Abstract

Heuristic Schrodinger equation of quantum mechanics (QM) perfectly describes the micro-world phenomena, but remains an axiomatic hypothesis having no logical explanations and properly formulated correspondence principle with classical mechanics. However, resent inves-tigations demonstrate immanent presence of this basic QM law in pure mathematics of hyper-complex (quaternion – Q) numbers. We outline the logics and stages of this discovery.

A deeper analysis of the Q-algebra (whose “imaginary” units behave as a vectors initiating a 3D Cartesian frame) involving the spectral theorem of the theory of matrices reveals existence of a pre-geometric (fractal) surface, a “square-root slice” of a 3D space. Simple distortions (os-cillation and stretching) of the surface violate the Q-algebra multiplication; a condition provid-ing the algebra’s “eternal” stability takes the shape of a continuity-type math (unit-less) equa-tion comprising an arbitrary vector. We chose this vector as the oscillation gradient; these pure math actions have amazing results.

The stability condition fractalizes, and when written in the physical micro-world units, it becomes precisely the Schrodinger equation of QM. The particle’s wave function then is image of oscillating 2D area of the pre-geometric space (predicted by Wheeler [1]). Respective 3D physical object is a massive point rotating about an axis. In the “lab” conditions, the fractal equation converts precisely into the Hamilton-Jacobi equation of classical mechanics, the oscil-lation phase acquiring sense of the action function. Finally, constancy of particle’s rotational velocity leads to the Einstein’s relativistic mechanics.

Thus, the math of Q-numbers unites quantum, classical, and relativistic mechanics in one theory.