  Recent experiments by OPERA with high energy neutrinos, as well as
astrophysics observation data, may possibly prove violations of underlying
principles of special relativity theory. This paper attempts to present an
elementary modification of relativistic mechanics that is consistent both with
the principles of mechanics and with Dirac's approach to derivation of
relativistic quantum equations. Our proposed hyper-relativistic model is based
on modified dispersion relations between energy and momentum of a particle.
Predictions of the new theory significantly differ from the standard model, as
the former implies large Lorentz gamma-factors (ratio of particle energy to its
mass). First of all, we study model relationships that describe hypothetical
motion of superluminal neutrinos. Next, we analyze characteristics of Cherenkov
radiation of photons and non-zero mass particles in vacuum. Afterwards, we
derive generalized Lorentz transformations for a hyper-relativistic case,
resulting in a radical change in the law of composition of velocities and
particle kinematics. Finally, we study a hyper-relativistic version of Dirac
equation and some of its properties. In present paper we attempted to use plain
language to make it accessible not only to scientists but to undergraduate
students as well.
