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Soviet physics. Doklady

Two-dimensional model of the relativistic quantum mechanics of a system of N particles with potential interaction

Skiff N. Sokolov

Published Apr 1, 1975• 0 citations
OpenAlex

Abstract

A two-dimensional model of relativistic quantum mechanics is suggested which permits any number of particles and a wide class of potentials and is as general as the two-dimensional nonrelativistic quantum mechanics. This proposed model is an example of the application of given axioms. (JFP)

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A nonrelativistic model is investigated that is a certain generalization of quantum mechanics of two particles interacting with a potential. A new point interaction of the particles is introduced, which involves their transition into another particle that has the same quantum numbers as the bound state of the first particles. In the theory thus developed, the mass of the introduced particle and its interaction constant are arbitrary. In the particular case when the mass of the introduced particle is chosen so that its mass exactly equals that of the bound state of the first particles, the theory is identical with ordinary quantum mechanics. The renormalization constants in this case vanish. (auth)
In this paper, some concepts of nonrelativistic many-particle quantum mechanics (e.g., product states, density matrix) are generalized to the relativistic domain using a framework called relativistic Schr\"odinger theory (RST). By using a general ansatz, the RST framework is simplified considerably and some of its field equations are solved directly. The RST approach is then compared with nonrelativistic quantum mechanics (QM) for the case of a product state (conventional QM) and its RST analog. It is shown that relativistic wave equations can be derived from the RST formalism, so that they coincide in the nonrelativistic limit with the well-known Hartree equations.
The idea of a single particle cannot be retained in relativistic quantum mechanics if there are interactions which lead to creation and annihilation of pair of particles. Therefore, the number of particles is not conserved in the relativistic theory. Most of the applications of quantum mechanics are in the realm of the three phases of matters: gases, liquids and solids. In these cases a nonrelativistic quantum theory is sufficient. However, in principle, a strictly correct treatment should include relativity. In 1928 Paul Adrian Maurice Dirac introduced a wave equation for a relativistic electron. The KG equation without mass corresponds to elastic stretching of an isolated vortex filament in the ideal fluid. The wave function has the meaning of the curve's position vector. The mass part of the KG equation describes the rotation of the helical curve about the screw axis due to the hydrodynamic self-inductor of the bent vortex filament.
In this article, the axioms presented in the first one are reformulated according to the special theory of relativity. Using these axioms, quantum mechanic's relativistic equations are obtained in the presence of electromagnetic fields for both the density function and the probability amplitude. It is shown that, within the present theory's scope, Dirac's second order equation should be considered the fundamental one in spite of the first order equation. A relativistic expression is obtained for the statistical potential. Axioms are again altered and made compatible with the general theory of relativity. These postulates, together with the idea of the statistical potential, allow us to obtain a general relativistic quantum theory for {\it ensembles} composed of single particle systems.
In this paper, we examine quantum systems with relativistic dynamics. We show that for a successful description of these systems, the application of Galilei invariant nonrelativistic Hamiltonian is necessary. To modify this Hamiltonian to relativistic dynamics, we require precise relativistic kinetic energy operators instead of nonrelativistic ones for every internal (Jacobi) coordinate. Finally, we introduce and investigate the Schrödinger equation with relativistic dynamics for two-particle systems with harmonic oscillator and Coulomb potentials.