On the Nonrelativistic Limit of Quantum Electrodynamics: From the Matter–Antimatter–Photon Quantum Field Hybrid to Charged, Massive and Spinning Particles Interacting with Photons

Abstract

Quantum electrodynamics (QED) is a quantum field theory describing light-matter interaction processes on all energy scales. In QED, not only photons are represented as occupied modes of a quantum field, but also electrons (matter) and positrons (antimatter). The necessity of the introduction of antimatter, which goes back to P.A.M. Dirac, has finally the consequence that the QED quantum field cannot be decomposed into its components (particles and photons), since in this sector of light-matter interactions there are processes allowed which violate the particle number conservation. This radically distinguishes QED from the low-energy sector of the light-matter interactions on the energy scale of atomic physics. Starting from the QED Hamiltonian in the Coulomb gauge, a unitary transformation is therefore sought that leads to a unitary equivalent QED Hamiltonian that manifestly preserves the particle number. This is feasible applying Wegner's flow equation in combination with perturbation theory, with the fine structure constant serving as the expansion parameter. In this way, first, high-energy photons, and second, modes of high matter and antimatter densities are eliminated from the QED quantum field up to second order. The resulting Hamiltonian preserves the particle number, is completely symmetric in all its constituents and in all interactions, and also includes terms which renormalize the bare mass of the electron (or the positron). Since this Hamiltonian is still in the Dirac representation, the matter and antimatter degrees of freedom must be decoupled from each other with the help of the Eriksen transformation. By this decoupling, the bare mass of the electron is consistently renormalized, and the anomalous magnetic moment is found in agreement with the Schwinger result. It is thus possible to establish the nonrelativistic limit of full QED as the low-energy sector of light-matter interaction processes in which the electron (or positron) can be interpreted as a classical point particle with the measurable attributes of mass, charge and spin, and in which the particle number is a conservation quantity.