Infinities in molecular quantum electrodynamics and generalized functions

The Power-Zienau-Woolley Hamiltonian for the quantum electrodynamics of atoms and molecules is written in terms of purely transverse electromagnetic field variables and so-called polarization fields for the charged particles. It is well known that the attempt at finding solutions to the coupled equations that arise from the Hamiltonian is marred by the occurrence of infinite “self-energies” for both particles and the field. Because of the occurrence of the Dirac $\delta$ function in the nonzero Poisson-brackets/commutation relation for the fields, and in the definition of the polarization fields, these variables, classical and quantum, must be identified as distributions, in the mathematical sense. The Schwartz “impossibility theorem” shows that there is no general multiplication rule for distributions, so one has to find a framework that gives meaning to the Hamiltonian The energy of the electric polarization field is analyzed in the Colombeau algebra and shown to be finite; in particular Coulomb's law ($1/r,r>0$) with a finite self-energy ($r=0$) is obtained. How these ideas could be extended to the free-field Hamiltonian is discussed. A finite zero-point energy for the electromagnetic field is to be expected. Relevant mathematical results are summarized in an Appendix.