Hot physics problems in early cosmology and ultrarelativistic heavy ion collisions call for an out of equilibrium treatment in quantum field theory. This takes the form of an initial value problem (or Cauchy problem). For strongly out of equilibrium situations with extreme energy densities (like in the early universe) we use nonperturbative large N methods. Otherwise, we can linearize the QFT evolution equations in the field expectation value (amplitude approximation). In this talk, I first review the scalar field evolution for in the case of a supercooled second order phase transition. That is, when the order parameter is initially close to the false vacuum. The role of the spinodal instabilities, the abundant particle production and the shut-off of the unstabilities by the non-linear interaction is reviewed.

I then report on the generation of a non-equilibrium plasma in scalar QED with N-charged scalar fields through spinodal instabilities in the case of a supercooled second order phase transition and parametric amplification when the order parameter oscillates with large amplitude around the minimum of the potential. The focus is to study the non-equilibrium electromagnetic properties of the plasma, such as photon production, electric and magnetic screening and conductivity. A novel kinetic equation is introduced to compute photon production far away from equilibrium in the large N limit and lowest order in the electromagnetic coupling. During the early stages of the dynamics the photon density grows exponentially and asymptotically the amplitude and frequency distribution becomes $ \sim \alpha \; m^2 / [\lambda^2 \; \omega^3] $ with $ \lambda $ the scalar self-coupling and $ m $ the scalar mass. In the case of a phase transition, electric and magnetic fields are correlated on distances $ \xi(t) \sim \sqrt{t} $ during the early stages and the power spectrum is peaked at low momentum. This aspect is relevant for the generation of primordial magnetic fields in the Early Universe and for photoproduction as a potential experimental signature of the chiral phase transition. Magnetic and Debye screening masses are defined out of equilibrium as generalizations of the equilibrium case. While the magnetic mass vanishes out of equilibrium in this abelian model, we introduce an effective time and wave-number magnetic mass that reveals the different processes that contribute to screening and their time scales. The Debye mass turns to be $ m^2_{Deb} \sim \alpha \; m^2/\lambda $ for a supercooled phase transition while in the case of an oscillating order parameter an interpolating time dependent Debye mass grows as $ \alpha \sqrt{m\,t}/\lambda $ due to a non-linear resonance at low momentum in the charged particle distribution. The build-up of the transverse electric conductivity is studied during the formation of the non-equilibrium plasma. Its long wavelength limit reaches a value $ \sigma_{k\approx 0} \sim \alpha \; m/\lambda $ at the end of the stage of linear instabilities. In the asymptotic regime it attains a form analogous to the equilibrium case but in terms of the non-equilibrium particle distribution functions.

Ref: D. Boyanovsky, H. J. de Vega and M. Simionato, hep-ph/9909259