Woit P. Quantum Theory, Groups and Representations: An Introduction

Springer International Publishing AG, 2017. — 616 p. — ISBN: 3319646109.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.Introduction and Overview
The Group U(1) and its Representations
Two-state Systems and SU(2)
Linear Algebra Review, Unitary and Orthogonal Groups
Lie Algebras and Lie Algebra Representations
The Rotation and Spin Groups in Three and Four Dimensions
Rotations and the Spin
Particle in a Magnetic Field
Representations of SU(2) and SO(3)
Tensor Products, Entanglement, and Addition of Spin
Momentum and the Free Particle
Fourier Analysis and the Free Particle
Position and the Free Particle
The Heisenberg group and the Schrödinger Representation
The Poisson Bracket and Symplectic Geometry
Hamiltonian Vector Fields and the Moment Map
Quadratic Polynomials and the Symplectic Group
Quantization
Semi-direct Products
The Quantum Free Particle as a Representation of the Euclidean Group
Representations of Semi-direct Products
Central Potentials and the Hydrogen Atom
The Harmonic Oscillator
Coherent States and the Propagator for the Harmonic Oscillator
The Metaplectic Representation and Annihilation and Creation Operators,
The Metaplectic Representation and Annihilation and Creation Operators, arbitrary
Complex Structures and Quantization
The Fermionic Oscillator
Weyl and Clifford Algebras
Clifford Algebras and Geometry
Anticommuting Variables and Pseudo-classical Mechanics
Fermionic Quantization and Spinors
A Summary: Parallels Between Bosonic and Fermionic Quantization
Supersymmetry, Some Simple Examples
The Pauli Equation and the Dirac Operator
Lagrangian Methods and the Path Integral
Multiparticle Systems: Momentum Space Description
Multiparticle Systems and Field Quantization
Symmetries and Non-relativistic Quantum Fields
Quantization of Infinite dimensional Phase Spaces
Minkowski Space and the Lorentz Group
Representations of the Lorentz Group
The Poincaré Group and its Representations
The Klein–Gordon Equation and Scalar Quantum Fields
Symmetries and Relativistic Scalar Quantum Fields
U(1) Gauge Symmetry and Electromagnetic Fields
Quantization of the Electromagnetic Field: the Photon
The Dirac Equation and Spin
An Introduction to the Standard Model