Magnetic Schr\"odinger Operators as the Quasi-Classical Limit of Pauli-Fierz-type Models

TL;DR

This paper investigates the quasi-classical limit of a quantum system with charged particles interacting with a radiation field, showing convergence to a magnetic Schr"odinger operator with a field-dependent potential.

Contribution

It establishes the resolvent convergence of the microscopic Hamiltonian to an effective magnetic Schr"odinger operator and links ground state energies to classical field configurations.

Findings

Convergence of the Hamiltonian in resolvent sense to an effective Schr"odinger operator.

Ground state energy convergence to the infimum over classical field configurations.

Effective magnetic and electric potentials depend on the classical field.

Abstract

We study the quasi-classical limit of the Pauli-Fierz model: the system is composed of finitely many non-relativistic charged particles interacting with a bosonic radiation field. We trace out the degrees of freedom of the field, and consider the classical limit of the latter. We prove that the partial trace of the full Hamiltonian converges, in resolvent sense, to an effective Schr\"odinger operator with magnetic field and a corrective electric potential that depends on the field configuration. Furthermore, we prove the convergence of the ground state energy of the microscopic system to the infimum over all possible classical field configurations of the ground state energy of the effective Schr\"odinger operator.