Quantum Electrodynamical Density-matrix Functional Theory and Group-theoretical Consideration of its Solution

For studying the group theoretical classification of the solutions of the density functional theory in relativistic framework, we propose quantum electrodynamical density-matrix functional theory (QED-DMFT). QED-DMFT gives the energy as a functional of a local one-body $4\times4$ matrix $Q(x)\equiv -<\psi(x)\bar{\psi}(x)>$, where $\psi$ and $\bar{\psi}$ are 4-component Dirac field and its Dirac conjugate, respectively. We examine some characters of QED-DMFT. After these preparations, by using Q(x), we classify the solutions of QED-DMFT under O(3) rotation, time reversal and spatial inversion. The behavior of Q(x) under nonrelativistic and ultrarelativistic limits are also presented. Finally, we give plans for several extensions and applications of QED-DMFT.