On geometry influence on the behavior of a quantum mechanical scalar particle with intrinsic structure in external magnetic and electric fields

TL;DR

This paper develops a relativistic theory for a non-point-like scalar particle with intrinsic structure in curved space-time, analyzing its quantum behavior under external electromagnetic fields across various geometrical backgrounds.

Contribution

It introduces a new relativistic framework for Cox's scalar particle with intrinsic structure, including non-minimal interactions and exact solutions in specific curved geometries.

Findings

Exact solutions for the wave equation in Minkowski space with external fields.

Modification of quantum oscillator frequency due to particle structure.

Analysis of quantum motion in Lobachevsky and Riemann spaces.

Abstract

Relativistic theory of the Cox's scalar not point-like particle with intrinsic structure is developed on the background of arbitrary curved space-time. It is shown that in the most general form, the extended Proca-like tensor first order system of equations contains non minimal interaction terms through electromagnetic tensor F_{\alpha \beta} and Ricci tensor R_{\alpha \beta}. In relativistic Cox's theory, the limiting procedure to non-relativistic approximation is performed in a special class of curved space-time models. This theory is specified in simple geometrical backgrounds: Euclid's, Lobachevsky's, and Rie\-mann's. Wave equation for the Cox's particle is solved exactly in presence of external uniform magnetic and electric fields in the case of Minkowski space. Non-trivial additional structure of the particle modifies the frequency of a quantum oscillator arising effectively in presence if external magnetic field. Extension of these problems to the case of the hyperbolic Lobachevsky space is examined. In presence of the magnetic field, the quantum problem in radial variable has been solved exactly; the quantum motion in z-direction is described by 1-dimensional Schr\"{o}dinger-like equation in an effective potential which turns out to be too difficult for analytical treatment. In the presence of electric field, the situation is similar. The same analysis has been performed for spherical Riemann space model.