Ultraviolet Renormalization of the Nelson Hamiltonian through Functional Integration

2013/04/24 by Massimiliano Gubinelli, Gubinelli, M., Fumio Hiroshima +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.1304.6662

openalex publication_date 2013/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Starting from the N-particle Nelson Hamiltonian defined by imposing an ultraviolet cutoff, we perform ultraviolet renormalization by showing that in the zero cutoff limit a self-adjoint operator exists after a logarithmically divergent term is subtracted from the original Hamiltonian. We obtain this term as the diagonal part of a pair interaction appearing in the density of a Gibbs measure derived from the Feynman-Kac representation of the Hamiltonian. Also, we show existence of a weak coupling limit of the renormalized Hamiltonian and derive an effective Yukawa interaction potential between the particles.

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