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  • © 2017

Quantum Theory, Groups and Representations

An Introduction

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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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  • ISBN: 978-3-319-64612-1
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Table of contents (49 chapters)

  1. Front Matter

    Pages i-xxii
  2. Introduction and Overview

    • Peter Woit
    Pages 1-14
  3. Two-state Systems and SU(2)

    • Peter Woit
    Pages 27-37
  4. Representations of SU(2) and SO(3)

    • Peter Woit
    Pages 103-124
  5. Momentum and the Free Particle

    • Peter Woit
    Pages 139-147
  6. Fourier Analysis and the Free Particle

    • Peter Woit
    Pages 149-164
  7. Position and the Free Particle

    • Peter Woit
    Pages 165-180
  8. Quantization

    • Peter Woit
    Pages 229-236
  9. Semi-direct Products

    • Peter Woit
    Pages 237-243

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 to the 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.

Keywords

  • Lie algebras
  • Lie groups
  • quantization
  • quantum fields
  • quantum mechanics
  • representation theory
  • Standard Model of particle physics
  • unitary group representations
  • two-state systems
  • Lie algebra representations
  • rotation and spin groups
  • momentum and free particle
  • fourier analysis and free particle
  • Schroedinger representation
  • Heisenberg group
  • Poisson bracket and symplectic geometry
  • Hamiltonian vector fields
  • quantum free particle
  • metaplectic representation
  • Fermionic oscillator

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.

Bibliographic Information

Buying options

eBook
USD 69.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-64612-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 89.99
Price excludes VAT (USA)
Hardcover Book
USD 89.99
Price excludes VAT (USA)