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arXiv:2509.14226 · Calculus of Variations and Partial Differential Equations

Adiabatic Klein–Gordon dynamics for the renormalized Nelson model

Morris Brooks, David Mitrouskas

Published Oct 1, 2026• 0 citations
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Abstract

Abstract 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 $$\varepsilon \ll 1$$ ε ≪ 1 . In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale $$t={\mathcal {O}}(\varepsilon ^{-2})$$ t = O ( ε - 2 ) . The natural semiclassical model is the coupled Schrödinger–Klein–Gordon (SKG) system, which encodes the time-scale separation through an explicit $$\varepsilon $$ ε -dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the $$\varepsilon $$ ε -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.

DOI
10.1007/s00526-026-03436-9

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Abstract 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 ε≪1\varepsilon \ll 1ε≪1 ε ≪ 1 . In this scaling the particle evolves on microscopic times, whereas the field exhibits nontrivial dynamics only on the macroscopic scale t=O(ε−2)t={\mathcal {O}}(\varepsilon ^{-2})t=O(ε−2) t = O ( ε - 2 ) . The natural semiclassical model is the coupled Schrödinger–Klein–Gordon (SKG) system, which encodes the time-scale separation through an explicit ε\varepsilon ε ε -dependence. Based on this scale separation in SKG, we apply the adiabatic principle to derive a new PDE for the classical field, the ε\varepsilon ε ε -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.
We consider the time evolution of the renormalized Nelson model, which describes N bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles N\gg 1 with coupling constant proportional to N^{-1/2} . First, we show that initial states exhibiting Bose–Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schrödinger–Klein–Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schrödinger–Klein–Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.
We study the time evolution of the Nelson model in a mean-field limit in which [Formula: see text] nonrelativistic bosons weakly couple (with respect to the particle number) to a positive or zero mass quantized scalar field. Our main result is the derivation of the Bogoliubov dynamics and higher-order corrections. More precisely, we prove the convergence of the approximate wave function to the many-body wave function in norm, with a convergence rate proportional to the number of corrections taken into account in the approximation. We prove an analogous result for the unitary propagator. As an application, we derive a simple system of partial differential equations describing the time evolution of the first- and second-order approximations to the one-particle reduced density matrices of the particles and the quantum field, respectively.
The validity of the semiclassical approximation is studied for a system comprising one quasiparticle coupled to a boson degree of freedom. Using a two-site Holstein model as an example, it is shown that the semiclassical approximation becomes exact in a nontrivial adiabatic limit. Furthermore, in the model's polaron regime, there exists a hierarchy of time scales that rationalizes the quantum dynamics of the Holstein model. For the single-mode case considered, the discrete nonlinear Schr\"odinger equation is found to be valid only in a highly limited antiadiabatic regime.
Recent advances in microscopic many-body methods have enabled increasingly accurate descriptions of equilibrium quantum states, yet large-scale simulations of emergent-order dynamics typically rely on phenomenological equations with prescribed free-energy functionals. We introduce the Adiabatic Variational Principle (AVP), a framework for deriving order-parameter dynamics directly from an interacting many-body Hamiltonian. When microscopic relaxation is fast compared with order-parameter evolution, the many-body state follows the collective configuration adiabatically, in analogy with the Born--Oppenheimer separation of electronic and nuclear motion. This assumption makes explicit the adiabatic closure implicit in conventional phase-field descriptions. Each instantaneous order-parameter configuration then defines a constrained quasi-stationary many-body state, and the Lagrange multipliers enforcing these constraints directly provide the thermodynamic forces driving the collective evolution. Nonlinear Mori--Zwanzig projection theory further connects the reversible dynamics to the microscopic algebra of the collective variables and relates dissipation and fluctuations to the eliminated degrees of freedom. Applications to charge- and spin-density-wave dynamics recover Allen--Cahn domain growth, resolve evolving domain networks together with their microscopic electronic structure, and yield generalized Landau--Lifshitz dynamics for itinerant magnetism. The AVP thus provides a solver-independent microscopic foundation for phase-field-like simulations of the nonequilibrium dynamics of emergent electronic order.
The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.
Several years ago Nelson succeeded in deriving the nonrelativistic Schrödinger equation within a stochastic model which included Newton’s second law as the fundamental dynamical rule. Unfortunately, the relativistic extension of Nelson’s work is not so straightforward as might at first be supposed. This paper examines the difficulties inherent in such a relativization and proposes supplemental axioms which resolve those difficulties. A stochastic derivation of the Klein–Gordon equation is then presented.