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
UR - http://www.scopus.com/inward/record.url?scp=85076059797&partnerID=8YFLogxK
M3 - Book
VL - 404
T3 - Asterisque
BT - Feynman-Kac formulas for the ultra-violet renormalized Nelson model
PB - Société Mathématique de France
ER -