Klein-Gordon equation for a particles in brane model

TL;DR

This paper derives a Klein-Gordon equation for a free particle in a brane universe model, linking conservation laws, symmetry, and chiral spin currents, highlighting spontaneous chiral symmetry breaking effects.

Contribution

It introduces a novel derivation of the Klein-Gordon equation in a brane context and connects it with spin current variations and chiral symmetry breaking.

Findings

Derived conservation laws on the brane using symmetry properties.

Established a form of the Klein-Gordon equation from the equations of motion.

Linked chiral spin current variation to spin connectivity derivatives.

Abstract

Brane model of universe is considered for a free particle. Conservation laws on the brane are obtained using the symmetry properties of the brane. Equation of motion is derived for a particle using variation principle from these conservation laws. This equation has a form of Klein-Gordon equation. Comparison of squared Dirac-Fok-Ivanenko equation for a spin particle with Klein-Gordon equation in curved space has given an expression for chiral spin current variation through the derivative of spin connectivity. This chiral spin current is anomalous spin current corresponding to spontaneous chiral symmetry breaking of massive particle in the space of KG equation solutions.