Duration of classicality of an inhomogeneous quantum field with repulsive contact self-interactions

Quantum fields with large degeneracy are often approximated as classical fields. Here, we show how the quantum and classical evolution of a highly degenerate quantum field with repulsive contact self-interactions differ fromeach other. Initially, the field is taken to be homogeneous except for a small plane-wave perturbation in only one mode. In quantum field theory, modes satisfying both momentum and energy conservation of the quasiparticles, grow exponentially with time. However, in the classical field approximation, the system is stable. We calculate the time scale after which the classical field description becomes invalid.