Quantum Theory, Groups and Representations

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.

Review

“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)

Contents

1 Introduction and Overview

2 The Group U(1) and its Representations

3 Two -state Systems and SU(2)

4 Linear Algebra Review, Unitary and Orthogonal Groups

5 Lie Algebras and Lie Algebra Representations

6 The Rotation and Spin Groups in Three and Four Dimensions

7 Rotations and the Spin 12 Particle in a Magnetic Field

8 Representations of SU(2) and SO(3)

9 Tensor Products, Entanglement, and Addition of Spin

10 Momentum and the Free Particle

11 Fourier Analysis and the Free Particle

12 Position and the Free Particle

13 The Heisenberg group and the Schrodinger Representation

14 The Poisson Bracket and Symplectic Geometry

15 Hamiltonian Vector Fields and the Moment Map

16 Quadratic Polynomials and the Symplectic Group

17 Quantization

18 Semi-direct Products

19 The Quantum Free Particle as a Representation of the Euclidean Group

20 Representations of Semi-direct Products

21 Central Potentials and the Hydrogen Atom

22 The Harmonic Oscillator

23 Coherent States and the Propagator for the Harmonic Oscillator

24 The Metaplectic Representation and Annihilation and Creation Operators, d=1

25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d

26 Complex Structures and Quantization

27 The Fermionic Oscillator

28 Wey! and Clifford Algebras

29 Clifford Algebras and Geometry

30 Anticommuting Variables and Pseudo -classical Mechanics

31 Fermionic Quantization and Spinors

32 A Summary: Parallels Between Bosonic and Fermionic Quantization

33 Supersymmetry, Some Simple Examples

34 The Pauli Equation and the Dirac Operator

35 Lagrangian Methods and the Path Integral

36 Multiparticle Systems: Momentum Space Description

37 Multiparticle Systems and Field Quantization

38 Symmetries and Non-relativistic Quantum Fields

39 Quantization of Infinite dimensional Phase Spaces

40 Minkowski Space and the Lorentz Group

41 Representations of the Lorentz Group

42 The Poincare Group and its Representations

43 The Klein–Gordon Equation and Scalar Quantum Fields

44 Symmetries and Relativistic Scalar Quantum Fields

45 U(1) Gauge Symmetry and Electromagnetic Fields

46 Quantization of the Electromagnetic Field: the Photon

46 Quantization of the Electromagnetic Field: the Photon

47 The Dirac Equation and Spin 12 Fields

48 An Introduction to the Standard Model

49 Further Topics

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.