  The general expectation that, in principle, time-dependent density functional
theory (TDDFT) be an exact formulation of the time-evolution of an interacting
N-electron system is critically reexamined. It is demonstrated that the
previous TDDFT foundation, resting on three theorems by Runge and Gross (RG)
[Phys.Rev.Lett.52, 997(1984)], is invalid because undefined phase factors
corrupt the RG action integral functionals. Our finding confirms much of a
previous analysis by van Leeuwen [Int. J. Mod. Phys. B15, 1969(2001)]. To
analyze the RG theorems and other aspects of TDDFT, an utmost simplification of
the KS theory is introduced, in which the density is obtained from a single KS
equation for one spatial (spin-less) particle. This radical KS formulation
allows us to analyze also the concept of KS-type equations derived without a
variational principle. We argue that such an approach can reproduce, but not
predict the time-development of the exact density of the interacting N-electron
system. Besides the issue of the formal exactness of TDDFT, it is shown that
both the static and time-dependent KS linear response equations neglect the
particle-particle (p-p) and hole-hole (h-h) matrix elements of the perturbing
operator. For a local (multiplicative) operator this does not lead to a loss of
information due to a remarkable general property of local operators.
Accordingly, no logical inconsistency arises here, because DFT requires
external potentials to be local.
