From Special Relativity to Feynman Diagrams.pdf

Excellent standard textbooks which have been for us a

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excellent standard textbooks which have been for us a precious guide for the preparation of the present work, referring the interested reader to them in order to deepen the understanding of the topics dealt with in this book. Riccardo D’Auria Mario Trigiante Preface ix
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Contents 1 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Galilean Relativity in Classical Mechanics . . . . . . . . 2 1.1.2 Invariance of Classical Mechanics Under Galilean Transformations . . . . . . . . . . . . . . . . . . . . . 7 1.2 The Speed of Light and Electromagnetism . . . . . . . . . . . . . . 10 1.3 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Kinematic Consequences of the Lorentz Transformations . . . . 21 1.5 Proper Time and Space–Time Diagrams . . . . . . . . . . . . . . . . 26 1.5.1 Space–Time and Causality . . . . . . . . . . . . . . . . . . . . 27 1.6 Composition of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.6.1 Aberration Revisited . . . . . . . . . . . . . . . . . . . . . . . . 33 1.7 Experimental Tests of Special Relativity . . . . . . . . . . . . . . . . 34 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1 Relativistic Energy and Momentum . . . . . . . . . . . . . . . . . . . 37 2.1.1 Energy and Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.1.2 Nuclear Fusion and the Energy of a Star . . . . . . . . . . 50 2.2 Space–Time and Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . 52 2.2.1 Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.2 Relativistic Theories and Poincaré Transformations . . 61 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3 The Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 Inertial and Gravitational Masses . . . . . . . . . . . . . . . . . . . . . 63 3.2 Tidal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 The Geometric Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 xi
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3.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.4.1 An Elementary Approach to the Curvature . . . . . . . . 78 3.4.2 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.3 Tidal Forces and Space–Time Curvature . . . . . . . . . . 81 3.5 Motion of a Particle in Curved Space–Time . . . . . . . . . . . . . 83 3.5.1 The Newtonian Limit . . . . . . . . . . . . . . . . . . . . . . . 85 3.5.2 Time Intervals in a Gravitational Field . . . . . . . . . . . 86 3.5.3 The Einstein Equation . . . . . . . . . . . . . . . . . . . . . . . 89 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4 The Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.1 Linear Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.1.1 Covariant and Contravariant Components . . . . . . . . . 95 4.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.4 Rotations in Three-Dimensions . . . . . . . . . . . . . . . . . . . . . . 108 4.5 Groups of Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.5.1 Lie Algebra of the SO(3) Group . . . . . . . . . . . . . . . . 116 4.6 Principle of Relativity and Covariance of Physical Laws . . . . . 121 4.7 Minkowski Space–Time and Lorentz Transformations . . . . . . 122 4.7.1 General Form of (Proper) Lorentz Transformations . . 129 4.7.2 The Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . . 134 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5 Maxwell Equations and Special Relativity . . . . . . . . . . . . . . . . . . 137 5.1 Electromagnetism in Tensor Form . . . . . . . . . . . . . . . . . . . . 137 5.2 The Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3 Behavior of E and B Under Lorentz Transformations . . . . . . . 144 5.4 The Four-Current and the Conservation of the Electric Charge 146 5.5 The Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . 149 5.6 The Four-Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.6.1 The Spin of a Plane Wave . . . . . . . . . . . . . . . . . . . . 159 5.6.2 Large Volume Limit . . . . . . . . . . . . . . . . . . . . . . . . 162 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6 Quantization of the Electromagnetic Field . . . . . . . . . . . . . . . . . . 165 6.1 The Electromagnetic Field as an Infinite System of Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.2 Quantization of the Electromagnetic Field . . . . . . . . . . . . . . . 171 6.3 Spin of the Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 xii Contents
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7 Group Representations and Lie Algebras . . . . . . . . . . . . . . . . . . 181 7.1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.3 Infinitesimal Transformations and Lie Algebras . . . . . . . . . . . 190 7.4 Representation of a Group on a Field . . . . . . . . . . . . . . . . . . 192 7.4.1 Invariance of Fields . . . . . . . . . . . . . . . . . . . . . . . . 197 7.4.2 Infinitesimal Transformations on Fields . . . . . . . . . . . 200 7.4.3 Application to Non-Relativistic Quantum Mechanics . 204 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8 Lagrangian and Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . 207 8.1 Dynamical System with a Finite Number of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.1.1 The Action Principle . . . . . . . . . . . . . . . . . . . . . . . . 207 8.1.2 Lagrangian of a Relativistic Particle . . . . . . . . . . . . . 212 8.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.2.1 The Noether Theorem for a System of Particles . . . . . 217 8.3
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