Effective Potentials Generated by Field Interaction in the Quasi-Classical Limit

TL;DR

This paper analyzes the quasi-classical limit of a quantum particle-field system, showing how the effective particle Hamiltonian converges to a Schrödinger operator with a field-dependent potential, including explicit potential expressions and ground state energy convergence.

Contribution

It provides a rigorous derivation of effective potentials in the quasi-classical limit for Nelson-type models, including explicit formulas and trapping conditions.

Findings

Effective Hamiltonian converges to a Schrödinger operator with a derived potential.

Explicit expression for the effective potential depending on the field state.

Ground state energy converges to an effective variational problem.

Abstract

We study the {\it quasi-classical limit} of a quantum system composed of finitely many non-relativistic particles coupled to a quantized field in Nelson-type models. We prove that, as the field becomes classical and the corresponding degrees of freedom are traced out, the effective Hamiltonian of the particles converges in resolvent sense to a self-adjoint Schr\"{o}dinger operator with an additional potential, depending on the state of the field. Moreover, we explicitly derive the expression of such a potential for a large class of field states and show that, for certain special sequences of states, the effective potential is trapping. In addition, we prove convergence of the ground state energy of the full system to a suitable effective variational problem involving the classical state of the field.