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Var."],"published-print":{"date-parts":[[2026,10]]},"abstract":"Abstract<\/jats:title>\n \n 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\n \n \n $$\\varepsilon \\ll 1$$<\/jats:tex-math>\n \n \n \u03b5<\/mml:mi>\n \u226a<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale\n \n \n $$t={\\mathcal {O}}(\\varepsilon ^{-2})$$<\/jats:tex-math>\n \n \n t<\/mml:mi>\n =<\/mml:mo>\n O<\/mml:mi>\n (<\/mml:mo>\n \n \u03b5<\/mml:mi>\n \n -<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . The natural semiclassical model is the coupled Schr\u00f6dinger\u2013Klein\u2013Gordon (SKG) system, which encodes the time-scale separation through an explicit\n \n \n $$\\varepsilon $$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the\n \n \n $$\\varepsilon $$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -free adiabatic Klein\u2013Gordon (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\u2013Nelson 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\u2013Oppenheimer type approximation of the renormalized Nelson model.\n <\/jats:p>","DOI":"10.1007\/s00526-026-03436-9","type":"journal-article","created":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T01:43:00Z","timestamp":1791769380000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adiabatic Klein\u2013Gordon dynamics for the renormalized Nelson model"],"prefix":"10.1007","volume":"65","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6249-0928","authenticated-orcid":false,"given":"Morris","family":"Brooks","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Mitrouskas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,12]]},"reference":[{"issue":"6","key":"3436_CR1","doi-asserted-by":"publisher","first-page":"5031","DOI":"10.1137\/17M1117598","volume":"49","author":"Z Ammari","year":"2016","unstructured":"Ammari, Z., Falconi, M.: Bohr\u2019s correspondence principle for the renormalized Nelson model. 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