The concept of a quasi-particle and the non-probabilistic interpretation of wave mechanics

In recent works of the author [Found. Phys. 36 (2006) 1701-1717; Math. Comput. Simul. 74 (2007) 93-103], the argument has been made that Hertz's equations of electrodynamics reflect the material invariance (indifference) of the latter. Then the principle of material invariance was postulated in lieu of Lorentz covariance, and the respective absolute medium was named the metacontinuum. Here, we go further to assume that the metacontinuum is a very thin but very stiff 3D hypershell in the 4D space. The equation for the deflection of the shell along the fourth dimension is the ''master'' nonlinear dispersive equation of wave mechanics whose linear part (Euler-Bernoulli equation) is nothing else but the Schrodinger wave equation written for the real or the imaginary part of the wave function. The wave function has a clear non-probabilistic interpretation as the actual amplitude of the flexural deformation. The ''master'' equation admits solitary-wave solutions/solitons that behave as quasi-particles (QPs). We stipulate that particles are our perception of the QPs (schaumkommen in Schrodinger's own words). We show the passage from the continuous Lagrangian of the field to the discrete Lagrangian of the centers of QPs and introduce the concept of (pseudo)mass. We interpret the membrane tension as an attractive (gravitational?) force acting between the QPs. Thus, a self-consistent unification of electrodynamics, wave mechanics, gravitation, and the wave-particle duality is achieved.