  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.
