Foundations of the Quaternion Quantum Mechanics.

Marek Danielewski,Lucjan Sapa

doi:10.3390/e22121424

Abstract

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

Highlights

Quantum mechanics is considered to be an irreducibly statistical theory, as a result unable to predict the behavior of individual processes
Quaternion quantum mechanics, quaternion quantum mechanics (QQM), presented by us is ontic in the sense that it answers the central question of interpretation of quantum mechanics
We present the fundamentals of the quaternion quantum theory, with the clear and precise specification of what the theory is basically about

Summary

Introduction

Quantum mechanics is considered to be an irreducibly statistical theory, as a result unable to predict the behavior of individual processes. It has been increasingly used, with stunning success, to gain control over individual objects on an atomic scale. This situation motivates the research into the foundations, leading to a variety of approaches towards an adequate theoretical justification of individual phenomena. It is not agreed whether such an interpretation requires a modification of the standard quantum formalism or whether it can be achieved within that formalism [1]. The main concepts of quaternion quantum mechanics (QQM) for both the general and mathematical audience are shown.

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