Feynman–Kac Formula and Asymptotic Behavior of the Minimal Energy for the Relativistic Nelson Model in Two Spatial Dimensions

Abstract

We consider the renormalized relativistic Nelson model in two spatial dimensions for a finite number of spinless, relativistic quantum mechanical matter particles in interaction with a massive scalar quantized radiation field. We find a Feynman–Kac formula for the corresponding semigroup and discuss some implications such as ergodicity and weighted Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} to Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document} bounds, for external potentials that are Kato decomposable in the suitable relativistic sense. Furthermore, our analysis entails upper and lower bounds on the minimal energy for all values of the involved physical parameters when the Pauli principle for the matter particles is ignored. In the translation invariant case (no external potential), these bounds permit to compute the leading asymptotics of the minimal energy in the three regimes where the number of matter particles goes to infinity, the coupling constant for the matter–radiation interaction goes to infinity and the boson mass goes to zero.