Relativistic Dynamics of a Point as an Emergent Phenomenon in a Standing Wave System

Abstract

In this paper, problems of relativistic kinematics are studied, and a formula of infinite string oscillations U(x, t) = cos Ф(1)(x, t) cos S(x, t) with the phase motion Ф(1) = 0 in accordance with the arbitrarily given law x = X(t) (|v| < c, where v = X; c is the sound velocity) is obtained. The obtained formula allows us to trace the emergence of one-dimensional relativistic dynamics laws for a material point. It is shown that S(x, t) is a solution of Hamilton Jacobi equation and can be considered as the action of the "particle" with Ф(1) = 0 phase. The phase motion is likely to be influenced by the potential V (x, t) = p(t)(X(t) − x) (where p is a momentum of the velocity v), and obeys the Newton equation and the Hamilton equations. The function ψ = exp (iS) is a solution of the Schrodinger equation with a relativistic Hamiltonian operator with the potential V (x, t), in which the operator √−c2∇2 + m2c4 is expanded into series. In a non-relativistic case with the particle speed v ≪ c, this equation coincides with the normal Schrodinger equation. It is demonstrated that the relativistic equation is linked with the one-dimensional Dirac equation in the Foldy Wouthuysen representation under condition of magnetic field absence. Further, the possibility of complex structure objects introduction in the framework of a linear wave model is investigated.

DOI 10.14258/izvasu(2015)1.1-02