Problems in Quantum Field Theory: With Fully-Worked Solutions

Description

This collection of problems in Quantum Field Theory, accompanied by their complete solutions, aims to bridge the gap between learning the foundational principles and applying them practically. The carefully chosen problems cover a wide range of topics, starting from the foundations of Quantum Field Theory and the traditional methods in perturbation theory, such as LSZ reduction formulas, Feynman diagrams and renormalization. Separate chapters are devoted to functional methods (bosonic and fermionic path integrals; worldline formalism), to non-Abelian gauge theories (Yang-Mills theory, Quantum Chromodynamics), to the novel techniques for calculating scattering amplitudes and to quantum field theory at finite temperature (including its formulation on the lattice, and extensions to systems out of equilibrium). The problems range from those dealing with QFT formalism itself to problems addressing specific questions of phenomenological relevance, and they span a broad range in difficulty, for graduate students taking their first or second course in QFT. Acknowledgements Contents Figures and Tables Preface Notation and Conventions 1. Quantum Field Theory Basics Introduction 1. Relationship between the Equations of Motion of φ and φ_{in} 2. Little-Group Elements for Massless Particles 3. S-matrix in Terms of φ_{in} and φ_{out} 4. Constraints on the S-matrix from Causality 5. Landau Equations for Soft and Collinear Singularities 6. Multi-loop Integration in D Dimensions 7. Weinberg Convergence Theorem 8. Electron Anomalous Magnetic Moment 9. Ward–Takahashi Identities and Lorentz Invariance 10. Equivalence Principle and Lorentz Invariance 11. Lee–Nauenberg Theorem 12. Classical External Field Approximation 13. Subleading Soft Radiation in Scalar QED 14. Low–Burnett–Kroll Theorem 15. Coherent States in Quantum Field Theory 16. Running Couplings in a Two-Field Scalar Field Theory 17. Solution of the Running Equation at Two Loops 2. Functional Methods Introduction 18. Weyl Quantization 19. Combinatorial Interpretation of the Legendre Transform 20. Coherent State Path Integral 21. Coherent State Path Integral (Continued) 22. Spin Coherent States and the Berry Phase 23. Zeta Function Regularization of the Klein–Gordon Operator 24. Casimir Zero-Point Energy 25. The Feynman Diagrams behind the Casimir Force 26. Gross–Neveu Model 27. D’Hoker–Gagné Formula 28. Stochastic Quantization 29. Complex Langevin Equation 3. Non-Abelian Fields Introduction 30. Basic su(n) Identities 31. More su(n) Identities 32. More su(n) Identities (Continued) 33. An su(2) Identity 34. Area Derivative of a Wilson Loop 35. Endpoint Derivative of a Wilson Line 36. Makeenko–Migdal Loop Equation 37. Classical Non-Abelian Particles and the Wong Equations 38. Wong Equations from the Worldline Formalism 39. Non-Abelian Stokes’s Theorem 40. QED in the ’t Hooft–Veltman Gauge 41. LSZ Reduction Formula and Gauge Invariance 42. Large-N Expansion in SU(N) Yang–Mills Theory 43. Soft Radiation and Antenna Effects in QCD 44. An Identity Obeyed by On-Shell QCD Amplitudes 45. Low–Burnett–Kroll Theorem in QCD 46. Globally Gauge Invariant Operators 47. Banks–Zaks Fixed Points 48. Instantons and Tunneling in Quantum Mechanics 49. Instantons as Transitions between Pure Gauges in Yang–Mills Theory 50. Vacua of Yang–Mills Theory 4. Scattering Amplitudes Introduction 51. Enumeration of Tree Graphs 52. Six-Point − − − + ++ Amplitude 53. Two Spinor-Helicity Identities 54. Auxiliary Vector Dependence of the Polarization Vectors 55. Polarization Sum in the Spinor-Helicity Formalism 56. Photon Decoupling Identity 57. Scalar–Antiscalar–Gluon Amplitudes 58. Color Decomposition with a Quark–Antiquark Pair 59. Quark–Antiquark–Gluon Amplitudes 60. Quark–Antiquark–Gluon Amplitudes (Continued) 61. Gluon–Gluon Scattering in the Spinor-Helicity Framework 62. Soft Limit of Gluon Amplitudes 63. Collinear Limit of Gluon Amplitudes 64. Enumeration of One-Loop Graphs 65. Ultraviolet Finiteness of One-Loop All-Plus Amplitudes 66. Rationality of One-Loop All-Plus Amplitudes 67. One-Loop Four-Point ++++ Amplitude from Generalized Unitarity 68. One-Loop Five-Point +++++ Amplitude from Generalized Unitarity 5. Lattice, Finite T , Strong Fields Introduction 69. Banks–Casher Relation 70. Nielsen–Ninomiya Theorem 72. Center Symmetry and Deconfinement 73. Photon Hard Thermal Loop from Classical Particles 74. Gluon Hard Thermal Loop from Colored Classical Particles 75. Landau–Pomeranchuk–Migdal Effect 76. Microcanonical Equilibration in φ⁴ Scalar Theory 77. Combinatorics of Multi-particle Production 78. Generating Functional for Particle Production 79. Correlations in the Schwinger Mechanism 80. Schwinger Mechanism from Bogoliubov Transformations 81. Unruh Effect, Hawking Radiation 82. Anderson Localization 83. Scaling Theory of Anderson Localization Index