Pre-geometric structure of quantum and classical particles in terms of quaternion spinors

TL;DR

This paper presents a quaternion-based geometric framework for particles, deriving quantum and classical equations from surface dyad transformations, linking geometric stability to fundamental physics equations like Schrödinger and Hamilton-Jacobi.

Contribution

It introduces a novel quaternion spinor model connecting surface dyads to quantum and classical particle equations through stability conditions.

Findings

Dyad vectors squared form quaternion units.

Stability condition yields Schrödinger, conservation, and Hamilton-Jacobi equations.

Particle dynamics include spin effects via vector field influence.

Abstract

It is shown that dyad vectors on a local domain of complex-number valued surface, when squared, form a set of four quaternion algebra units. A model of proto-particle is built by the dyad's rotation and stretching; this transformation violates metric properties of the surface, but the defect is cured by a stability condition for normalization functional over an abstract space. If the space is the physical one then the stability condition is precisely Schrodinger equation; separated real and imaginary parts of the condition are respectively equation of mass conservation and Hamilton-Jacoby equation. A 3D particle (composed of the proto-particle's parts) has to be conceived as a rotating massive point, its Lagrangian automatically becoming that of relativistic classical particle, energy and momentum proportional to Planck constant. In uence of a vector field onto the particle's propagation causes automatic appearance of Pauli spin term in Schrodinger equation.