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Providing an easily accessible introduction to quantum field theory via Feynman rules and calculations in particle physics, the aim of this text is to clarify the physical foundations of present day field theory and the physical content of Feynman rules, and to outline their domain of applicability.

Martinus Justinus Godefriedus Veltman (born June 27, 1931 in Waalwijk) is a Dutch theoretical physicist. He shared the 1999 Nobel Prize in physics with his former student Gerardus 't Hooft for their work on particle theory.

Martinus J.G. Veltman was born in Waalwijk, Netherlands on June 27, 1931. He started studying mathematics and physics at Utrecht University in 1948. He obtained his PhD in theoretical physics in 1963 and became professor at Utrecht University in 1966.

In 1963/64, during an extended stay at SLAC he designed the computer program Schoonschip for symbolic manipulation of mathematical equations, which is now considered the very first Computer algebra system.

In 1971, Gerardus 't Hooft, who was completing his PhD under the supervision of Veltman, renormalized Yang-Mills theory. They showed that if the symmetries of Yang-Mills theory were to be broken according to the method suggested by Guralnik, Hagen, Kibble, Higgs, Brout, and Englert, then Yang-Mills theory can be renormalized.[1][2] Renormalization of Yang-Mills theory is one of the biggest achievements of twentieth century physics.

In 1981, Veltman left Utrecht University for the University of Michigan-Ann Arbor, frustrated by the recognition his student 't Hooft got for his PhD thesis. Veltman felt that he had done most of the preliminary work and written the program which made the dissertation possible. However, most of the credit went to 't Hooft.[3]

But eventually, in 1999, he was awarded the Nobel Prize for Physics in 1999 together with 't Hooft, "for elucidating the quantum structure of electroweak interactions in physics".[4] Veltman and 't Hooft joined in the celebrations at Utrecht University when the prize was awarded.

Veltman is now retired and holds a position of Emeritus Professor at the University of Michigan. Asteroid 9492 Veltman is named in his honor.

In 2003, Veltman published a book about particle physics for a broad audience, entitled Facts and Mysteries in Elementary Particle Physics.

Introduction
1 Lorentz and Poincare Invariance 1
1.1 Lorentz Invariance 1
1.2 Structure of the Lorentz Group 7
1.3 Poincare Invariance 10
1.4 Maxwell Equations 10
1.5 Notations and Conventions 12
2 Relativistic Quantum Mechanics of Free Particles 15
2.1 Hilbert Space 15
2.2 Matrices in Hilbert Space 22
2.3 Fields 25
2.4 Structure of Hilbert Space 29
3 Interacting Fields 32
3.1 Physical System 32
3.2 Hilbert Space 33
3.3 Magnitude of Hilbert Space 34
3.4 U-matrix, S-matrix 35
3.5 Interpolating Fields 39
3.6 Feynman Rules 46
3.7 Feynman Propagator 54
3.8 Scattering Cross Section 55
3.9 Lifetime 60
3.10 Numerical Evaluation 62
3.11 Schrodinger Equation, Bound States 62
4 Particles with Spin 68
Representations of the Lorentz Group 68
The Dirac Eauation 76
4.3 Fermion Fields 79
4.4 The E.M. Field 85
4.5 Quantum Electrodynamics 87
4.6 Charged Vector Boson Fields 91
4.7 Electron-Proton Scattering. The Rutherford Formula 92
5 Explorations 98
5.1 Scattering Cross Section for e+e~ —> ^ pT 98
5.2 Pion Decay. Two Body Phase Space. Cabibbo Angle 101
5.3 Vector Boson Decay 105
5.4 Muon Decay. Fiertz Transformation 108
5.5 Hyperon Leptonic Decay 117
5.6 Pion Decay and PCAC 124
5.7 Neutral Pion Decay and PCAC 131
6 Renormalization 137
6.1 Introduction 137
6.2 Loop Integrals 137
6.3 Self Energy 145
6.4 Power Counting 148
6.5 Quantum Electrodynamics 151
6.6 Renormalizable Theories 153
6.7 Radiative Corrections: Lamb Shift 154
6.8 Radiative Corrections: Top Correction to p-Parameter 157
6.9 Neutral Pion Decay and the Anomaly 162
7 Massive and Massless Vector Fields 169
7.1 Subsidiary Condition Massive Vector Fields 169
7.2 Subsidiary Condition Massless Vector Fields 171
7.3 Photon Helicities 173
7.4 Propagator and Polarization Vectors of Massive
Vector Particles 174
7.5 Photon Propagator 177
7.6 Left Handed Photons 180
8 Unitarity 183
8.1 [/-matrix 183
8.2 Largest Time Equation 185
8.3 Cutting Equations 187
8.4 Unitarity and Cutting Equation 191
8.5 Unitarity: General Case
8.6 Kallen-Lehmann Representation, Dispersion
Relation 200
8.7 Momenta in Propagators 204
9 Quantum Electrodynamics: Finally 207
9.1 Unitarity 207
9.2 Ward Identities 208
Appendix A Complex Spaces, Matrices, CBH
Equation 213
A.I Basics 213
A.2 Differentiation of Matrices 217
A.3 Functions of Matrices 218
A.4 The CBH Equation 220
Appendix B Traces 224
B.I General 224
B.2 Multi-Dimensional 7-Matrices 230
B.3 Frequently Used Equations 231
Appendix C Dimensional Regularization 233
Appendix D Summary. Combinatorial Factors 243
D.I Summary 243
D.2 External Lines, Spin Sums, Propagators 245
D.3 Combinatorial Factors 247
Appendix E Standard Model 249
E.I Lagrangian 249
E.2 Feynman Rules 258
Appendix F Metric and Conventions 273
F.I General Considerations 273
F.2 Translation Examples 279
F.3 Translation Dictionary 281
Index 282
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