B Relativistic Homogeneous Electron Gas

In this appendix we summarise some properties of the relativistic homogeneous electron gas (RHEG) in order to illustrate the renormalisation of ground state energies (indicated in Section 2) and to provide the details for the RLDA and the relativistic gradient expansion (RGE), which are discussed in Section 4 and Appendix D, respectively. For simplicity we restrict the discussion to the unpolarised RHEG. 

We start by noting the basic difference between the perturbative approaches to the RHEG and the QED vacuum (discussed in Appendix A). As a consequence of the difference between the ground state 0 of the RHEG and the homogeneous vacuum 0 the fermion propagator 

While the other two basic elements of perturbation theory, the free photon propagator (A.9) and the simple vertex, remain unchanged, it seems worth pointing out that the full photon propagator 

Most information concerning the RHEG required in the present context is contained in the response functions of the RHEG. In our notation the time-ordered current response functions (n-point functions) are defined as 

For the time-independent systems of interest here a partial Fourier transformation of is advantageous, 

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