The relativistic ground state energy

The correct frame of description of interacting relativistic electrons is quantum electrodynamics (QED) where the matter field is the four-component operator-valued electron-positron field acting in the Fock space and depending on space-time = (ct, r) (x = (ct, —r)). Electron-electron interaction takes place via a photon field which is described by an operatorvalued four-potential A x ). Additionally, the system is subject to a static external classical (Bose condensed, c-number) field F , given by the four-potential (distinguished by the missing hat) 

Of course, the static condition refers to a certain distinguished reference frame (due to this static field of nuclei at rest there is no Lorentz invariance and hence also no natural preference any more of a Lagrangian formalism against a Hamiltonian one). We further assume spatial periodicity in a large periodic spatial volume V with respect to that reference frame, and refer all integrated quantities to that volume V (toroidal three-space). 

The colons stand for normal order of the Fock space operators, and 

For the following it is furthermore crucial that normal order in (8, 9) refers to the creation and annihilation operators of the (renormalized) asymptotic fields of the homogeneous system, i.e. with zero external four-potential A, so that this intrinsic Hamiltonian density is independent of the external four-potential. 

In the rest frame of the nuclei, only stationary Fock space states are considered in which all observables are stationary, that is, 

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