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PART I Mathematical Foundations |
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1 Lie Groups and Lie Algebras: Basic Concepts |
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3 | (80) |
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1.1 Topological Groups and Lie Groups |
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5 | (15) |
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1.2 Linear Groups and Symmetry Groups of Vector Spaces |
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20 | (13) |
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1.3 Homomorphisms of Lie Groups |
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33 | (2) |
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35 | (4) |
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1.5 From Lie Groups to Lie Algebras |
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39 | (13) |
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1.6 *From Lie Subalgebras to Lie Subgroups |
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52 | (3) |
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55 | (12) |
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1.8 *Cartan's Theorem on Closed Subgroups |
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67 | (9) |
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1.9 Exercises for Chap. 1 |
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76 | (7) |
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2 Lie Groups and Lie Algebras: Representations and Structure Theory |
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83 | (44) |
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84 | (22) |
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2.2 Invariant Metrics on Lie Groups |
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106 | (2) |
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108 | (2) |
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2.4 *Semisimple and Compact Lie Algebras |
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110 | (7) |
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2.5 * Ad-Invariant Scalar Products on Compact Lie Groups |
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117 | (3) |
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2.6 *Homotopy Groups of Lie Groups |
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120 | (2) |
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2.7 Exercises for Chap. 2 |
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122 | (5) |
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127 | (66) |
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3.1 Transformation Groups |
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128 | (1) |
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3.2 Definition and First Properties of Group Actions |
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129 | (7) |
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3.3 Examples of Group Actions |
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136 | (5) |
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3.4 Fundamental Vector Fields |
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141 | (6) |
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3.5 The Maurer-Cartan Form and the Differential of a Smooth Group Action |
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147 | (2) |
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3.6 Left or Right Actions? |
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149 | (1) |
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150 | (12) |
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162 | (6) |
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3.9 *Stiefel and Grassmann Manifolds |
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168 | (2) |
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3.10 * The Exceptional Lie Group G2 |
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170 | (8) |
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3.11 *Godement's Theorem on the Manifold Structure of Quotient Spaces |
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178 | (7) |
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3.12 Exercises for Chap. 3 |
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185 | (8) |
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193 | (64) |
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4.1 General Fibre Bundles |
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195 | (12) |
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4.2 Principal Fibre Bundles |
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207 | (14) |
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4.3 * Formal Bundle Atlases |
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221 | (2) |
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223 | (2) |
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225 | (8) |
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4.6 The Clutching Construction |
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233 | (4) |
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4.7 Associated Vector Bundles |
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237 | (12) |
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4.8 Exercises for Chap. 4 |
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249 | (8) |
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5 Connections and Curvature |
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257 | (62) |
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5.1 Distributions and Connections |
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258 | (3) |
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261 | (4) |
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5.3 Gauge Transformations |
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265 | (5) |
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5.4 Local Connection 1-Forms and Gauge Transformations |
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270 | (3) |
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273 | (5) |
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5.6 Local Curvature 2-Forms |
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278 | (7) |
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5.7 `Generalized Electric and Magnetic Fields on Minkowski Spacetime of Dimension 4 |
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285 | (1) |
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286 | (3) |
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5.9 The Covariant Derivative on Associated Vector Bundles |
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289 | (7) |
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5.10 `Parallel Transport and Path-Ordered Exponentials |
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296 | (5) |
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5.11 `Holonomy and Wilson Loops |
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301 | (1) |
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5.12 The Exterior Covariant Derivative |
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302 | (6) |
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5.13 Forms with Values in Ad(P) |
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308 | (3) |
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5.14 *A Second and Third Version of the Bianchi Identity |
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311 | (1) |
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5.15 Exercises for Chap. 5 |
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312 | (7) |
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319 | (82) |
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6.1 The Pseudo-Orthogonal Group O(s, t) of Indefinite Scalar Products |
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320 | (7) |
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327 | (8) |
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6.3 The Clifford Algebras for the Standard Symmetric Bilinear Forms |
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335 | (11) |
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6.4 The Spinor Representation |
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346 | (2) |
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348 | (11) |
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359 | (4) |
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6.7 `Spin Invariant Scalar Products |
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363 | (10) |
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6.8 Explicit Formulas for Minkowski Spacetime of Dimension 4 |
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373 | (3) |
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6.9 Spin Structures and Spinor Bundles |
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376 | (7) |
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6.10 The Spin Covariant Derivative |
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383 | (6) |
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6.11 Twisted Spinor Bundles |
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389 | (3) |
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6.12 Twisted Chiral Spinors |
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392 | (2) |
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6.13 Exercises for Chap. 6 |
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394 | (7) |
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PART II The Standard Model of Elementary Particle Physics |
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7 The Classical Lagrangians of Gauge Theories |
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401 | (44) |
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7.1 Restrictions on the Set of Lagrangians |
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403 | (3) |
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7.2 The Hodge Star and the Codifferential |
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406 | (7) |
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7.3 The Yang-Mills Lagrangian |
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413 | (8) |
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7.4 Mathematical and Physical Conventions for Gauge Theories |
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421 | (3) |
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7.5 The Klein-Gordon and Higgs Lagrangians |
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424 | (5) |
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429 | (8) |
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437 | (1) |
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7.8 Dirac and Majorana Mass Terms |
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438 | (2) |
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7.9 Exercises for Chap. 7 |
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440 | (5) |
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8 The Higgs Mechanism and the Standard Model |
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445 | (82) |
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8.1 The Higgs Field and Symmetry Breaking |
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446 | (12) |
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8.2 Mass Generation for Gauge Bosons |
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458 | (6) |
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8.3 Massive Gauge Bosons in the SU(2) × U(1)-Theory of the Electroweak Interaction |
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464 | (9) |
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8.4 The SU(3)-Theory of the Strong Interaction (QCD) |
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473 | (5) |
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8.5 The Particle Content of the Standard Model |
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478 | (17) |
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8.6 Interactions Between Fermions and Gauge Bosons |
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495 | (10) |
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8.7 Interactions Between Higgs Bosons and Gauge Bosons |
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505 | (7) |
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8.8 Mass Generation for Fermions in the Standard Model |
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512 | (10) |
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8.9 The Complete Lagrangian of the Standard Model |
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522 | (1) |
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8.10 Lepton and Baryon Numbers |
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522 | (2) |
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8.11 Exercises for Chap. 8 |
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524 | (3) |
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9 Modern Developments and Topics Beyond the Standard Model |
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527 | (76) |
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9.1 Flavour and Chiral Symmetry |
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527 | (5) |
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532 | (14) |
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9.3 C, P and CP Violation |
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546 | (11) |
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9.4 Vacuum Polarization and Running Coupling Constants |
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557 | (7) |
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9.5 Grand Unified Theories |
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564 | (26) |
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9.6 A Short Introduction to the Minimal Supersymmetric Standard Model (MSSM) |
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590 | (8) |
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9.7 Exercises for Chap. 9 |
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598 | (5) |
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A Background on Differentiable Manifolds |
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603 | (1) |
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603 | (13) |
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616 | (9) |
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B Background on Special Relativity and Quantum Field Theory |
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625 | (1) |
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B.1 Basics of Special Relativity |
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625 | (3) |
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B.2 A Short Introduction to Quantum Field Theory |
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628 | (13) |
| References |
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641 | (6) |
| Index |
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647 | |
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