Furry representation and S-matrix theory

In the nonrelativistic many-body problem one wishes to solve the Schrbdinger equation for the Hamiltonian 

While variational methods allow this to be solved to very high accuracy for helium (the ground state energy has been determined to 28 digits by Korobov [20]), and to under the microHartree level for lithium [21], [22], these methods cannot be applied to cesium. Instead, one splits the Hamiltonian into two parts, H = Hq + Vc, with 

The new term in Ho approximates the actual interaction of an electron with the other electrons with a mean field U r), chosen to do so as accurately as possible. Applying standard Rayleigh-Schrddinger perturbation theory to Vc then gives the MBPT expansion. It is also possible to generalize to the relativistic case by introducing the instantaneous Breit interaction. 

Rather than review MBPT, which will be dealt with in considerable detail when cesium PNC is treated later, it is useful at this point to instead directly introduce this same breakup in the framework of QED, which is done with a particular kind of interaction representation known as Furry representation [4]. The basic idea is to divide the complete Hamiltonian of QED into a noninteracting part Hq and an interaction term Hj that 

We illustrate the use of the effective charge with one of the potentials we will use, the core-Hartree (CH) potential, defined by 

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