On the relativistic effect in the Dirac--Fock theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910226834259968 |
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| author | Meng, Long |
| author_facet | Meng, Long |
| contents | In this paper, we study the error bound between the Dirac--Fock ground-state energy and the Hartree--Fock ground-state energy, a quantity known as the relativistic effect in quantum mechanics. We confirm that the relativistic effect in the Dirac--Fock ground-state energy is of the order $\mathcal{O}(c^{-2})$ with $c$ being the speed of light. Furthermore, if the potential between electrons and nuclei is regular, we get the well-known leading order relativistic correction -- the Breit--Pauli term, which is the sum of the mass-velocity term, the Darwin term, and the spin-orbit term. As a consequence, we also show that the same relativistic effects and leading order relativistic correction also hold in a QED model introduced by Mittleman when the vacuum polarization -- a term of the order $\mathcal{O}(c^{-3})$ -- is neglected. To our knowledge, this is the first time in mathematics that the leading-order relativistic correction has been obtained from nonlinear Dirac ground-state energy problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21405 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the relativistic effect in the Dirac--Fock theory Meng, Long Mathematical Physics Analysis of PDEs Atomic and Molecular Clusters 81V55, 35Q40, 81Q15 In this paper, we study the error bound between the Dirac--Fock ground-state energy and the Hartree--Fock ground-state energy, a quantity known as the relativistic effect in quantum mechanics. We confirm that the relativistic effect in the Dirac--Fock ground-state energy is of the order $\mathcal{O}(c^{-2})$ with $c$ being the speed of light. Furthermore, if the potential between electrons and nuclei is regular, we get the well-known leading order relativistic correction -- the Breit--Pauli term, which is the sum of the mass-velocity term, the Darwin term, and the spin-orbit term. As a consequence, we also show that the same relativistic effects and leading order relativistic correction also hold in a QED model introduced by Mittleman when the vacuum polarization -- a term of the order $\mathcal{O}(c^{-3})$ -- is neglected. To our knowledge, this is the first time in mathematics that the leading-order relativistic correction has been obtained from nonlinear Dirac ground-state energy problems. |
| title | On the relativistic effect in the Dirac--Fock theory |
| topic | Mathematical Physics Analysis of PDEs Atomic and Molecular Clusters 81V55, 35Q40, 81Q15 |
| url | https://arxiv.org/abs/2503.21405 |