Feynman-Kac formulas for the ultra-violet renormalized Nelson model

Oliver Matte; Jacob Schach Møller

Feynman-Kac formulas for the ultra-violet renormalized Nelson model

Oliver Matte, Jacob Schach Møller

Department of Mathematics

Research output: Book/anthology/dissertation/report › Book › Research › peer-review

8 Citations (Scopus)

Abstract

— We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in a non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, then the spectrum of the N-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal constant times g ⁴N ³, for all values of the coupling constant g. A variational argument also yields an upper bound of the same form for large g ²N. We further verify that the semi-groups generated by the ultra-violet renormalized Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a natural self-dual cone, if the Pauli principle is ignored. In another application we discuss continuity properties of elements in the range of the semi-group of the renormalized Nelson Hamiltonian.

Original language	English

Publisher	Société Mathématique de France
Volume	404
Number of pages	150
ISBN (Electronic)	978-2-85629-893-0
Publication status	Published - 2018

Series	Asterisque
ISSN	0303-1179

Keywords

Feynman-Kac formula
Nelson model
Non-Fock representation
Perron-Frobenius arguments
Renormalization

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https://smf.emath.fr/publications/formules-de-feynmann-kac-pour-le-modele-de-nelson-ultra-violet-renormalisee

Cite this

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title = "Feynman-Kac formulas for the ultra-violet renormalized Nelson model",

abstract = "— We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in a non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, then the spectrum of the N-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal constant times g 4N 3, for all values of the coupling constant g. A variational argument also yields an upper bound of the same form for large g 2N. We further verify that the semi-groups generated by the ultra-violet renormalized Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a natural self-dual cone, if the Pauli principle is ignored. In another application we discuss continuity properties of elements in the range of the semi-group of the renormalized Nelson Hamiltonian.",

keywords = "Feynman-Kac formula, Nelson model, Non-Fock representation, Perron-Frobenius arguments, Renormalization",

author = "Oliver Matte and M{\o}ller, {Jacob Schach}",

year = "2018",

language = "English",

volume = "404",

series = "Asterisque",

publisher = "Soci{\'e}t{\'e} Math{\'e}matique de France",

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TY - BOOK

T1 - Feynman-Kac formulas for the ultra-violet renormalized Nelson model

AU - Matte, Oliver

AU - Møller, Jacob Schach

PY - 2018

Y1 - 2018

N2 - — We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in a non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, then the spectrum of the N-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal constant times g 4N 3, for all values of the coupling constant g. A variational argument also yields an upper bound of the same form for large g 2N. We further verify that the semi-groups generated by the ultra-violet renormalized Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a natural self-dual cone, if the Pauli principle is ignored. In another application we discuss continuity properties of elements in the range of the semi-group of the renormalized Nelson Hamiltonian.

AB - — We derive Feynman-Kac formulas for the ultra-violet renormalized Nelson Hamiltonian with a Kato decomposable external potential and for corresponding fiber Hamiltonians in the translation invariant case. We simultaneously treat massive and massless bosons. Furthermore, we present a non-perturbative construction of a renormalized Nelson Hamiltonian in a non-Fock representation defined as the generator of a corresponding Feynman-Kac semi-group. Our novel analysis of the vacuum expectation of the Feynman-Kac integrands shows that, if the external potential and the Pauli-principle are dropped, then the spectrum of the N-particle renormalized Nelson Hamiltonian is bounded from below by some negative universal constant times g 4N 3, for all values of the coupling constant g. A variational argument also yields an upper bound of the same form for large g 2N. We further verify that the semi-groups generated by the ultra-violet renormalized Nelson Hamiltonian and its non-Fock version are positivity improving with respect to a natural self-dual cone, if the Pauli principle is ignored. In another application we discuss continuity properties of elements in the range of the semi-group of the renormalized Nelson Hamiltonian.

KW - Feynman-Kac formula

KW - Nelson model

KW - Non-Fock representation

KW - Perron-Frobenius arguments

KW - Renormalization

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M3 - Book

VL - 404

T3 - Asterisque

BT - Feynman-Kac formulas for the ultra-violet renormalized Nelson model

PB - Société Mathématique de France

ER -

Feynman-Kac formulas for the ultra-violet renormalized Nelson model

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Keywords

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