Quantum Theory, Groups and Representations: An Introduction

Description

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. Front Matter ....Pages i-xxii Introduction and Overview (Peter Woit)....Pages 1-14 The Group U(1) and its Representations (Peter Woit)....Pages 15-26 Two-state Systems and SU(2) (Peter Woit)....Pages 27-37 Linear Algebra Review, Unitary and Orthogonal Groups (Peter Woit)....Pages 39-53 Lie Algebras and Lie Algebra Representations (Peter Woit)....Pages 55-71 The Rotation and Spin Groups in Three and Four Dimensions (Peter Woit)....Pages 73-86 Rotations and the Spin \(\frac{1}{2}\) Particle in a Magnetic Field (Peter Woit)....Pages 87-101 Representations of SU(2) and SO(3) (Peter Woit)....Pages 103-124 Tensor Products, Entanglement, and Addition of Spin (Peter Woit)....Pages 125-138 Momentum and the Free Particle (Peter Woit)....Pages 139-147 Fourier Analysis and the Free Particle (Peter Woit)....Pages 149-164 Position and the Free Particle (Peter Woit)....Pages 165-180 The Heisenberg group and the Schrödinger Representation (Peter Woit)....Pages 181-188 The Poisson Bracket and Symplectic Geometry (Peter Woit)....Pages 189-198 Hamiltonian Vector Fields and the Moment Map (Peter Woit)....Pages 199-214 Quadratic Polynomials and the Symplectic Group (Peter Woit)....Pages 215-227 Quantization (Peter Woit)....Pages 229-236 Semi-direct Products (Peter Woit)....Pages 237-243 The Quantum Free Particle as a Representation of the Euclidean Group (Peter Woit)....Pages 245-257 Representations of Semi-direct Products (Peter Woit)....Pages 259-273 Central Potentials and the Hydrogen Atom (Peter Woit)....Pages 275-285 The Harmonic Oscillator (Peter Woit)....Pages 287-298 Coherent States and the Propagator for the Harmonic Oscillator (Peter Woit)....Pages 299-311 The Metaplectic Representation and Annihilation and Creation Operators, \(d=1\) (Peter Woit)....Pages 313-324 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d (Peter Woit)....Pages 325-339 Complex Structures and Quantization (Peter Woit)....Pages 341-356 The Fermionic Oscillator (Peter Woit)....Pages 357-363 Weyl and Clifford Algebras (Peter Woit)....Pages 365-372 Clifford Algebras and Geometry (Peter Woit)....Pages 373-381 Anticommuting Variables and Pseudo-classical Mechanics (Peter Woit)....Pages 383-393 Fermionic Quantization and Spinors (Peter Woit)....Pages 395-410 A Summary: Parallels Between Bosonic and Fermionic Quantization (Peter Woit)....Pages 411-412 Supersymmetry, Some Simple Examples (Peter Woit)....Pages 413-420 The Pauli Equation and the Dirac Operator (Peter Woit)....Pages 421-431 Lagrangian Methods and the Path Integral (Peter Woit)....Pages 433-445 Multiparticle Systems: Momentum Space Description (Peter Woit)....Pages 447-463 Multiparticle Systems and Field Quantization (Peter Woit)....Pages 465-475 Symmetries and Non-relativistic Quantum Fields (Peter Woit)....Pages 477-491 Quantization of Infinite dimensional Phase Spaces (Peter Woit)....Pages 493-501 Minkowski Space and the Lorentz Group (Peter Woit)....Pages 503-513 Representations of the Lorentz Group (Peter Woit)....Pages 515-525 The Poincaré Group and its Representations (Peter Woit)....Pages 527-539 The Klein–Gordon Equation and Scalar Quantum Fields (Peter Woit)....Pages 541-559 Symmetries and Relativistic Scalar Quantum Fields (Peter Woit)....Pages 561-572 U(1) Gauge Symmetry and Electromagnetic Fields (Peter Woit)....Pages 573-583 Quantization of the Electromagnetic Field: the Photon (Peter Woit)....Pages 585-603 The Dirac Equation and Spin \(\frac{1}{2}\) Fields (Peter Woit)....Pages 605-620 An Introduction to the Standard Model (Peter Woit)....Pages 621-626 Further Topics (Peter Woit)....Pages 627-629 Back Matter ....Pages 631-668