From special relativity to feynman diagrams: introduction to special relativity and to classical and quantum field theory PDF
Preview From special relativity to feynman diagrams: introduction to special relativity and to classical and quantum field theory
UNITEXT Collana di Fisica e Astronomia Subseries Editors Michele Cini Stefano Forte Inguscio Massimo G. Montagna Oreste Nicrosini Franco Pacini Luca Peliti Alberto Rotondi For furthervolumes: http://www.springer.com/series/4467 Riccardo D’Auria Mario Trigiante • From Special Relativity to Feynman Diagrams A Course of Theoretical Particle Physics for Beginners 123 Prof.Riccardo D’Auria Prof.Mario Trigiante Department of Physics Department of Physics Politecnico ofTorino Politecnico ofTorino Corso Duca degli Abruzzi24 Corso Duca degli Abruzzi24 10129Torino 10129Torino Italy Italy e-mail: [email protected] e-mail: [email protected] ISSN 2038-5730 e-ISSN2038-5765 ISBN 978-88-470-1503-6 e-ISBN978-88-470-1504-3 DOI 10.1007/978-88-470-1504-3 SpringerMilanHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber:2011920257 (cid:2)Springer-VerlagItalia2012 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcast- ing, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationorpartsthereofispermittedonlyundertheprovisionsoftheItalianCopyrightLawinits currentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliableto prosecutionundertheItalianCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelawsandregulationsandthereforefreeforgeneraluse. Coverdesign:SimonaColombo,Milan Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Anna Maria R.D’A. To my parents M.T. Preface ThisbookhasdevelopedfromthelecturenotesofacourseinAdvancedQuantum Mechanics held by the authors at the Politecnico of Torino for students of ‘‘physical engineering’’, that is students who, even though oriented towards applied physics and technology, were interested in acquiring a fair knowledge of modern fundamental physics. Although originally conceived for students of engineering, we have eventually extended the target of this book to also include students of physics who may be interested in a comprehensive and concise treatmentofthemainsubjectsoftheirtheoreticalphysicscourses.Whatunderlies our choice of topics is the purpose of giving a consistent presentation of the theoretical ideas which have been developed since the very beginning of the last century, namely special relativity and quantum mechanics, up to the first consis- tent and experimentally validated quantum field theory, namely quantum elec- trodynamics. This theory provides a successful description of the interaction between photons and electrons and dates back to the middle of the last century. Consistently with this purpose (and also for keeping the book within a rea- sonable size), we have refrained from dealing with the many important ideas that have been developed in the context of quantum field theory in the second part of thelastcentury,althoughtheseareessentialforasatisfactoryunderstandingofthe current status of elementary particle physics. A prominent example of such developmentsistheso-calledstandardmodel,whereforthefirsttimeallthe(non- gravitational)interactionsandthefundamentalparticles(quarksandleptons)were coherentlydescribedwithinaunifiedfieldtheoryframework.Lookingatthepast, however,onerecognizesthatthisachievementhasitsveryfoundationsinthetwo building blocks of any modern physical theory: special relativity and quantum mechanics,whichhavebeenleftessentiallyunaffectedbythelaterdevelopments. Quantumelectrodynamicshasprovidedabasicreferencefortheformulationof the standard model and in general of any field theory description of the fundamental interactions. In particular a major role is played in quantum elec- trodynamicsbytheconcept ofgauge symmetrywhichis theguidingprinciple for the correct description of the interaction. Likewise, the standard model too, as a vii viii Preface quantum field theory, is based on a suitable gauge symmetry, which is a non-abelian extension of the one present in quantum electrodynamics. On the basis of these considerations, we hope the concise account of quantum electrodynamics that we give at the end of our book can provide the interested reader with the necessary background to cope with more advanced topics in theoretical particle physics, in particular with the standard model. The present book is intended to be accessible to students with only a basic knowledge of non-relativistic quantum mechanics. We start with a concise, but (hopefully) comprehensive exposition of special relativity,towhichwehaveaddedachapterontheimplicationsoftheprincipleof equivalence. Here we have a principle whose importance can be hardly overes- timated since it is at the very basis of the general theory of relativity, but whose discussion in a class, however, requires no more than a couple of hours. Nevertheless this issue and its main implications are rarely dealt with even in graduate courses of physics. Can general relativity be totally absent from the scientificeducation ofastudentofphysics orengineering?Ofcourseitcan asfar as the full geometrical formulation of theory is concerned. However it is well known that many technological devices, mainly the GPS, require for their proper functioningtoconsiderthecorrectionsimpliedbytheEinstein’stheoriesofspecial andgeneralrelativity.Ouraccountoftheprincipleofequivalenceandofitsmain implications will allow us to derive in a rather non-rigorous but intuitive way the conceptsofconnection,curvature,geodesiclines,etc.,emphasizingtheirintimate connection to gravitational physics. Thereafter,inChaps.4and7,wegivethebasicsofthetheoryofgroupsandLie algebras, discussing the group of rotations, the Lorentz and the Poincaré group. Wealsogiveaconciseaccountofrepresentationtheoryandoftensorcalculus,in view of its application to the formulation of relativistically covariant physical laws. These include Maxwell’s equations, which we discuss, in their manifestly covariant form, in Chap. 5. In Chap. 6, anticipating part of the analysis which will be later developed, we discuss the quantization of the electromagnetic field in the radiation gauge. We thought it worth illustrating this important example earlier since it clarifies how the concepts of photon and of its spin emerge quite naturally from a straightforward application of special relativity and quantum theory in a field theoretical framework. In Chap. 8 we review the essentials of the Lagrangian and Hamiltonian formalisms, first considering systems with a finite number of degrees offreedom, and then extending the discussion to fields. Particular importance is given to the relation between the symmetry properties of a physical system and conservation laws. The last four chapters are devoted to the development of the quantum field theory. In Chap. 9 we recall the basic construction of quantum mechanics in the Dirac notation. Eventually in Chap. 10 we study the quantum relativistic wave equations emphasizing their failure to represent the wave function evolution in a consistent way. Preface ix In Chap. 11 we perform the quantization of the free scalar, spin 1/2 and electromagnetic fields in the relativistically covariant approach. The final goal of thisanalysisistogiveanaccountofthequantumrelativisticdescriptionoffieldsin interaction,withparticularreferencetotheinteractionbetweenspin1/2fields(like anelectron)andtheelectromagneticone(quantumelectrodynamics).Thisisdone in Chap. 12 where the graphical description of interaction processes by means of Feynman diagrams is introduced. After the classical example of the tree-level processes, we start analyzing the one-loop ones where infinities make their appearance. We then discuss how one can circumvent this difficulty through the process of renormalization, in order to obtain sensible results. We shall however limit ourselves to give only a short preliminary account of the renormalization program and its implementation at one-loop level. Asthereadercanrealize,thereisscarcelyanyambitiononoursidetodevelop thevarioustopicsinanoriginalway.Ourgoal,aspointedoutearlier,istogiveina single 1 year course the main concepts which are the basis of the contemporary theoretical physics. A note on the Bibliography. It is almost impossible to give an even short accountofthemanytextbookscoveringsomeofthetopicswhicharedealtwithin this book.Any textbook onrelativity orelementary particle theorycoversat least part of the content of our book. We therefore limit ourselves to quote those excellent standard textbooks which have been for us a precious guide for the preparationofthepresentwork,referringtheinterestedreadertotheminorderto deepen the understanding of the topics dealt with in this book. Riccardo D’Auria Mario Trigiante 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 xii Contents 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
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