Renormalized Bogoliubov theory for the Nelson model

Falconi M, Lampart J, Leopold N, Mitrouskas DJ. 2025. Renormalized Bogoliubov theory for the Nelson model. Annales de l’Institut Henri Poincaré C.

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Author
Falconi, Marco; Lampart, Jonas; Leopold, Nikolai; Mitrouskas, David JohannesISTA
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Abstract
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≫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.
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Date Published
2025-05-06
Journal Title
Annales de l'Institut Henri Poincaré C
Publisher
European Mathematical Society
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Falconi M, Lampart J, Leopold N, Mitrouskas DJ. Renormalized Bogoliubov theory for the Nelson model. Annales de l’Institut Henri Poincaré C. 2025. doi:10.4171/aihpc/154
Falconi, M., Lampart, J., Leopold, N., & Mitrouskas, D. J. (2025). Renormalized Bogoliubov theory for the Nelson model. Annales de l’Institut Henri Poincaré C. European Mathematical Society. https://doi.org/10.4171/aihpc/154
Falconi, Marco, Jonas Lampart, Nikolai Leopold, and David Johannes Mitrouskas. “Renormalized Bogoliubov Theory for the Nelson Model.” Annales de l’Institut Henri Poincaré C. European Mathematical Society, 2025. https://doi.org/10.4171/aihpc/154.
M. Falconi, J. Lampart, N. Leopold, and D. J. Mitrouskas, “Renormalized Bogoliubov theory for the Nelson model,” Annales de l’Institut Henri Poincaré C. European Mathematical Society, 2025.
Falconi M, Lampart J, Leopold N, Mitrouskas DJ. 2025. Renormalized Bogoliubov theory for the Nelson model. Annales de l’Institut Henri Poincaré C.
Falconi, Marco, et al. “Renormalized Bogoliubov Theory for the Nelson Model.” Annales de l’Institut Henri Poincaré C, European Mathematical Society, 2025, doi:10.4171/aihpc/154.
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