# The General Theory of Particle Mechanics. A Special Course: results of more than 30-year research are published

To reach the goal he selects a specific way atypical for formulation regularities of the physical world; it has nothing to do with experiment or at least with variation procedures normally utilized by theoreticians to derive equations of motion or of a field (although no one can convincingly substantiate reckless use of these procedures). Instead, the author takes for the basis of his considerations a pure mathematical object, a set of quaternions, exotic numbers discovered in XIX century; they have four units and non-symmetric multiplication.

Two reasons support use of quaternions. They form the last good – exclusive – algebra hence deserve attention. Moreover, quaternions are “very geometric” since their three vector units in a way describe the 3D space as a coordinate system.

At this point, the author makes an unexpected move. He finds a unique two-dimensional subspace whose elements in square combinations provide restoration of the 3D space, and consequently the whole quaternion algebra. Then he slightly deforms the subspace and looks for a condition leaving the algebra stable.

The most strange (even incredible) fact is that the sought for condition, a pure mathematical differential equation, when written in physical units takes precise shape of the Schrodinger equation of quantum mechanics! Moreover, a picture of the deformed subspace elements acquire the sense of a geometric image of the wave function, traditionally an abstract “faceless” entity.

Then triggering to the macro-length scale the author reduces the stability condition to equations of classical mechanics. It becomes possible only if the phase of the subspace deformation nearly stops changing, probably taking a minimal value. Necessarily, this leads to identifying the phase with the so-called “action”, suggested by Pierre Maupertuis in XVIII century; up to now, it has been remained an enigmatic function providing derivation of Newton’s dynamic equations within the realm of analytical mechanics, which the Nobel Prize winner Eugene Wigner used to call incomprehensible.

In the Yefremov’s math-born “synthetic mechanics”, the basic correlations of analytical mechanics arise automatically together with a strange “dual” model of a particle. Here the classical massive particle has at once two “faces”.

The first one is imaged as an unusual structure “hardly working” in the subspace; resembling a conic-gearing mechanism, it pumps over (with a certain frequency) a specific “load” from a real sector of the subspace to its imaginary sector, and vice versa. Objectively speaking, this picture interpreted by the author as a new image of a complex (or quaternion) number, in fact is a (rarely clear) display of an oscillating 2D spinor.

The second particle’s “face” is built out of “squared” spinor elements. We see it in the 3D space as a mass confined in a very small volume, nearly a point; however, this point-like particle rotates about one axis with frequency twice as great as that of the spinor oscillation.

At this stage, the author undertakes another non-conventional step. He “takes a microscope” and starts examining the moving in space and rotating about itself “nearly-point mass” from a closer distance. He discovers that a border-point of the uniformly moving particle depicts a helix line, and if the point’s speed is fixed, then the helix arc is described precisely as the Einstein’s space-time interval giving birth to special relativistic mechanics.

Geometry of the rotating and non-uniformly moving particle generates a dynamic equation characteristic to general relativity. In the small speed approximation, this theory expectedly reduces to classical mechanics but as well gives – and it is really a surprising fact – the energy-frequency and momentum-wave vector correlations postulated by De Broglie at the earliest times of quantum mechanics.

On this, the circle closes. From the math, saving quaternion algebra, the author deduces equation for a quantum particle, analyzing it at the lab scale he arrives to Newtonian dynamics, the classical particle’s fast rotation leads to relativity, which in its weak version returns the basic classical-quantum correlations. By the way, in this theory the rest-mass energy exists due to the particle’s proper rotation.

The logic of this theory looks good although some actions and interpretations differ from the generally accepted. So, all aspects of the traditional axiomatic approach to quantum mechanics is fully ignored. Nothing is said about the properties of operators; the wave function is claimed to describe a certain “square root of relative mass”, not a probability amplitude. The Schrodinger equation decay into an interior mass-distribution equation and exterior classical Hamilton-Jacobi equations (on the lab scale) looks somewhat artificial. The geometric “helix-line approach” to relativity introduces a cylindrical spiral space line instead of the habitual fourth coordinate thus deleting time from the list of real physical directions.

However, the author evidently does not regard his theory as the ultimate truth rather taking it for an interesting phenomenon and an acceptable way to learn mechanics cursory. That is why he composes his book as a “special course”, aimed, first to display the study of quaternion, and then, as a consequence, to consider the laws of mechanics born from the math. In each chapter, he inserts detailed methodical supplements, numerous typical and test problems and questions (may be too much).

As a whole, the book makes a strange impression. Correct mathematically it nonetheless too radically touches on the most fundamental positions in conventional physics. Normally such “crazy ideas” can hardly break through traditional time-tested approaches to apprehend the scientific facts. Generations of teachers and students have to be gone until such “an easy way” to study physics as suggested in the Yefremov’s book could be accepted. If that ever happens.