Abstract
AbstractWe 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 $$L^p$$
L
p
to $$L^q$$
L
q
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.
Funder
Danmarks Frie Forskningsfond
Friedrich-Schiller-Universität Jena
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Nuclear and High Energy Physics,Statistical and Nonlinear Physics
Reference33 articles.
1. Alvarez, B., Møller, J.S.: Ultraviolet renormalisation of a quantum field toy model I. Preprint (2021). arXiv:2103.13770
2. Cambridge Studies in Advanced Mathematics;D Applebaum,2009
3. Arai, A.: Analysis on Fock Spaces and Mathematical Theory of Quantum Fields. World Scientific, New Jersey (2018). https://doi.org/10.1142/10367
4. Broderix, K., Hundertmark, D., Leschke, H.: Continuity properties of Schrödinger semigroups with magnetic fields. Rev. Math. Phys. 12(2), 181–225 (2000). https://doi.org/10.1142/S0129055X00000083. arXiv:math-ph/9808004
5. Bley, G.A.: A lower bound on the renormalized Nelson model. J. Math. Phys. 59(6), 061901 (2018). https://doi.org/10.1063/1.5008831. arXiv:1609.08590