The conic-gearing image of a complex number and a spinor-born surface geometry

A.P. Yefremov1

Abstract

Quaternion (Q-) mathematics formally contains many fragments of physical laws; in particular, the Hamiltonian for the Pauli equation automatically emerges in a space with Q-metric. The eigenfunction method shows that any Q-unit has an interior structure consisting of spinor functions; this helps us to represent any complex number in an orthogonal form associated with a novel geometric image (the conic-gearing picture). Fundamental Q-unit-spinor relations are found, revealing the geometric meaning of spinors as Lame coefficients (dyads) locally coupling the base and tangent surfaces.

References

  1. P. Fjelstad, Extending special relativity via the perplex numbers, Am. J. Phys. 54 (5), 416-422 (1986).
  2. A. P. Yefremov, Quaternions: Algebra, geometry, and physical theories, Hypercomplex Numbers in Geometry and Physics, 1, 104-119 (2004). Available online in < http://hypercomplex.xpsweb.com/articles/147/en/pdf/01-10-e.pdf>.
  3. A. P. Yefremov, Quaternion model of relativity: solutions for non-inertial motions and new effects, Adv. Sci. Lett. 1, 179-186 (2008).
  4. W. R. Hamilton, Lectures on Quaternions (Dublin, Hodges & Smith, 1853). Available online in < http://www.archive.org/details/lecturesonquater00hami>.
  5. A. P. Yefremov, Structure of hypercomplex units and exotic numbers as sections of bi-quaternions, Adv. Sci. Lett. 3, 537-542 (2010).
  6. R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comm. Math. Helv. 4, 9-20 (1932).
  7. A. P. Yefremov, Quaternionic multiplication rule and a local Q-metric, Lett. Nuovo Cim. 37 (8), 315-316 (1983).
For more information about this paper please visit Springer's Home Page of this paper.


Back to The Contents Page