Adiabatic Klein-Gordon Dynamics for the Renormalized Nelson Model
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912590819491840 |
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| author | Brooks, Morris Mitrouskas, David |
| author_facet | Brooks, Morris Mitrouskas, David |
| contents | We study the renormalized Nelson model in a semiclassical regime where the field becomes classical while the particle remains quantum. The degree of classicality is measured by a small parameter $\varepsilon \ll 1$. In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale $t=\mathcal{O}(\varepsilon^{-2})$. The natural semiclassical model is the coupled Schrödinger-Klein-Gordon (SKG) system, which encodes the time-scale separation through an explicit $\varepsilon$-dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the $\varepsilon$-free adiabatic Klein-Gordon (aKG) equation, where the field is driven by the instantaneous ground state of the particle. Our main result is a norm approximation of the Nelson dynamics by the aKG solution corrected by a quasi-free fluctuation dynamics around the classical field, generated by a renormalized Bogoliubov-Nelson Hamiltonian. As a corollary, we obtain convergence of the reduced one-body densities for both subsystems, where the fluctuation correction vanishes, thereby justifying aKG as a semiclassical Born-Oppenheimer type approximation of the renormalized Nelson model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adiabatic Klein-Gordon Dynamics for the Renormalized Nelson Model Brooks, Morris Mitrouskas, David Mathematical Physics We study the renormalized Nelson model in a semiclassical regime where the field becomes classical while the particle remains quantum. The degree of classicality is measured by a small parameter $\varepsilon \ll 1$. In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale $t=\mathcal{O}(\varepsilon^{-2})$. The natural semiclassical model is the coupled Schrödinger-Klein-Gordon (SKG) system, which encodes the time-scale separation through an explicit $\varepsilon$-dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the $\varepsilon$-free adiabatic Klein-Gordon (aKG) equation, where the field is driven by the instantaneous ground state of the particle. Our main result is a norm approximation of the Nelson dynamics by the aKG solution corrected by a quasi-free fluctuation dynamics around the classical field, generated by a renormalized Bogoliubov-Nelson Hamiltonian. As a corollary, we obtain convergence of the reduced one-body densities for both subsystems, where the fluctuation correction vanishes, thereby justifying aKG as a semiclassical Born-Oppenheimer type approximation of the renormalized Nelson model. |
| title | Adiabatic Klein-Gordon Dynamics for the Renormalized Nelson Model |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2509.14226 |