Quantum and Classical Behavior in Interacting Bosonic Systems
  It is understood that in free bosonic theories, the classical field theory
accurately describes the full quantum theory when the occupancy numbers of
systems are very large. However, the situation is less understood in
interacting theories, especially on time scales longer than the dynamical
relaxation time. Recently there have been claims that the quantum theory
deviates spectacularly from the classical theory on this time scale, even if
the occupancy numbers are extremely large. Furthermore, it is claimed that the
quantum theory quickly thermalizes while the classical theory does not. The
evidence for these claims comes from noticing a spectacular difference in the
time evolution of expectation values of quantum operators compared to the
classical micro-state evolution. If true, this would have dramatic consequences
for many important phenomena, including laboratory studies of interacting BECs,
dark matter axions, preheating after inflation, etc. In this work we critically
examine these claims. We show that in fact the classical theory can describe
the quantum behavior in the high occupancy regime, even when interactions are
large. The connection is that the expectation values of quantum operators in a
single quantum micro-state are approximated by a corresponding classical
ensemble average over many classical micro-states. Furthermore, by the ergodic
theorem, a classical ensemble average of local fields with statistical
translation invariance is the spatial average of a single micro-state. So the
correlation functions of the quantum and classical field theories of a single
micro-state approximately agree at high occupancy, even in interacting systems.
Furthermore, both quantum and classical field theories can thermalize, when
appropriate coarse graining is introduced, with the classical case requiring a
cutoff on low occupancy UV modes. We discuss applications of our results.
