Symmetry of Differential Equations and Quantum Theory

TL;DR

This paper explores the relationship between symmetry groups of differential equations in physics and the mathematical structures used in quantum mechanics, proposing new quaternion-based approaches.

Contribution

It demonstrates the connection between symmetry groups and the mathematical nature of physical quantities, and introduces correct quaternion-based methods for quantum mechanics.

Findings

Main postulate of quantum mechanics proven via symmetry analysis

High symmetry of Maxwell equations linked to quaternion matrices

Incorrect application of quaternions due to non-abelian group properties

Abstract

The symmetry study of main differential equations of mechanics and electrodynamics has shown, that differential equations, which are invariant under transformations of groups, which are symmetry groups of mathematical numbers (considered within the frames of the number theory) determine the mathematical nature of the quantities, incoming in given equations. It allowed to proof the main postulate of quantum mechanics, consisting in that, that to any mechanical quantity can be set up into the correspondence the Hermitian matrix by quantization. High symmetry of Maxwell equations allows to show, that to quantities, incoming in given equations can be set up into the correspondence the Quaternion (twice-Hermitian) matrix by their quantization. It is concluded, that the equations of the dynamics of mechanical systems are not invariant under transformations of quaternion multiplicative group and, consecuently, direct application of quaternions with usually used basis \{e, i, j, k \} to build the new version of quantum mechanics, which was undertaken in the number of modern publications, is incorrect. It is the consequence of non-abelian character of given group. At the same time we have found the correct ways for the creation of the new versions of quantum mechanics on the quaternion base by means of choice of new bases in quaternion ring, from which can be formed the bases for complex numbers under multiplicative groups of which the equations of the dynamics of mechanical systems are invariant.