A Pedestrian Introduction to the Mathematical Concepts of Quantum Physics

TL;DR

This paper provides an accessible introduction to the core mathematical structures of quantum physics, emphasizing algebraic and Hilbert space methods, with examples like the quantum harmonic oscillator and gauge symmetries.

Contribution

It offers a pedagogical overview focusing on algebraic and geometric aspects of quantum theory, bridging to path integrals and gauge invariance, suitable for beginners.

Findings

Clarifies the algebraic structure of quantum mechanics.

Connects quantum concepts to gauge symmetries and path integrals.

Prepares readers for relativistic quantum field theories.

Abstract

These notes offer a basic introduction to the primary mathematical concepts of quantum physics, and their physical significance, from the operator and Hilbert space point of view, highlighting more what are essentially the abstract algebraic aspects of quantisation in contrast to more standard treatments of such issues, while also bridging towards the path integral formulation of quantisation. A discussion of the (first) Noether theorem and Lie symmetries is also included to complement the presentation. Emphasis is put throughout, as illustrative examples threading the presentation, on the quantum harmonic oscillator and the dynamics of a charged particle coupled to the electromagnetic field, with the ambition to bring the reader onto the threshold of relativistic quantum field theories with their local gauge invariances as a natural framework for describing relativistic quantum particles in interaction and carrying specific conserved charges.