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Mathematical Aspects of Quantum Field Theory

Book Description

Over the last century quantum field theory has made a significant impact on the formulation and solution of mathematical problems and inspired powerful advances in pure mathematics. However, most accounts are written by physicists, and mathematicians struggle to find clear definitions and statements of the concepts involved. This graduate-level introduction presents the basic ideas and tools from quantum field theory to a mathematical audience. Topics include classical and quantum mechanics, classical field theory, quantization of classical fields, perturbative quantum field theory, renormalization, and the standard model. The material is also accessible to physicists seeking a better understanding of the mathematical background, providing the necessary tools from differential geometry on such topics as connections and gauge fields, vector and spinor bundles, symmetries and group representations.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Foreword
  7. Preface
  8. 1. Classical mechanics
    1. 1.1 Newtonian mechanics
    2. 1.2 Lagrangian mechanics
    3. 1.3 Hamiltonian mechanics
    4. 1.4 Poisson brackets and Lie algebra structure of observables
    5. 1.5 Symmetry and conservation laws: Noether’s theorem
  9. 2. Quantum mechanics
    1. 2.1 The birth of quantum theory
    2. 2.2 The basic principles of quantum mechanics
    3. 2.3 Canonical quantization
    4. 2.4 From classical to quantum mechanics: the C* algebra approach
    5. 2.5 The Weyl C* algebra
    6. 2.6 The quantum harmonic oscillator
    7. 2.7 Angular momentum quantization and spin
    8. 2.8 Path integral quantization
    9. 2.9 Deformation quantization
  10. 3. Relativity, the Lorentz group, and Dirac’s equation
    1. 3.1 Relativity and the Lorentz group
    2. 3.2 Relativistic kinematics
    3. 3.3 Relativistic dynamics
    4. 3.4 The relativistic Lagrangian
    5. 3.5 Dirac’s equation
  11. 4. Fiber bundles, connections, and representations
    1. 4.1 Fiber bundles and cocycles
    2. 4.2 Principal bundles
    3. 4.3 Connections
    4. 4.4 The gauge group
    5. 4.5 The Hodge ⋆ operator
    6. 4.6 Clifford algebras and spinor bundles
    7. 4.7 Representations
  12. 5. Classical field theory
    1. 5.1 Introduction
    2. 5.2 Electromagnetic field
    3. 5.3 Conservation laws in field theory
    4. 5.4 The Dirac field
    5. 5.5 Scalar fields
    6. 5.6 Yang–Mills fields
    7. 5.7 Gravitational fields
  13. 6. Quantization of classical fields
    1. 6.1 Quantization of free fields: general scheme
    2. 6.2 Axiomatic field theory
    3. 6.3 Quantization of bosonic free fields
    4. 6.4 Quantization of fermionic fields
    5. 6.5 Quantization of the free electromagnetic field
    6. 6.6 Wick rotations and axioms for Euclidean QFT
    7. 6.7 The CPT theorem
    8. 6.8 Scattering processes and LSZ reduction
  14. 7. Perturbative quantum field theory
    1. 7.1 Discretization of functional integrals
    2. 7.2 Gaussian measures and Wick’s theorem
    3. 7.3 Discretization of Euclidean scalar fields
    4. 7.4 Perturbative quantum field theory
    5. 7.5 Perturbative Yang–Mills theory
  15. 8. Renormalization
    1. 8.1 Renormalization in perturbative QFT
    2. 8.2 Constructive field theory
  16. 9. The Standard Model
    1. 9.1 Particles and fields
    2. 9.2 Particles and their quantum numbers
    3. 9.3 The quark model
    4. 9.4 Non-abelian gauge theories
    5. 9.5 Lagrangian formulation of the standard model
    6. 9.6 The intrinsic formulation of the Lagrangian
  17. Appendix A: Hilbert spaces and operators
    1. A.1 Hilbert spaces
    2. A.2 Linear operators
    3. A.3 Spectral theorem for compact operators
    4. A.4 Spectral theorem for normal operators
    5. A.5 Spectral theorem for unbounded operators
    6. A.6 Functional calculus
    7. A.7 Essential self-adjointness
    8. A.8 A note on the spectrum
    9. A.9 Stone’s theorem
    10. A.10 The Kato–Rellich theorem
  18. Appendix B: C* algebras and spectral theory
    1. B.1 Banach algebras
    2. B.2 C* algebras
    3. B.3 The spectral theorem
    4. B.4 States and GNS representation
    5. B.5 Representations and spectral resolutions
    6. B.6 Algebraic quantum field theory
  19. Bibliography
  20. Index