Time-dependent Ginzburg-Landau theory for a weak-coupling superconductor

Using a functional approach to the Keldysh formalism we develop a time-dependent Ginzburg-Landau equation for the long-wavelength and low-frequency dynamics of a weak-coupling superconductor, which is also valid in situations that deviate significantly from thermal equilibrium. The theory takes the interactions between quasiparticles and Cooper pairs into account exactly and is explicitly Galilean-invariant at zero temperature. However, the temporal and spatial dependence of the order parameter is assumed to be such that the effects of Landau damping can be neglected. We also consider the screening properties of the superconductor and show that the inclusion of the polarizability of the electron gas is necessary for a complete description.