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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
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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
