Foundations of the Quaternion Quantum Mechanics

Abstract

We show that the quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing 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.

On Non-Associative Rings

“…According to quantum mechanics, every quantum system corresponds to a separable extended Hilbert space. In [8,9], the Hilbert space is replaced with Hilbert module over the ring of all quaternions. Motivated by such extension of the mathematical formulation of quantum mechanics, i.e., by replacing the Hilbert space with the quaternionic Hilbert module as the state space of quantum mechanics, we will formulate in the next article another possible extension of quantum mechanics by replacing the Hilbert space with a Hilbert module over a nonassociative ring and investigate the implications of such replacement.…”