Operator formalism of quantum mechanics

TL;DR

This paper introduces a novel operator formalism for quantum mechanics based on symmetry principles, deriving standard representations without classical mechanics references, and explains particle properties via group theory.

Contribution

It presents an unconventional approach to quantum mechanics using symmetry principles and covariance systems, offering a new mathematical foundation and interpretation.

Findings

Derives standard quantum mechanics representation from symmetry principles

Explains spin and mass as labels of Galilei group representations

Provides a new operator formalism based on covariance systems

Abstract

This is the first chapter of a new and unconventional textbook on quantum mechanics and quantum field theory. The chapter introduces standard quantum mechanics by means of a symmetry principle, without reference to classical mechanics. The mathematical foundation of this approach comes from a recent paper of Naudts and Kuna on covariance systems. The standard representation of quantum mechanics is derived. Next, spin and mass of a quantum particle are explained as labels of projective representations of the Galilei group.