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

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 Schrödinger 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.

minor revision, Ann. H. Poincaré in press, 41 pages, pdfLaTex

References in corpus (7)

• Many-Body Physics with Ultracold Gases
• The time-dependent Born-Oppenheimer approximation
• On the dynamics of polarons in the strong-coupling limit
• Introduction to the physics of artificial gauge fields
• Effective dynamics for particles coupled to a quantized scalar field
• Self-Adjointness criterion for operators in Fock spaces
• Semiclassical polaron dynamics of impurities in ultracold gases

Cited by in corpus (8)

• Magnetic Schrödinger Operators as the Quasi-Classical Limit of Pauli-Fierz-type Models
• Ground State Properties in the Quasi-Classical Regime
• Mean-field Dynamics for the Nelson Model with Fermions
• Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model
• Microscopic Derivation of Time-dependent Point Interactions
• Quasi-classical Limit of a Spin Coupled to a Reservoir
• Quantum Point Charges Interacting with Quasi-classical Electromagnetic Fields
• Convergence of states for polaron models in the classical limit