## An Introduction to Quantum Field TheoryThis is a systematic presentation of Quantum Field Theory from first principles, emphasizing both theoretical concepts and experimental applications. Starting from introductory quantum and classical mechanics, this book develops the quantum field theories that make up the "Standard Model" of elementary processes. It derives the basic techniques and theorems that underly theory and experiment, including those that are the subject of theoretical development. Special attention is also given to the derivations of cross sections relevant to current high-energy experiments and to perturbative quantum chromodynamics, with examples drawn from electron-positron annihilation, deeply inelastic scattering and hadron-hadron scattering. The first half of the book introduces the basic ideas of field theory. The discussion of mathematical issues is everywhere pedagogical and self contained. Topics include the role of internal symmetry and relativistic invariance, the path integral, gauge theories and spontaneous symmetry breaking, and cross sections in the Standard Model and the parton model. The material of this half is sufficient for an understanding of the Standard Model and its basic experimental consequences. The second half of the book deals with perturbative field theory beyond the lowest-order approximation. The issues of renormalization and unitarity, the renormalization group and asymptotic freedom, infrared divergences in quantum electrodynamics and infrared safety in quantum chromodynamics, jets, the perturbative basis of factorization at high energy and the operator product expansion are discussed. Exercises are included for each chapter, and several appendices complement the text. |

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### Contents

SCALAR FIELDS | 3 |

12 Relativistic scalar fields | 5 |

13 Invariance and conservation | 10 |

14 Lie groups and internal symmetries | 16 |

15 The Poincaré group and its generators | 20 |

Exercises | 26 |

Canonical quantization | 29 |

22 Quantum symmetries | 32 |

92 Wick rotation in perturbation theory | 250 |

93 Dimensional regularization | 252 |

94 Poles at 𝑛 4 | 261 |

95 Timeordered perturbation theory | 266 |

96 Unitarity | 271 |

Exercises | 278 |

Introduction to renormalization | 280 |

102 Power counting and renormalizability | 288 |

23 The free scalar field as a system of harmonic oscillators | 38 |

24 Particles and Green functions | 46 |

25 Interacting fields and scattering | 49 |

Exercises | 55 |

Path integrals perturbation theory and Feynman rules | 58 |

32 The path integral and coherent states | 71 |

33 Coherent state construction of the path integral in field theory | 76 |

34 Feynman diagrams and Feynman rules | 82 |

Exercises | 92 |

Scattering and cross sections for scalar fields | 94 |

42 The 𝑆matrix | 98 |

43 Cross sections | 100 |

44 The charged scalar Held | 107 |

Exercises | 114 |

Spinors vectors and gauge invariance | 119 |

52 Spinor equations and Lagrangians | 125 |

53 Vector fields and Lagrangians | 131 |

54 Interactions and local gauge invariance | 134 |

Exercises | 144 |

Spin and canonical quantization | 147 |

62 Unitary representations of the Poincaré group | 148 |

63 Solutions with mass | 152 |

64 Massless solutions | 158 |

65 Quantization | 162 |

66 Parity and leptonic weak interactions | 170 |

Exercises | 174 |

Path integrals for fermions and gauge fields | 176 |

72 Fermions in an external field | 182 |

73 Gauge vectors and ghosts | 189 |

74 Reduction formulas and cross sections | 199 |

Exercises | 203 |

Gauge theories at lowest order | 204 |

82 Cross sections with photons | 217 |

83 Weak interactions of leptons | 223 |

84 Quantum chromodynamics and quarkquark scattering | 227 |

85 Gluons and ghosts | 231 |

86 Partonmodel interpretation of QCD cross sections | 237 |

Exercises | 242 |

PART III REMORMALIZATION | 247 |

103 Oneloop counterterms for ϕ³₆ | 290 |

104 Renormalization at two loops and beyond | 300 |

105 Introduction to the renormalization group | 309 |

Exercises | 317 |

Renormalization and unitarity of gauge theories | 319 |

112 Renormalization and unitarity in QED | 334 |

113 Ward identities and the 𝑆matrix in QCD | 348 |

114 The axial anomaly | 358 |

Exercises | 364 |

THE NATURE OF PRETURBATIVE CROSS SECTIONS | 369 |

122 Ordera infrared bremsstrahlung | 378 |

123 Infrared divergences to all orders | 384 |

124 Infrared safety and renormalization in QCD | 394 |

125 Jet cross sections at order 𝑎 in 𝐞𝐞 annihilation | 404 |

Exercises | 408 |

Analytic structure and infrared finiteness | 411 |

132 The twopoint function | 417 |

133 Massless particles and infrared power counting | 422 |

134 The threepoint function and collinear power counting | 431 |

135 The KinoshitaLeeNauenberg theorem | 440 |

Exercises | 447 |

Factorization and evolution in highenergy scattering | 449 |

142 Deeply inelastic scattering for massless quarks | 454 |

143 Factorization and parton distributions | 459 |

144 Evolution | 475 |

145 The operator product expansion | 484 |

Exercises | 490 |

Epilogue Bound states and the limitations of perturbation theory | 492 |

APPENDICES | 502 |

Symmetry factors and generating functionals | 509 |

The standard model | 514 |

T C and CPT | 523 |

The Goldstone theorem and 𝛑⁰2𝑦 | 532 |

Groups algebras and Dirac matrices | 539 |

Cross sections and Feynman rules | 545 |

551 | |

562 | |

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### Common terms and phrases

5-matrix algebra amplitude annihilation anticommuting arbitrary behavior bosons calculation cancel classical collinear compute conjugate contribution corresponding counterterms coupling covariant cross section cut diagrams deeply inelastic scattering defined delta function derive dimensions Dirac equation eigenstates energy evaluate exercise expansion external momenta factor fermion loop Feynman diagrams Feynman rules field theory gamma matrices gauge invariance gauge theories gluon graph Green functions hadronic Hamiltonian infinitesimal infrared divergences interaction Lagrange density Lagrangian leptonic logarithm Lorentz group mass massless matrix element momentum normal notation on-shell one-loop operators parameters parity particles parton path integral perturbation theory photon Phys physical pinch surface polarization pole propagator quantization quantum quark reduced diagram relation renormalization representation result rotation scalar field self-energy shown in Fig solutions space spin spinor structure functions subdiagram symmetry tensor theorem time-ordered tion ultraviolet divergences unitary vacuum vanish variables vertex vertices Ward identities