Group Theory and Representation Theory for Identical Particles

TL;DR

This paper provides a comprehensive introduction to the group theory and representation theory underlying the quantum mechanics of identical particles, bridging concepts across condensed matter, quantum chemistry, and quantum computing.

Contribution

It offers a detailed development of the mathematics of identical particles and their descriptions in both first and second quantization, serving as a foundational resource.

Findings

Clarifies the role of symmetries in quantum systems of identical particles

Develops the mathematical framework for describing identical particles in various quantization schemes

Provides educational material for graduate-level understanding of quantum symmetries

Abstract

Few, if any, applications of quantum technology are as widely known as the quantum simulation of quantum matter. Consequently, many interesting questions have been sparked at the intersection of condensed matter, quantum chemistry, and quantum computing. Given the common mathematical foundation of these subjects, we walk through the necessary group theory and representation theory serving as background in all of these fields. Our discussion will include a full development of the mathematics of identical particles and the mechanics of describing systems of identical particles in both first and second quantization schemes. This chapter is an offshoot of a larger work that provides a graduate-level introduction to quantum information science. This chapter is being released separately because it is not explicitly focused on quantum information. It has grown beyond a short digression into a full-fledged journey into the symmetries and representations of identical particles that we invite you, the reader, to join.