Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908971352195072 |
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| author | Hinrichs, Benjamin Matte, Oliver |
| author_facet | Hinrichs, Benjamin Matte, Oliver |
| contents | In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09005 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions Hinrichs, Benjamin Matte, Oliver Mathematical Physics In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian. |
| title | Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2309.09005 |