Field-Theoretical Approach to the Non-Relativistic Quantum Electrodynamics

Abstract

We give here the field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This derivation however illustrates the proper use of the Euler–Lagrange equations and the canonical formalism that fails if one tries to quantize the classical theory of point-like particles interacting with the electromagnetic fields. In the same time it confirms the 1∕c ² result of the preceding chapter on states without photons.