Preface -- 1 Introduction and Overview -- 2 The Group U(1) and its R epresentations -- 3 Two-state Systems and SU(2) -- 4 Linear Algebra Rev iew, Unitary and Orthogonal Groups -- 5 Lie Algebras and Lie Algebra Re presentations -- 6 The Rotation and Spin Groups in 3 and 4 Dimensions - - 7 Rotations and the Spin 1/2 Particle in a Magnetic Field -- 8 Repres entations of SU(2) and SO(3) -- 9 Tensor Products, Entanglement, and Ad dition of Spin -- 10 Momentum and the Free Particle -- 11 Fourier Analy sis and the Free Particle -- 12 Position and the Free Particle -- 13 Th e Heisenberg group and the Schrödinger Representation -- 14 The Poisso n Bracket and Symplectic Geometry -- 15 Hamiltonian Vector Fields and t he Moment Map -- 16 Quadratic Polynomials and the Symplectic Group -- 1 7 Quantization -- 18 Semi-direct Products -- 19 The Quantum Free Partic le 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 Annih ilation and Creation Operators, d = 1 -- 25 The Metaplectic Representat ion and Annihilation and Creation Operators, arbitrary d -- 26 Complex Structures and Quantization -- 27 The Fermionic Oscillator -- 28 Weyl a nd Clifford Algebras -- 29 Clifford Algebras and Geometry -- 30 Anticom muting Variables and Pseudo-classical Mechanics -- 31 Fermionic Quantiz ation and Spinors -- 32 A Summary: Parallels Between Bosonic and Fermio nic Quantization -- 33 Supersymmetry, Some Simple Examples -- 34 The Pa uli Equation and the Dirac Operator -- 35 Lagrangian Methods and the Pa th 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 dimensio nal Phase Spaces -- 40 Minkowski Space and the Lorentz Group -- 41 Repr esentations of the Lorentz Group -- 42 The Poincaré Group and its Repr esentations -- 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 Electroma gnetic Field: the Photon -- 47 The Dirac Equation and Spin-1/2 Fields - - 48 An Introduction to the Standard Model -- 49 Further Topics -- A Co nventions -- B Exercises -- Index.

This text systematically presents the basics of quantum mechanics, e mphasizing the role of Lie groups, Lie algebras, and their unitary repr esentations. The mathematical structure of the subject is brought to th e fore, intentionally avoiding significant overlap with material from s tandard 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 t he mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of th e subject. The latter portions of the book focus on central mathematica l objects that occur in the Standard Model of particle physics, underli ning the deep and intimate connections between mathematics and the phys ical world. While an elementary physics course of some kind would be he lpful to the reader, no specific background in physics is assumed, maki ng this book accessible to students with a grounding in multivariable c alculus 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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