Mapping from current densities to vector potentials in time-dependent current-density functional theory

We show that the time-dependent particle density $n(\vec r,t)$ and the current density ${\vec j}(\vec r,t)$ of a many-particle system that evolves under the action of external scalar and vector potentials $V(\vec r,t)$ and $\vec A(\vec r,t)$ and is initially in the quantum state $|\psi (0)>$, can always be reproduced (under mild assumptions) in another many-particle system, with different two-particle interaction, subjected to external potentials $V'(\vec r,t)$ and $\vec A'(\vec r,t)$, starting from an initial state $|\psi' (0)>$, which yields the same density and current as $|\psi (0)>$. Given the initial state of this other many-particle system, the potentials $V'(\vec r,t)$ and $\vec A'(\vec r,t)$ are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a new and simpler proof of the Runge-Gross theorem for time-dependent current density functional theory. This theorem provides a formal basis for the application of time-dependent current density functional theory to transport problems.