131 p.

This thesis is devoted to the formulation and implications of the time-dependent density functional theory (TDDFT). The work is divided into two main parts. In the first part we develop rigorous theorems for the density-potential mapping in quantum many-body systems on a lattice. We prove the uniqueness of the TDDFT map and demonstrate that a given density is v-representable if the initial many-body state and the density satisfy certain well defined conditions. In particular, we show that for a system evolving from its ground state any density with a continuous second time derivative is v-representable. Then the lattice TDDFT formulation is extended to cover system of interacting lattice electrons strongly coupled to cavity photons.We prove that under some mathematical conditions the electron-photon wave function is a unique functional of the electronic density and the expectation value of the photonic coordinate. Then we further generalize the ground state v representability theorem to include the ground state of a general lattice electron-photon Hamiltonian.