Saturday, July 20, 2019

Quantum Theory, Groups and Representations: An Introduction

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This book began as course notes prepared for a class taught at Columbia University during the 2012-13 academic year. The intent was to cover the basics of quantum mechanics, up to and including relativistic quantum eld theory of free elds, from a point of view emphasizing the role of unitary representations of Lie groups in the foundations of the subject. It has been signi cantly rewritten and extended since that time, partially based upon experience teaching the same material during 2014-15.

The approach to this material is simultaneously rather advanced, using crucially some fundamental mathematical structures discussed, if at all, only in graduate mathematics courses, while at the same time trying to do this in as elementary terms as possible. The Lie groups needed are (with one crucial exception) ones that can be described simply in terms of matrices. Much of the representation theory will also just use standard manipulations of matrices. The only prerequisite for the course as taught was linear algebra and multivariable calculus (while a full appreciation of the topics covered would bene t from quite a bit more than this). My hope is that this level of presentation will simultaneously be useful to mathematics students trying to learn something about both quantum mechanics and Lie groups and their representations, as well as to physics students who already have seen some quantum mechanics, but would like to know more about the mathematics underlying the subject, especially that relevant to exploiting symmetry principles.

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 In nite 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
47. The Dirac Equation and Spin 1/2 Fields
48. An Introduction to the Standard Model
49. Further Topics
A. Conventions
B. Exercises

Author Details
"Peter Woit"

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