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

*Alexander P. Yefremov*^{1}

(1) Institute of Gravitation and Cosmology of Peoples' Friendship University of Russia, ul. Miklukho-Maklaya, 6, Moscow, 117198, Russia

### Abstract

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

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