Pre-geometric structure of quantum and classical particles in terms of quaternion spinors
Alexander P. Yefremov1
(1) Institute of Gravitation and Cosmology of Peoples' Friendship University of Russia, ul. Miklukho-Maklaya, 6, Moscow, 117198, Russia
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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- J. A. Wheeler, "Pregeometry: motivations and prospects". In: A. R. Marlov (ed.), Quantum Theory and Gravitation (New York, Academic Press, 1980), p. 1–11.
- W. R. Hamilton, Lectures on Quaternions (Hodges and Smith, Dublin, 1853).
- A. P. Yefremov, The conic-gearing image of a complex number and a spinor-born surface geometry, Grav. Cosmol. 17(1), 1 (2011).
- A. P. Yefremov, The Schro"dinger equations as a consequence of the quaternion algebra stability, Grav. Cosmol. 18(4), 239 (2012).
- A. Sommerfeld, Atombau und Spectrallilien (Friedr., Vieweg und Sohn, Braunschweig, 1951).
- A. P. Yefremov, Quaternionic multiplication rule and a localQ-metric, Lett. Nuovo Cim. 37(8), 315 (1983).
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