Quantum electrodynamics with nonrelativistic sources. I. Transformation to the multipolar formalism for second-quantized electron and Maxwell interacting fields

The multipolar formalism is commonly used as the starting point in chemical physics and quantum optics for discussion of the interaction of radiation with atoms and molecules. The relationship of the multipolar to the minimal-coupling formalism is examined when both the electron and the radiation are second-quantized fields. Both the Lagrangian and Hamiltonian formulations are considered: in the former the transformation between the two is a point transformation on the electron field coordinates, while in the latter it is a canonical transformation. The resulting equations of motion are Maxwell's equations, in terms of the basic and auxiliary fields, for the electromagnetic field and Schr\"odinger equations for charges in an electromagnetic field with the coupling given through the multipole moments. That the Schr\"odinger equation is different from that which arises in the minimal-coupling formalism is a natural consequence of the use of new field coordinates. The theory is extended to a system of molecules anticipating the discussion of intermolecular energies in paper III (the second succeeding paper).