  Starting from a generic generally covariant classical theory we introduce the
logarithmic correction to the quantum wave equation. We demonstrate the
emergence of the evolution time from the group of automorphisms of the von
Neumann algebra governed by this non-linear correction. It turns out that such
time parametrization is essentially energy-dependent and becomes global only
asymptotically - when the energies get very small comparing to the effective
quantum gravity scale. Similar thing happens to the Lorentz invariance - in the
resulting theory it becomes an asymptotic low-energy phenomenon. We show how
the logarithmic non-linearity deforms the vacuum wave dispersion relations and
explains certain features of the astrophysical data coming from recent
observations of high-energy cosmic rays. In general, the estimates imply that
ceteris paribus the particles with higher energy propagate slower than those
with lower one, therefore, for a high-energy particle the mean free path,
lifetime in a high-energy state and, therefore, travel distance from the source
can be significantly larger than one would expect from the conventional theory.
Apart from this, we discuss also the possibility and conditions of the
transluminal phenomena in the physical vacuum such as the Cherenkov-type shock
waves.
