  An overview is given of the limitations of Luttinger liquid theory in
describing the real time equilibrium dynamics of critical one-dimensional
systems with nonlinear dispersion relation. After exposing the singularities of
perturbation theory in band curvature effects that break the Lorentz invariance
of the Tomonaga-Luttinger model, the origin of high frequency oscillations in
the long time behaviour of correlation functions is discussed. The notion that
correlations decay exponentially at finite temperature is challenged by the
effects of diffusion in the density-density correlation due to umklapp
scattering in lattice models.
