Abstract
Quantum fields with large degeneracy are often approximated as classical fields. Here, we show how quantum and classical evolution of a highly degenerate quantum field with repulsive contact self-interactions differ from each other. Initially, the field is taken to be homogeneous except for a small plane wave perturbations in only one mode. In quantum field theory, modes satisfying both momentum and energy conservation of the quasi-particles, 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.
Highlights
The duration of classicality tcl is the longest time for which the dynamics of a quantum system can be accurately approximated by the dynamics of an analogous classical system
This work was motivated by the following question: how long can a highly degenerate quantum scalar field be accurately described by classical field equations? Intuitively, this time scale cannot be longer than the thermalization time scale
A generic formalism to calculate the duration of classicality was developed in Ref. [4]
Summary
The duration of classicality tcl is the longest time for which the dynamics of a quantum system can be accurately approximated by the dynamics of an analogous classical system. One could choose initial conditions such that hN jð0Þi ≈ Njð0Þ, and trust that the quantum and classical equations will make similar predictions for an indefinitely long time, up to small corrections. Λljmk 1⁄4 0, these differences are not relevant, as both equations are linear and the operators ajðtÞ and the amplitudes AjðtÞ have the same time dependence This implies that, if hN jðtÞi 1⁄4 NjðtÞ initially, it will always be so. [4], the quantum treatment of the initial time evolution of homogeneous fields was given for the cases of attractive contact interactions and for gravitational self-interactions In both cases, an estimate for tcl was provided. III, we solve the linearized Heisenberg equations of motion and obtain an analytical expression for tcl
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