Quantum Field Theory and Particle Physics

435 Pages·2013·2.38 MB·English

Quantum Field Theory
and
Particle Physics
Badis Ydri
Département de Physique, Faculté des Sciences
Université d’Annaba
Annaba, Algerie
2 YDRI QFT
Contents
1 Introduction and References 1
I Free Fields, Canonical Quantization and Feynman Diagrams 5
2 Relativistic Quantum Mechanics 7
2.1 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Relativistic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Lorentz Transformations: Boosts . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.4 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.5 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Free Solutions of The Dirac Equation. . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Lorentz Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Canonical Quantization of Free Fields 29
3.1 Classical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 D’Alembert Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 Lagrange’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.3 Hamilton’s Principle: The Principle of Least Action . . . . . . . . . . . . 33
3.1.4 The Hamilton Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Classical Free Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 The Klein-Gordon LagrangianDensity . . . . . . . . . . . . . . . . . . . . 37
3.2.2 The Dirac LagrangianDensity . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Canonical Quantization of a Real Scalar Field . . . . . . . . . . . . . . . . . . . . 40
3.4 Canonical Quantization of Free Spinor Field . . . . . . . . . . . . . . . . . . . . . 44
3.5 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Scalar Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.2 Dirac Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6.2 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.6.3 Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.7 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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4 The S Matrix and Feynman Diagrams For Phi-Four Theory 61
−
4.1 Forced Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.1 Asymptotic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.2 The Schrodinger, Heisenberg and Dirac Pictures . . . . . . . . . . . . . . 63
4.1.3 The S Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
−
4.1.4 Wick’s Theorem For Forced Scalar Field . . . . . . . . . . . . . . . . . . . 67
4.2 The Φ Four Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
−
4.2.1 The LagrangianDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 The S Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
−
4.2.3 The Gell-Mann Low Formula . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.4 LSZ Reduction Formulae and Green’s Functions . . . . . . . . . . . . . . 74
4.3 Feynman Diagrams For φ Four Theory . . . . . . . . . . . . . . . . . . . . . . . 76
−
4.3.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Wick’s Theorem For Green’s Functions . . . . . . . . . . . . . . . . . . . 77
4.3.3 The 2 Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
−
4.3.4 Connectedness and Vacuum Energy . . . . . . . . . . . . . . . . . . . . . 83
4.3.5 Feynman Rules For Φ Four Theory . . . . . . . . . . . . . . . . . . . . . 86
−
4.4 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
II Quantum Electrodynamics 93
5 The Electromagnetic Field 95
5.1 CovariantFormulation of Classical Electrodynamics . . . . . . . . . . . . . . . . 95
5.2 Gauge Potentials and Gauge Transformations . . . . . . . . . . . . . . . . . . . . 98
5.3 Maxwell’s LagrangianDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 PolarizationVectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.5 Quantization of The Electromagnetic Gauge Field . . . . . . . . . . . . . . . . . 103
5.6 Gupta-Bleuler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.7 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.8 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 Quantum Electrodynamics 113
6.1 LagrangianDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Review of φ4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 Wick’s Theorem for Forced Spinor Field . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.1 Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3.2 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4 Wick’s Theorem for Forced Electromagnetic Field. . . . . . . . . . . . . . . . . . 121
6.5 The LSZ Reduction fromulas and The S Matrix . . . . . . . . . . . . . . . . . . 122
−
6.5.1 The LSZ Reduction fromulas . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5.2 The Gell-Mann Low Formula and the S Matrix . . . . . . . . . . . . . . 126
−
6.5.3 Perturbation Theory:Tree Level . . . . . . . . . . . . . . . . . . . . . . . . 129
6.5.4 Perturbation Theory: One-Loop Corrections . . . . . . . . . . . . . . . . 131
6.6 LSZ Reduction formulas for Photons . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.6.1 Example II: e +γ e +γ . . . . . . . . . . . . . . . . . . . . . . . . 137
− −
−→
6.6.2 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.7 Feynman Rules for QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.8 Cross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
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6.9 Tree Level Cross Sections: An Example . . . . . . . . . . . . . . . . . . . . . . . 146
6.10 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7 Renormalization of QED 155
7.1 Example III: e +µ e +µ . . . . . . . . . . . . . . . . . . . . . . . . . 155
− − − −
−→
7.2 Example IV : Scattering From External Electromagnetic Fields . . . . . . . . . . 156
7.3 One-loop Calculation I: Vertex Correction . . . . . . . . . . . . . . . . . . . . . . 159
7.3.1 Feynman Parameters and Wick Rotation . . . . . . . . . . . . . . . . . . 159
7.3.2 Pauli-Villars Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.3.3 Renormalization (Minimal Subtraction) and Anomalous Magnetic Moment 166
7.4 Exact Fermion 2 Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
−
7.5 One-loop Calculation II: Electron Self-Energy . . . . . . . . . . . . . . . . . . . . 171
7.5.1 Electron Mass at One-Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.5.2 The Wave-Function RenormalizationZ . . . . . . . . . . . . . . . . . . . 174
2
7.5.3 The Renormalization Constant Z . . . . . . . . . . . . . . . . . . . . . . 175
1
7.6 Ward-TakahashiIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.7 One-Loop Calculation III: Vacuum Polarization . . . . . . . . . . . . . . . . . . . 180
7.7.1 The Renormalization Constant Z and Renormalization of the Electric
3
Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.7.2 Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.8 Renormalization of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.9 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
III Path Integrals, Gauge Fields and Renormalization Group 191
8 Path Integral Quantization of Scalar Fields 193
8.1 Feynman Path Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.2 Scalar Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2.1 Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.2.2 The Free 2 Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . 199
−
8.2.3 Lattice Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.3 The Effective Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.3.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.3.2 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.3.3 Analogy with Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . 208
8.4 The O(N) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.4.1 The 2 Point and 4 Point Proper Vertices . . . . . . . . . . . . . . . . . 210
− −
8.4.2 Momentum Space Feynman Graphs . . . . . . . . . . . . . . . . . . . . . 211
8.4.3 Cut-off Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.4.4 Renormalization at 1 Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 215
−
8.5 Two-LoopCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.5.1 The Effective Action at 2 Loop . . . . . . . . . . . . . . . . . . . . . . . 216
−
8.5.2 The Linear Sigma Model at 2 Loop . . . . . . . . . . . . . . . . . . . . . 217
−
8.5.3 The 2 Loop Renormalization of the 2 Point Proper Vertex. . . . . . . . 219
− −
8.5.4 The 2 Loop Renormalization of the 4 Point Proper Vertex. . . . . . . . 224
− −
8.6 Renormalized Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.7 Effective Potential and Dimensional Regularization . . . . . . . . . . . . . . . . . 228
8.8 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
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8.8.1 Example: The O(N) Model . . . . . . . . . . . . . . . . . . . . . . . . . . 232
8.8.2 Glodstone’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9 Path Integral Quantization of Dirac and Vector Fields 241
9.1 Free Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
9.1.1 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
9.1.2 Fermionic Path Integral and Grassmann Numbers . . . . . . . . . . . . . 242
9.1.3 The Electron Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.2 Free Abelian Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
9.2.1 Maxwell’s Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
9.2.2 Gauge Invariance and Canonical Quantization . . . . . . . . . . . . . . . 250
9.2.3 Path Integral Quantization and the Faddeev-Popov Method . . . . . . . . 252
9.2.4 The Photon Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.3 Gauge Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
9.3.1 Spinor and Scalar Electrodynamics: Minimal Coupling . . . . . . . . . . . 255
9.3.2 The Geometry of U(1) Gauge Invariance. . . . . . . . . . . . . . . . . . . 257
9.3.3 Generalization: SU(N) Yang-Mills Theory . . . . . . . . . . . . . . . . . 260
9.4 Quantization and Renormalization at 1 Loop . . . . . . . . . . . . . . . . . . . . 266
−
9.4.1 The Fadeev-Popov Gauge Fixing and Ghost Fields . . . . . . . . . . . . . 266
9.4.2 Perturbative Renormalization and Feynman Rules . . . . . . . . . . . . . 270
9.4.3 The Gluon Field Self-Energy at 1 Loop . . . . . . . . . . . . . . . . . . . 273
−
9.4.4 The Quark Field Self-Energy at 1 Loop . . . . . . . . . . . . . . . . . . . 283
−
9.4.5 The Vertex at 1 Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
−
10 The Renormalization Group 293
10.1 Critical Phenomena and The φ4 Theory . . . . . . . . . . . . . . . . . . . . . . . 293
10.1.1 Critical Line and Continuum Limit . . . . . . . . . . . . . . . . . . . . . . 293
10.1.2 Mean Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
10.1.3 Critical Exponents in Mean Field . . . . . . . . . . . . . . . . . . . . . . . 302
10.2 The Callan-Symanzik Renormalization Group Equation . . . . . . . . . . . . . . 307
10.2.1 Power Counting Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . 307
10.2.2 Renormalization Constants and Renormalization Conditions. . . . . . . . 311
10.2.3 Renormalization Group Functions and Minimal Subtraction . . . . . . . 314
10.2.4 CS Renormalization Group Equation in φ4 Theory . . . . . . . . . . . . . 317
10.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
10.3 Renormalization Constants and RenormalizationFunctions at Two-Loop . . . . . 325
10.3.1 The Divergent Part of the Effective Action . . . . . . . . . . . . . . . . . 325
10.3.2 Renormalization Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 330
10.3.3 Renormalization Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 332
10.4 Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
10.4.1 Critical Theory and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . 333
10.4.2 Scaling Domain (T >T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
c
10.4.3 Scaling Below T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
c
10.4.4 Critical Exponents from 2 Loop and Comparisonwith Experiment . . . 345
−
10.5 The Wilson Approximate Recursion Formulas . . . . . . . . . . . . . . . . . . . . 348
10.5.1 Kadanoff-Wilson Phase Space Analysis. . . . . . . . . . . . . . . . . . . . 348
10.5.2 Recursion Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
10.5.3 The Wilson-Fisher Fixed Point . . . . . . . . . . . . . . . . . . . . . . . . 359
10.5.4 The Critical Exponents ν . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
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10.5.5 The Critical Exponent η . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
10.6 Exercises and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
A Exams 371
B Problem Solutions 387
8 YDRI QFT
1
Introduction and References
This book-broject contains my lectures on quantum field theory (QFT) which were delivered
during the academic years 2010-2011, 2011-2012 and 2012-2013 at the University of Annaba to
first year and second year master students in theoretical physics. Each part of the book covers
roughly a semester consisting of 13 weeks of teaching with 2 lectures and 1 recitation per week.
In our master program quantum field theory is formally organized as an anual course so either
part I and part II can be used as the material for the course or part I and part III. Another
possibilityistomergepartIIandpartIIIinsuchawaythatthecontentfitswithinonesemester
as we will discuss further below.
PartIisessentialsincewelayinitthefoundationsandthelanguageofQFT,althoughIthink
now the third chapter of this part should be shortened in some fashion. Part II and part III are
independentunitessowecandoeitheroneinthesecondsemester. PartIIdealsmainlywiththe
problem of quantization and renormalization of electrodynamics using the canonical approach
while partIII dealswithpathintegralformulation,gaugetheoryandthe renormalizationgroup.
[The last chapter on the renormalization group was not actually covered with the other two
chapters of part III in a single semester. In fact it was delivered informally to master and
doctoral students].
InmyviewnowamergerofpartIIandpartIIIinwhichthelastchapterontherenormaliza-
tion groupis completely suppressed(althoughin my opinion it is the most importantchapter of
this book), the other two chapters of part III and the last two chapters of part II are shortened
considerably may fit within one single semester. Our actual experience has, on the other hand,
been as shown on table (1).
The three main and central references of this book were: Strathdee lecture notes for part
I and chapter two of part II, Peskin and Schroeder for part II especially the last chapter and
the secondchapter of partIII andZinn-Justin for the lastchapter on the renormalizationgroup
of part III. Chapter one of part II on the canonical quantization of the electromagnetic field
followsGreinerandReinhardt. Chapterone ofpartIII onthe pathintegralformulationandthe
effective action follows Randjbar-Daemi lecture notes. I have also benefited from many other
books and reviews; I only mention here A.M.Polyakov and J.Smit books and K.Wilson and J.
Kogut review. A far from complete list of references is given in the bibliography.
2 YDRI QFT
Year Spring Fall
Part I with the exception Part II with the exception
2011
of section 3.6. of sections 6.6, 7.4 and 7.8.
Part I with the exception Part III with the exception
2012
of section 3.6. of section 8.5 and chapter 10.