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Quantum Theory, Groups and Representations An Introduction
Author
Title
Quantum Theory, Groups and Representations [electronic resource] : An Introduction / by Peter Woit.
ISBN
9783319646121
Publication
Cham : Springer International Publishing : Imprint: Springer, 2017.
Physical Description
XXII, 668 p. 27 illus : online resource.
Local Notes
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Summary
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.
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Format
Books / Online
Language
English
Added to Catalog
December 01, 2017
Contents
Preface
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 3 and 4 Dimensions
7 Rotations and the Spin 1/2 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 Schrödinger 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 Weyl 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 Multi-particle Systems: Momentum Space Description
37 Multi-particle 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 Poincaré 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 Field
46 Quantization of the Electromagnetic Field: the Photon
47 The Dirac Equation and Spin-1/2 Fields
48 An Introduction to the Standard Model
49 Further Topics
A Conventions
B Exercises
Index.
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