Electromagnetic interactions Coulomb gauge

As was shown in [13], to include the relativistic recoil corrections in calculations of the energy levels, we must add to the standard Hamiltonian of the electron-positron field interacting with the quantized electromagnetic field and with the Coulomb field of the nucleus Vc an additional term. In the Coulomb gauge, this term is given by... 

This approach clearly distinguishes two ranges of interaction At the scale r < A, where the electrostatic interaction dominates, and at the scale r > /, where the retardation effects dominate. This scale property justifies the separation, implicit in the Coulomb gauge, between instantaneous terms and retarded terms. However, the electric-dipole gauge shows that these two distinct aspects of the electromagnetic interaction are physically undissoci-able, even though it is possible in many problems to omit retardation effects. 

A large class of molecular properties arise from the interaction of molecules with electromagnetic fields. As emphasized previously, the external fields are treated as perturbations and so one considers only the effect of the fields on the molecule and not the effect of the molecule on the field. The electromagnetic fields introduced into the electronic wave equation is accordingly those of free space. From (79) one observes that in the absence of sources the electric field has zero divergence, and so both the electric and magnetic fields are purely transversal. It follows that the scalar potential is a constant and can be set to zero. In Coulomb gauge the vector potential is found from the equation... 

The precise form of this correction depends upon the gauge condition used to describe the electromagnetic field. In the Coulomb gauge, which has been employed more often in relativistic atomic structure, the electron-electron interactions come from one-photon exchange process and is sum of instantaneous Coulomb interaction and the transverse photon interaction. 

Our goal now is to determine the interaction part of the hamiltonian, that is, the part that describes the interaction of the charged particle with the external electromagnetic field, assuming that the latter is a small perturbation in the motion of the free particle. We will also specialize the discussion to electrons, in which case m = Me,q = —e. In the Coulomb gauge, we express the vector potential as... 

Electromagnetic interaction due to the transverse degrees of freedom of the electromagnetic field (as opposed to the Coulomb interaction, which is due to the longitudinal degrees of freedom, when Coulomb gauge is employed). Two-component spinor... 

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