A New Formulation of Quantum Mechanics

TL;DR

This paper introduces a novel quantum mechanics formulation using differential commutator brackets, deriving a wave equation for fermions and exploring invariance properties of fundamental equations.

Contribution

It develops a new formalism based on differential commutator brackets, unifying aspects of Klein-Gordon, Schrödinger, and Dirac equations, and analyzing invariance under specific transformations.

Findings

Derived a wave equation for fermionic particles.

Showed invariance of Dirac and Maxwell equations under certain transformations.

Connected the new formalism with quaternionic formulations.

Abstract

A new formulation of quantum mechanics based on differential commutator brackets is developed. We have found a wave equation representing the fermionic particle. In this formalism, the continuity equation mixes the Klein-Gordon and Schrodinger probability density while keeping the Klein -Gordon and Schrodinger current unaltered. We have found time and space transformations under which Dirac's equation is invariant. The invariance of Maxwell's equations under these transformations shows that the electric and magnetic fields of a moving charged particle are perpendicular to the velocity of the propagating particle. This formulation agrees with the quaternionic formulation recently developed by Arbab.