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Quantum Theory, Groups and Representations

An Introduction

  • Textbook
  • © 2017

Overview

Authors:
  1. Peter Woit

    1. Department of Mathematics, Columbia University, New York, USA

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  • Systematically emphasizes the role of Lie groups, Lie algebras, and their unitary representation theory in the foundations of quantum mechanics
  • Introduces fundamental structures and concepts of representation theory in an elementary, physically relevant context
  • Gives a careful treatment of the mathematical subtleties of quantum theory, without obscuring its global mathematical shape
  • Includes supplementary material: sn.pub/extras

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Table of contents (49 chapters)

  1. Front Matter

    Pages i-xxii
    Download chapter PDF

  2. Introduction and Overview

    • Peter Woit
    Pages 1-14

  3. The Group U(1) and its Representations

    • Peter Woit
    Pages 15-26

  4. Two-state Systems and SU(2)

    • Peter Woit
    Pages 27-37

  5. Linear Algebra Review, Unitary and Orthogonal Groups

    • Peter Woit
    Pages 39-53

  6. Lie Algebras and Lie Algebra Representations

    • Peter Woit
    Pages 55-71

  7. The Rotation and Spin Groups in Three and Four Dimensions

    • Peter Woit
    Pages 73-86

  8. Rotations and the Spin \(\frac{1}{2}\) Particle in a Magnetic Field

    • Peter Woit
    Pages 87-101

  9. Representations of SU(2) and SO(3)

    • Peter Woit
    Pages 103-124

  10. Tensor Products, Entanglement, and Addition of Spin

    • Peter Woit
    Pages 125-138

  11. Momentum and the Free Particle

    • Peter Woit
    Pages 139-147

  12. Fourier Analysis and the Free Particle

    • Peter Woit
    Pages 149-164

  13. Position and the Free Particle

    • Peter Woit
    Pages 165-180

  14. The Heisenberg group and the Schrödinger Representation

    • Peter Woit
    Pages 181-188

  15. The Poisson Bracket and Symplectic Geometry

    • Peter Woit
    Pages 189-198

  16. Hamiltonian Vector Fields and the Moment Map

    • Peter Woit
    Pages 199-214

  17. Quadratic Polynomials and the Symplectic Group

    • Peter Woit
    Pages 215-227

  18. Quantization

    • Peter Woit
    Pages 229-236

  19. Semi-direct Products

    • Peter Woit
    Pages 237-243

  20. The Quantum Free Particle as a Representation of the Euclidean Group

    • Peter Woit
    Pages 245-257
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Keywords

About this book

This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful tothe reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.

Reviews

“The book presents a large variety of important subjects, including the basic principles of quantum mechanics … . This good book is recommended for mathematicians, physicists, philosophers of physics, researchers, and advanced students in mathematics and physics, as well as for readers with some elementary physics, multivariate calculus and linear algebra courses.” (Michael M. Dediu, Mathematical Reviews, June, 2018)

Authors and Affiliations

  • Department of Mathematics, Columbia University, New York, USA

    Peter Woit

About the author

Peter Woit is a Senior Lecturer of Mathematics at Columbia University. His general area of research interest is the relationship between mathematics, especially representation theory, and fundamental physics, especially quantum field theories like the Standard Model.

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