From Special Relativity to Feynman Diagrams.pdf

Möller scattering 475 1252 a comment on the role of

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Möller Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 475 12.5.2 A Comment on the Role of Virtual Photons . . . . . . . 480 12.5.3 Bhabha and Electron-Muon Scattering . . . . . . . . . . . 482 12.5.4 Compton Scattering and Feynman Rules . . . . . . . . . . 486 12.5.5 Gauge Invariance of Amplitudes . . . . . . . . . . . . . . . 489 12.5.6 Interaction with an External Field . . . . . . . . . . . . . . . 492 12.6 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 12.6.1 The Bahbha Scattering . . . . . . . . . . . . . . . . . . . . . . 495 12.6.2 The Compton Scattering . . . . . . . . . . . . . . . . . . . . . 499 12.7 Divergent Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 12.8 A Pedagogical Introduction to Renormalization . . . . . . . . . . . 507 12.8.1 Power Counting and Renormalizability . . . . . . . . . . . 511 12.8.2 The Electron Self-Energy Part . . . . . . . . . . . . . . . . . 516 12.8.3 The Photon Self-Energy . . . . . . . . . . . . . . . . . . . . . 521 12.8.4 The Vertex Part . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 12.8.5 One-Loop Renormalized Lagrangian . . . . . . . . . . . . . 531 12.8.6 The Electron Anomalous Magnetic Moment . . . . . . . 531 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 Appendix A: The Eotvös’ Experiment . . . . . . . . . . . . . . . . . . . . . . . . 537 Appendix B: The Newtonian Limit of the Geodesic Equation . . . . . . . 539 Appendix C: The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Appendix D: Jacobi Identity for Poisson Brackets . . . . . . . . . . . . . . . 545 Appendix E: Induced Representations and Little Groups . . . . . . . . . . 547 Contents xv
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Appendix F: SU(2) and SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Appendix G: Gamma Matrix Identities . . . . . . . . . . . . . . . . . . . . . . . 557 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 xvi Contents
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Chapter 1 Special Relativity 1.1 The Principle of Relativity The aim of physics is to describe the laws underlying physical phenomena. This description would be devoid of a universal character if its formulation were different for different observers, that is for different reference frames, and, as such, it could not deserve the status of an objective law of nature. Any physical theory should therefore fulfil the following requirement: The laws of physics should not depend on the reference frame used by the observer. This statement is referred to as the principle of relativity , and is really at the heart of any physical theory aiming at the description of the physical world. Actually, the physical laws are described in the language of mathematics, that is by means of one or more equations involving physical quantities, whose value in general will depend on the reference frame (RF) used for their measure. As a consequence of this, any change in the reference frame results in a change in the physical quantities appearing in the equations, so that in general these will satisfy new equations, called transformed equations . The requirement that the transformed equations be equivalent to the original ones, so that they describe the same physical law, allows us to give a more precise formulation of the principle of relativity: The equations of a physical theory must preserve the same form under transfor- mations induced by a change in the reference frame. By preserving the same form we mean that if the physical law is given in terms of a single equation, the transformed equation will have exactly the same form, albeit in terms of the transformed variables. If we have a system of equations, we can allow the transformed system to be a linear combination of the old ones. Obviously, in both cases, the physical content of the original and transformed equations would be exactly the same.
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