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

Under reasonable assumptions the time-dependent particle density n(r⃗,t) and the current density j⃗(r⃗,t) of a many-particle system that evolves under the action of external scalar and vector potentials V(r⃗,t) and A⃗(r⃗,t) and is initially in the quantum state ∣ψ(0)⟩ can be reproduced in another many-particle system with a different two-particle interaction, subjected to external potentials V^′(r⃗,t) and A⃗^′(r⃗,t) and starting from an initial state ∣ψ^′(0)⟩, which yields the same density and current as ∣ψ(0)⟩. Here we show that given the initial state of this other many-particle system, the potentials V^′(r⃗,t) and A⃗^′(r⃗,t), if they exist, are uniquely determined up to gauge transformations that do not alter the initial state. As a special case, we obtain a 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.