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Group Representations & Quantum Theory

Vehbi Emrah Paksoy

Published Oct 24, 2024• 0 citations
OpenAlex

Abstract

What is the common thing between some integers and symmetries of an equilateral triangle? How studying rotations on a circle helps us understand quantum mechanics? Groups and their representations provide an amazing framework for many physics and chemistry theories. Understanding how certain groups act on physical systems, both classical and quantum, helps us to build models and make predictions. In this talk, we discover how abstract mathematical notions such as groups and representation theory and counterintuitive nature of quantum mechanics come together to guide us in our journey to explore the world we live in. This talk is designed to be accessible to any person with some understanding of calculus and a little bit of linear algebra.

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What is the common thing between some integers and symmetries of an equilateral triangle? How studying rotations on a circle helps us understand quantum mechanics? Groups and their representations provide an amazing framework for many physics and chemistry theories. Understanding how certain groups act on physical systems, both classical and quantum, helps us to build models and make predictions. In this talk, we discover how abstract mathematical notions such as groups and representation theory and counterintuitive nature of quantum mechanics come together to guide us in our journey to explore the world we live in. This talk is designed to be accessible to any person with some understanding of calculus and a little bit of linear algebra.
Abstract Group theory is a branch of mathematics appropriate for studying symmetries of physical systems. A symmetry is a natural attribute of the physical world and therefore is the starting point of any physical law. The foundation of dynamics is in Lagrangian theory or equivalently in Hamiltonian theory. A symmetry of a physical system is expressed by the invariance of its Lagrangian or Hamiltonian, or ultimately of its equations of motion, under certain transformations. When these transformations form groups it is convenient and useful to study symmetries through group theory. Group theory can explain observed regularities and helps to simplify and unify the physics of apparently different systems. It is a valuable tool that helps in predicting the behaviour of systems for which a symmetry has been found.
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.
Abstract This book provides an introduction to quantum theory, primarily for mathematics students. It assumes a knowledge of basic algebra and elementary group theory, with little or no familiarity with more advanced topics. Although it takes a traditional approach, the book exploits ideas of linear algebra and points out some of the mathematical subtleties of the theory. It also covers such topics as Bell's inequalities and coherent and squeezed states, and introduces group representation theory, algebraic quantum theory, and quantum statistical mechanics. Later chapters discuss relativistic wave equations and elementary particle symmetries from a group-theoretical standpoint.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
Group theory, originating from algebraic structures in mathematics, has long been a powerful tool in many areas of physics, chemistry and other applied sciences, but it has seldom been covered in a manner accessible to undergraduates. This book from renowned educator Robert Kolenkow introduces group theory and its applications starting with simple ideas of symmetry, through quantum numbers, and working up to particle physics. It features clear explanations, accompanying problems and exercises, and numerous worked examples from experimental research in the physical sciences. Beginning with key concepts and necessary theorems, topics are introduced systematically including: molecular vibrations and lattice symmetries; matrix mechanics; wave mechanics; rotation and quantum angular momentum; atomic structure; and finally particle physics. This comprehensive primer on group theory is ideal for advanced undergraduate topics courses, reading groups, or self-study, and it will help prepare graduate students for higher-level courses.
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.