Standard derivations of the functional integral in non-equilibrium quantum field theory are based on the discrete time representation. In this work we derive the non-equilibrium functional integral for non-interacting bosons and fermions using a cont
inuum time approach by accounting for the statistical distribution through the boundary conditions and using them to evaluate the Greens function.
We scrutinize the hydrodynamic approach for calculating dynamical correlations in one-dimensional superfluids near integrability and calculate the characteristic time scale {tau} beyond which this approach is valid. For time scales shorter than {tau}
hydrodynamics fails and we develop an approach based on kinetics of fermionic quasiparticles described as mobile impurities. New universal results for the dynamical structure factor relevant to experiments in ultracold atomic gases are obtained.
