Quantization in relativistic classical mechanics: the Stueckelberg
  equation, neutrino oscillation and large-scale structure of the Universe
  Based on the Chetaev theorem on stable dynamical trajectories in the presence
of dissipative forces, we obtain the generalized condition for stability of
relativistic classical Hamiltonian systems (with an invariant evolution
parameter) in the form of the Stueckelberg equation. As is known, this equation
is the basis of a competing paradigm known as parametrized relativistic quantum
mechanics (pRQM). It is shown that the energy of dissipative forces, which
generate the Chetaev generalized condition of stability, coincides exactly with
Bohmian relativistic quantum potential. We show that the squared amplitude of a
wave function in the Stueckelberg equation is equivalent to the probability
density function for the number of particle trajectories, relative to which the
velocity and the position of the particle are not hidden parameters.
  The conditions for reasonableness of trajectory interpretation of pRQM are
discussed. On basis of analysis of a general formalism for vacuum-flavor mixing
of neutrino within the context of the standard and pRQM models we show that the
corresponding expressions for the probability of transition from one neutrino
flavor to another differ appreciably, but they are experimentally testable: the
estimations of absolute value for neutrino mass based on modern experimental
data for solar and atmospheric neutrinos show that the pRQM results have a
preference. It is noted that the selection criterion of mass solutions relies
on proximity between the average size of condensed neutrino clouds, which is
described by the Muraki formula (29th ICRC, 2005) and depends on the neutrino
mass, and the average size of typical observed void structure (dark matter +
hydrogen gas), which plays the role of characteristic dimension of large-scale
structure of the Universe.
