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Topics in contemporary mathematical physics / Kai S. Lam..

By: Material type: TextTextPublication details: Singapore : World Scientific, ©2016.Edition: 2nd edDescription: xviii, 832 p. : illustrations ; 26 cmISBN:
  • 9789814667807 (pbk. : alk. paper)
Subject(s): DDC classification:
  • 530.15 23 L213
Contents:
1. Vectors and Linear Transformations -- 2. Tensors -- 3. Symmetry and Conservation: the Angular Momentum -- 4. The Angular Momentum as Generators of Rotations: Lie Groups and Lie Algebras -- 5. Algebraic Structures -- 6. Basic Group Concepts -- 7. Basic Lie Algebra Concepts -- 8. Inner Products, Metrics, and Dual Spaces -- 9. SO(4) and the Hydrogen Atom -- 10. Adjoints and Unitary Transformations -- 11. The Lorentz Group and SL(2,C) -- 12. The Dirac Bracket Notation in Quantum Theory -- 13. The Quantum Mechanical Simple Harmonic Oscillator -- 14. Fourier Series and Fourier Transforms, the Dirac Delta Function, Green's Functions -- 15. The Continuous Spectrum and Non-normalizable States -- 16. Skew-Symmetric Tensors and Determinants -- 17. Eigenvalue Problems -- 18. Group Representation Theory -- 19. The Dihedral Group D6 and the Benzene Molecule --20. Representations of the Symmetric Groups and the General Linear Groups, Young Diagram -- 21. Irreducible Representations of U(n), SL(n), SU(n) and O(n) -- 22. Irreducible Representations of SU(2) and SO(3) -- 23. The Spherical Harmonics -- 24. The Structure of Semisimple Lie Algebras -- 25. The Representations of Semisimple Lie Algebras -- 26. SU(3) and the Strong Interaction -- 27. Clifford Algebras -- 28. Exterior Products -- 29. The Hodge-Star Operator -- 30. Differential Forms and Exterior Differentiation -- 31. Moving Frames and Curvilinear Coordinates in R3 -- 32. Integrals of Differential Forms and the Stokes Theorem -- 33. Homology and De Rham Cohomology -- 34. The Geometry of Lie Groups -- 35. Connections and Curvatures on a Vector Bundle -- 36. Yang-Mills Equations -- 37. Connections on a Principal Bundle -- 38. Magnetic Monopoles and Molecular Dynamics -- 39. Riemannian Geometry -- 40. Complex Manifolds -- 41. Characteristic Classes -- 42. Chern-Simons Forms -- 43. The Atiyay-Singer Index Theorem -- 44. Symplectic Structures and Hamiltonian Mechanics -- 45. Quantization via path integration -- 46. Euclideanization, the Wiener Measure and the Feynman Kac Formula -- 47. Functional integrals of the Gaussian type -- 48. Quantum fields, the generating functional, and the Feynman propagator -- 49. Correlation (Green's) functions and the functional derivative -- 50. Perturbative expansions and Feynman diagrams -- 51. Holomorphic quantization -- 52. Perturbative renormalization -- 53. Regularization schemes for Feynman integrals -- 54. The Callan-Symanzik Equation and the Renormalization group -- 55. The Wilsonian Approach to the Renormalization group -- 56. Critical phonemena, the Landau-Ginasburg theory, Mean-Field theory -- 57. The Wilsonian approach and critical exponents -- 58. The Callan-Symanzik Equation and critical exponents.
Summary: This 2nd edition contains a general treatment of quantum field theory (QFT) in a simple scalar field setting in addition to the modern material on the applications of differential geometry and topology, group theory, and the theory of linear operators to physics found in the first edition. All these are introduced without assuming more background on the part of the reader than a good foundation in undergraduate (junior) level mathematical physics. The new material entirely focuses on an introduction to quantum field theory, emphasizing the Feynman path (functional integral) approach to QFT and the renormalization group. With respect to the latter, the focus is on an introduction of its application to critical phenomena in statistical physics, following the outgrowth of the Callan-Symanzik equation originally developed in the context of high energy physics, and the seminal contributions of Kenneth Wilson. One of the overriding aims of the new material is also to draw students' attention to the deep connections between high energy physics and statistical mechanics. The unavoidable technical aspects are explained with a minimum of prerequisite material and jargon, and conceptual understanding is always given prominence before mastery of technical details, but the importance of the latter is never underestimated. Derivational details and motivational discussions are provided in abundance in order to ensure continuity of reading, and to avoid trying the readers' patience.
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Includes bibliographical references and index.

1. Vectors and Linear Transformations --
2. Tensors --
3. Symmetry and Conservation: the Angular Momentum --
4. The Angular Momentum as Generators of Rotations: Lie Groups and Lie Algebras --
5. Algebraic Structures --
6. Basic Group Concepts --
7. Basic Lie Algebra Concepts --
8. Inner Products, Metrics, and Dual Spaces --
9. SO(4) and the Hydrogen Atom --
10. Adjoints and Unitary Transformations --
11. The Lorentz Group and SL(2,C) --
12. The Dirac Bracket Notation in Quantum Theory --
13. The Quantum Mechanical Simple Harmonic Oscillator --
14. Fourier Series and Fourier Transforms, the Dirac Delta Function, Green's Functions --
15. The Continuous Spectrum and Non-normalizable States --
16. Skew-Symmetric Tensors and Determinants --
17. Eigenvalue Problems --
18. Group Representation Theory --
19. The Dihedral Group D6 and the Benzene Molecule --20. Representations of the Symmetric Groups and the General Linear Groups, Young Diagram --
21. Irreducible Representations of U(n), SL(n), SU(n) and O(n) --
22. Irreducible Representations of SU(2) and SO(3) --
23. The Spherical Harmonics --
24. The Structure of Semisimple Lie Algebras --
25. The Representations of Semisimple Lie Algebras --
26. SU(3) and the Strong Interaction --
27. Clifford Algebras --
28. Exterior Products --
29. The Hodge-Star Operator --
30. Differential Forms and Exterior Differentiation --
31. Moving Frames and Curvilinear Coordinates in R3 --
32. Integrals of Differential Forms and the Stokes Theorem --
33. Homology and De Rham Cohomology --
34. The Geometry of Lie Groups --
35. Connections and Curvatures on a Vector Bundle --
36. Yang-Mills Equations --
37. Connections on a Principal Bundle --
38. Magnetic Monopoles and Molecular Dynamics --
39. Riemannian Geometry --
40. Complex Manifolds --
41. Characteristic Classes --
42. Chern-Simons Forms --
43. The Atiyay-Singer Index Theorem --
44. Symplectic Structures and Hamiltonian Mechanics --
45. Quantization via path integration --
46. Euclideanization, the Wiener Measure and the Feynman Kac Formula --
47. Functional integrals of the Gaussian type --
48. Quantum fields, the generating functional, and the Feynman propagator --
49. Correlation (Green's) functions and the functional derivative --
50. Perturbative expansions and Feynman diagrams --
51. Holomorphic quantization --
52. Perturbative renormalization --
53. Regularization schemes for Feynman integrals --
54. The Callan-Symanzik Equation and the Renormalization group --
55. The Wilsonian Approach to the Renormalization group --
56. Critical phonemena, the Landau-Ginasburg theory, Mean-Field theory --
57. The Wilsonian approach and critical exponents --
58. The Callan-Symanzik Equation and critical exponents.

This 2nd edition contains a general treatment of quantum field theory (QFT) in a simple scalar field setting in addition to the modern material on the applications of differential geometry and topology, group theory, and the theory of linear operators to physics found in the first edition. All these are introduced without assuming more background on the part of the reader than a good foundation in undergraduate (junior) level mathematical physics. The new material entirely focuses on an introduction to quantum field theory, emphasizing the Feynman path (functional integral) approach to QFT and the renormalization group. With respect to the latter, the focus is on an introduction of its application to critical phenomena in statistical physics, following the outgrowth of the Callan-Symanzik equation originally developed in the context of high energy physics, and the seminal contributions of Kenneth Wilson. One of the overriding aims of the new material is also to draw students' attention to the deep connections between high energy physics and statistical mechanics. The unavoidable technical aspects are explained with a minimum of prerequisite material and jargon, and conceptual understanding is always given prominence before mastery of technical details, but the importance of the latter is never underestimated. Derivational details and motivational discussions are provided in abundance in order to ensure continuity of reading, and to avoid trying the readers' patience.

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