An Introduction
Peter Woit

#Quantum_Theory
#Representations
#mathematics
#calculations
#physics
#algebras
#quantum_mechanics
“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
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.









