Effective Hamiltonians in Nonrelativistic Quantum Electrodynamics

In this paper, we consider some second-order effective Hamiltonians describing the interaction of the quantum electromagnetic field with atoms or molecules in the nonrelativistic limit. Our procedure is valid only for off-energy-shell processes, specifically virtual processes such as those relevant for ground-state energy shifts and dispersion van der Waals and Casimir-Polder interactions, while on-energy-shell processes are excluded. These effective Hamiltonians allow for a considerable simplification of the calculation of radiative energy shifts, dispersion, and Casimir-Polder interactions, including in the presence of boundary conditions. They can also provide clear physical insights into the processes involved. We clarify that the form of the effective Hamiltonian depends on the field states considered, and consequently different expressions can be obtained, each of them with a well-defined range of validity and possible applications. We also apply our results to some specific cases, mainly the Lamb shift, the Casimir-Polder atom-surface interaction, and the dispersion interactions between atoms, molecules, or, in general, polarizable bodies.

A New Family of Interactions between Clothed Particles in QED

The method of unitary clothing transformations (UCTs) has been applied to the quantum electrodynamics (QED) by using the clothed particle representation (CPR). Within CPR, the Hamiltonian for interacting electromagnetic and electron-positron fields takes the form in which the interaction operators responsible for such two-particle processes as e−e− → e−e−, e+e+ → e+e+, e−e+ → e−e+, e−e+ → yy, yy → e−e+, ye− → ye−, and ye+ → ye+ are obtained on the same physical footing. These novel interactions include the off-energy-shell and recoil effects (the latter without any expansion in (v/c)2-series) and their on-energy shell matrix elements reproduce the well-known results derived within the perturbation theory based on the Dyson expansion for the S-matrix (in particular, the Møller formula for the e−e−-scattering, the Bhabha formula for e−e+-scattering, and the Klein–Nishina one for the Compton scattering).