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

Alexander P. Yefremov1

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.

References

  1. J. A. Wheeler, "Pregeometry: motivations and prospects". In: A. R. Marlov (ed.), Quantum Theory and Gravitation (New York, Academic Press, 1980), p. 111.
  2. W. R. Hamilton, Lectures on Quaternions (Hodges and Smith, Dublin, 1853).
  3. A. P. Yefremov, The conic-gearing image of a complex number and a spinor-born surface geometry, Grav. Cosmol. 17(1), 1 (2011).
  4. A. P. Yefremov, The Schro"dinger equations as a consequence of the quaternion algebra stability, Grav. Cosmol. 18(4), 239 (2012).
  5. A. Sommerfeld, Atombau und Spectrallilien (Friedr., Vieweg und Sohn, Braunschweig, 1951).
  6. A. P. Yefremov, Quaternionic multiplication rule and a localQ-metric, Lett. Nuovo Cim. 37(8), 315 (1983).
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