Foundations of the Quaternion Quantum Mechanics

Symmetry

We present a quaternion representation of quantum mechanics that allows its ontological interpretation. The correspondence between classical and quaternion quantum equations permits one to consider the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase energy density, and as a result, generate gravity forcing and affect the wave speed. Consequently, gravity could be described by an index of refraction.

Advances in Manufacturing Science and Technology, 2020

Developed by French mathematician Augustin-Louis Cauchy, the classical theory of elasticity is the starting point to show the value and the physical reality of quaternions. The classical balance equations for the isotropic, elastic crystal, demonstrate the usefulness of quaternions. The family of wave equations and the diffusion equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic solid. Using the quaternion algebra, we present the derivation of the quaternion form of the multiple wave equations. The fundamental consequences of all derived equations and relations for physics, chemistry, and future prospects are presented.

From time to time, some eminent physicists commenced to ask: What is the reality behind quantum mechanical predictions? Is there a realism interpretation of Quantum Physics? This paper is intended to explore such a possibility of a realism interpretation of QM, based on a derivation of Maxwell equations in Quaternion Space. In this regards, we begin with Quaternion space and its respective Quaternion Relativity (it also may be called as Rotational Relativity) as it has been discussed in several papers including [1]. The purpose of the present paper is to review our previous derivation of Maxwell equations in Q-space [17], with discussion on some implications. First, we will review our previous results in deriving Maxwell equations using Dirac decomposition, introduced by Gersten (1999). Then we will shortly make a few remark on helical solutions of Maxwell equations, Smarandache’s Hypothesis and possible cosmological entanglement. Further observations are of course recommended to refute or verify some implications of this proposition.

We present quaternion quantum mechanics and its ontological interpretation. The theory combines the Cauchy model of the elastic continuum with the Planck-Kleinert crystal hypothesis. In this model, the universe is an ideal elastic solid where the elementary particles are soliton-like waves. Tension induced by the compression and twisting of the continuum affects its energy density and generates the force of gravity, as density changes alters the wave speed and hence gravity could be described by an index of refraction.

International Journal of Theoretical Physics, 1997

This paper is an attempt to simplify and clarify the mathematical language used to express quaternionic quantum mechanics (QQM). In our quaternionic approach the choice of “complex” geometries allows an appropriate definition of momentum operator and gives the possibility to obtain consistent formulations of standard theories. Barred operators represent the key to realizing a set of translation rules between quaternionic and complex quantum mechanics (QM). These translations enable us to obtain a rapid quaternionic counterpart of standard quantum mechanical results.

A quaternionic version of Quantum Mechanics is constructed using the Schwinger's formulation based on measurements and a Variational Principle. Commutation relations and evolution equations are provided, and the results are compared with other formulations.

2018

In the last article, an approach was developed to form an analogy of the wave function and derive analogies for both the mathematical forms of the Dirac and Klein-Gordon equations. The analogies obtained were the transformations from the classical real model forms to the forms in complex space. The analogous of the Klein-Gordon equation was derived from the analogous Dirac equation as in the case of quantum mechanics. In the present work, the forms of Dirac and Klein-Gordon equations were derived as a direct transformation from the classical model. It was found that the Dirac equation form may be related to a complex velocity equation. The Dirac's Hamiltonian and coefficients correspond to each other in these analogies. The Klein-Gordon equation form may be related to the complex acceleration equation. The complex acceleration equation can explain the generation of the flat spacetime. Although this approach is classical, it may show a possibility of unifying relativistic quantum mechanics and special relativity in a single model and throw light on the undetectable aether.

Quaternionic Quantum Wave Equation and Applications, 2019

The perception of duality nature in physics is extended to covers the idea of superposition of field and its dual field by assuming there is resultant field emerges as a superposition when a field and its dual field are combined. Quaternion formulation is used in this model. This generalization enables in constructing a general plane wave to generalize, Klein Gordon equation, Helmholtz's equation. It can be applied to electromagnetic and Meissner effect.

Provides an introduction to the concept and derivation of the Quaternions. Then derives several basic algebraic relationships involving quaternions, culminating with how quaternions can be used to describe 3d rotations. Then discusses the computational advantages that quaternions have over other 3d rotation methods, and provides an example of a classical mechanics rotation problem being solved with quaternions. Then discusses Frobenius's Theorem and the potential physical implication that it has. Overall aim is to provide a good educational resource for the intuition of quaternions and their applications.

2002

The present article is the first in a program that aims at generalizing quantum mechanics by keeping its structure essentially intact, but constructing the Hilbert space over a new number system much richer than the field of complex numbers. We call this number system ``the Quantionic Algebra''. It is eight dimensional like the algebra of octonions, but, unlike the latter, it is associative. It is not a division algebra, but "almost" one (in a sense that will be evident when we come to it). It enjoys the minimum of properties needed to construct a Hilbert space that admits quantum-mechanical interpretations (like transition probabilities), and, moreover, it contains the local Minkowski structure of space-time. Hence, a quantum theory built over the quantions is inherently relativistic. The algebra of quantions has been discovered in two steps. The first is a careful analysis of the abstract structure of quantum mechanics (the first part of the present work), the ...