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Neutrino oscillations in the early Universe prior to the epoch of primordialnucleosynthesis is studied by implementing real time non-equilibrium fieldtheory methods. We focus on two flavors of Dirac neutrinos, however, theformulation is general. We obtain the equations of motion for neutrinowavepackets of either chirality and helicity in the plasma allowing for CPasymmetry. Contributions non-local in space-time to the self-energy dominateover the asymmetry for $T \gtrsim 3-5 \mathrm{MeV}$ if the lepton and neutrinoasymmetries are of the same order as the baryon asymmetry. We find a newcontribution which cannot be interpreted as the usual effective potential. Themixing angles and dispersion relations in the medium depend on \emph{helicity}.We find that resonant transitions are possible in the temperature range $ 10\lesssim T \ll 100 \mathrm{MeV} $. Near a resonance in the mixing angle, theoscillation time scale in the medium as compared to the vacuum is\emph{slowed-down} substantially for small vacuum mixing angle. The time scaleof oscillations \emph{speeds-up} for off resonance high energy neutrinos forwhich the mixing angle becomes vanishingly small. The equations of motionreduce to the familiar oscillation formulae for negative helicityultrarelativistic neutrinos, but include consistently both the \emph{mixingangle and the oscillation frequencies in the medium}. These equations of motionalso allow to study the dynamics of right handed as well as positive helicityneutrinos.Comment: 31 pages 2 figures. Version to appear in Phys. Rev.

Neutrino oscillations in the early universe: A real-time formulation