  We show that Hamiltonian nonlinear dispersive wave systems with cubic
nonlinearity and random initial data develop, during their evolution, anomalous
correlators. These are responsible for the appearance of "ghost" excitations,
i.e. those characterized by negative frequencies, in addition to the positive
ones predicted by the linear dispersion relation. We use generalization of the
Wick's decomposition and the wave turbulence theory to explain theoretically
the existence of anomalous correlators. We test our theory on the celebrated
$\beta$-Fermi-Pasta-Ulam-Tsingou chain and show that numerically measured
values of the anomalous correlators agree, in the weakly nonlinear regime, with
our analytical predictions. We also predict that similar phenomena will occur
in other nonlinear systems dominated by nonlinear interactions, including
surface gravity waves. Our results pave the road to study phase correlations in
the Fourier space for weakly nonlinear dispersive wave systems.
