following Berezin, they are quadratic in the fermionic field and in this way well-defined self-adjoint operators acting on the fermionic Fock space. We analyze their diagonalization by applying a novel elliptic operator-valued differential equations studied in a companion paper. This allows for their ($\mathrm{N}$-) diagonalization under much weaker assumptions than before. Last but not least, in 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, as soon as the vacuum state belongs to the domain of definition of these Hamiltonians. This second outcome is demonstrated to be reminiscent to the celebrated Shale-Stinespring condition on Bogoliubov transformations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_11697 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quadratic Hamiltonians in Fermionic Fock Spaces Bru, Jean-Bernard Metraud, Nathan Mathematical Physics 81Q10, 34A34, 47D06 Quadratic Hamiltonians are important in quantum field theory and quantum statistical mechanics. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case studied here. Following Berezin, they are quadratic in the fermionic field and in this way well-defined self-adjoint operators acting on the fermionic Fock space. We analyze their diagonalization by applying a novel elliptic operator-valued differential equations studied in a companion paper. This allows for their ($\mathrm{N}$-) diagonalization under much weaker assumptions than before. Last but not least, in 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, as soon as the vacuum state belongs to the domain of definition of these Hamiltonians. This second outcome is demonstrated to be reminiscent to the celebrated Shale-Stinespring condition on Bogoliubov transformations. |
| title | Quadratic Hamiltonians in Fermionic Fock Spaces |
| topic | Mathematical Physics 81Q10, 34A34, 47D06 |
| url | https://arxiv.org/abs/2305.11697 |