Quantum electrodynamics with nonrelativistic sources. IV. Poynting vector, energy densities, and other quadratic operators of the electromagnetic field

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Abstract
The quantized electromagnetic field in interaction with nonrelativistic sources, i.e., atoms and molcules, has been the subject of study in the previous papers in this series. The Heisenberg formalism is used to calculate the electric- and magnetic-field operators arising from charges and currents expressed in multipolar form. In this paper, quadratic operators in these fields such as the Thompson energy density and the Poynting vector and their expectation values for specified states are found. The expectation values are, in general, time dependent. For initial conditions given at t=0 it is shown that the fields are causal, i.e., for t<r/c the source-dependent fields are zero and the quadratic operators have only their zero-point contributions. For t>r/c they have both time-independent and time-dependent terms. The time-dependent terms, though transient, are shown to obey Poynting’s theorem. The steady-state part of the Poynting vector is related to the Einstein coefficients. The corresponding electric-energy density is related to the Casimir-Polder potential for a polarizable test body in the field of the source molecules. Similarly the magnetic-energy density is derived and is used in the calculation of energy shifts. Quadratic operators referring to different field points are also discussed.