Symmetries in Fundamental Physics (Fundamental Theories of Physics Book 176)

Over the course of the last century it has become clear that both elementary particle physics and relativity theories are based on the notion of symmetries. These symmetries become manifest in that the "laws of nature" are invariant under spacetime transformations and/or gauge transformations. The consequences of these symmetries were analyzed as early as in 1918 by Emmy Noether on the level of action functionals. Her work did not receive due recognition for nearly half a century, but can today be understood as a recurring theme in classical mechanics, electrodynamics and special relativity, Yang-Mills type quantum field theories, and in general relativity. As a matter of fact, as shown in this monograph, many aspects of physics can be derived solely from symmetry considerations. This substantiates the statement of E.P. Wigner "... if we knew all the laws of nature, or the ultimate Law of nature, the invariance properties of these laws would not furnish us new information." Thanks to Wigner we now also understand the implications of quantum physics and symmetry considerations: Poincare invariance dictates both the characteristic properties of particles (mass, spin, ...) and the wave equations of spin 0, 1/2, 1, ... objects. Further, the work of C.N. Yang and R. Mills reveals the consequences of internal symmetries as exemplified in the symmetry group of elementary particle physics. Given this pivotal role of symmetries it is thus not surprising that current research in fundamental physics is to a great degree motivated and inspired by considerations of symmetry. The treatment of symmetries in this monograph ranges from classical physics to now well-established theories of fundamental interactions, to the latest research on unified theories and quantum gravity. Preface 6 Acknowledgments 12 Notation and Conventions 13 Contents 18 1 Introduction 26 1.1 Symmetry: Argument, Principle, and Leitmotif 26 1.2 Operations and Invariants 27 1.3 ``Symmetries'' in ``Fundamental Physics'' 29 1.3.1 What is Meant by ``Fundamental Physics''? 29 1.3.2 ``Physics'' on Which Level of Description? 30 1.3.3 Which Kind of ``Symmetry''? 34 1.4 The Scope of Symmetries 36 1.4.1 Ontology of Symmetries 36 1.4.2 Symmetry Groups in Fundamental Physics 38 1.4.3 The Use of Symmetries 40 1.5 Bibliographical Notes 42 2 Classical Mechanics 43 2.1 Newtonian and Analytical Mechanics 44 2.1.1 Newtonian Mechanics 44 2.1.2 Lagrange Form of Mechanics 46 2.1.3 Hamiltonian Formulation 48 2.1.4 Principle of Stationary Action 51 2.1.5 *Classical Mechanics in Geometrical Terms 56 2.2 Symmetries and Conservation Laws 61 2.2.1 Conservation Laws 61 2.2.2 Noether Theorem--A First Glimpse 67 2.2.3 Symmetry and Canonical Transformations 75 2.2.4 Conservation Laws and Symmetries 77 2.2.5 *Noether--Geometrically 87 2.3 Galilei Group 88 2.3.1 Transformations and Invariants of Classical Mechanics 88 2.3.2 Structure of the Galilei Group 89 2.3.3 Lie Algebra of the Galilei Group 90 2.4 Concluding Remarks and Bibliographical Notes 91 3 Electrodynamics and Special Relativity 93 3.1 Electrodynamics à la Maxwell 93 3.1.1 Maxwell Equations 93 3.1.2 Lorentz Boosts 94 3.2 Special Relativity 96 3.2.1 ``Deriving'' Special Relativity 96 3.2.2 Minkowski Geometry 100 3.2.3 Relativistic Mechanics 105 3.2.4 Relativistic Field Theory 108 3.3 Noether Theorems 114 3.3.1 Variational Symmetries in Field Theories 115 3.3.2 Global Symmetries and 1st Noether Theorem 118 3.3.3 Local Symmetries and 2nd Noether Theorem 125 3.3.4 Further Topics Relating to Variational Symmetries 130 3.4 Poincaré Transformations 137 3.4.1 Poincaré and Lorentz Groups 137 3.4.2 Poincaré Algebra 139 3.4.3 Galilei and Bargmann Algebra 141 3.4.4 Forms of Relativistic Dynamics 142 3.4.5 Kinematical Groups and Their Mutual Contractions 143 3.5 *Generalizations of Poincaré Symmetry 149 3.5.1 Conformal Symmetry 149 3.5.2 de Sitter Group 158 3.6 On the Validity of Special Relativity 162 3.7 Concluding Remarks and Bibliographical Notes 164 4 Quantum Mechanics 166 [DELETE] 166 4.1 Principles of Quantum Mechanics 166 4.1.1 Hilbert Space 167 4.1.2 Operators 167 4.1.3 States, Observables, and Measurements 169 4.1.4 Time Evolution 171 4.2 Symmetry Transformations in Quantum Mechanics 173 4.2.1 Wigner Theorem 174 4.2.2 Symmetry Transformations and Observables 176 4.2.3 ``Noether Theorem of Quantum Mechanics'' 176 4.2.4 Symmetries and Superselection Rules 177 4.3 Quantum Physics and Group Representation 179 4.3.1 Why Group Representation? 179 4.3.2 Galilei Operators 179 4.3.3 Bargmann Group 183 4.3.4 Symmetries of the Schrödinger Equation 185 4.4 Concluding Remarks and Bibliographical Notes 191 5 Relativistic Field Theory 192 5.1 Representations of the Poincaré Group 194 5.1.1 Global Structure of ISO(3, 1) 195 5.1.2 Transformation of the Generators 195 5.1.3 The ``Little Group'' 196 5.1.4 Classification of Particles 199 5.2 Symmetry and Quantum Field Theory 204 5.2.1 Lorentz Symmetry Rules Field Variants 204 5.2.2 Representations of SL(2, mathbbC) 205 5.2.3 Field Variants 206 5.2.4 Quantum-Field Theoretical Symmetry Transformations 208 5.3 Actions 209 5.3.1 Requirements on a QFT Action 209 5.3.2 Scalar Fields 211 5.3.3 Spinor Actions 217 5.3.4 Gauge Vector Fields 226 5.3.5 Higher-Spin Fields 241 5.4 Spontaneous Symmetry Breaking 244 5.4.1 Goldstone Bosons 245 5.4.2 Nambu-Goldstone Model 247 5.4.3 Higgs Mechanism 250 5.5 Discrete Symmetries 253 5.5.1 General Preliminary Remarks and Definition of Terms 253 5.5.2 Space Inversion P 255 5.5.3 Time Reversal T 258 5.5.4 Charge Conjugation C 260 5.5.5 CPT Theorem 261 5.6 Effective Field Theories 263 5.6.1 EFT: The Very Idea 263 5.6.2 Historical Examples 264 5.6.3 Renormalization (Group) 266 5.6.4 Chain of Effective Theories 271 5.7 Concluding Remarks and Bibliographical Notes 274 6 Particle Physics 281 6.1 Particles and Interactions 281 6.1.1 Standard Model Constituents 281 6.1.2 Quarks as Building Blocks of Hadrons 284 6.1.3 Interaction Processes 292 6.1.4 Lagrangian of the Standard Model 295 6.2 Strong Interactions 296 6.2.1 Lagrangian of Quantum Chromo Dynamics 296 6.2.2 Symmetries of QCD 297 6.2.3 Theoretical Consistency and Experimental Support 299 6.3 Weak and Electromagnetic Interaction 300 6.3.1 Fermi-Type Model of Weak Interactions 302 6.3.2 Current Algebra 303 6.3.3 Glashow-Salam-Weinberg Model 307 6.3.4 Theoretical Consistency and Experimental Support 312 6.4 Paralipomena on the Standard Model 313 6.4.1 Limits of the Standard Model 313 6.4.2 Massive Neutrinos 315 6.4.3 Anomalies 319 6.4.4 Strong CP Problem 321 6.4.5 Standard Model and Effective Field Theories 323 6.5 Concluding Remarks and Bibliographical Notes 328 7 General Relativity and Gravitation 330 7.1 Introductory Remarks 330 7.2 Equivalence Principle 332 7.2.1 Different Versions of the Equivalence Principle 332 7.2.2 Reference Systems and Gravitation 335 7.2.3 Geodesics 336 7.2.4 The ``Principle'' of General Covariance 340 7.3 Riemann-Cartan Geometry 340 7.3.1 Tensors 341 7.3.2 Affine Connection and Covariant Derivative 342 7.3.3 Torsion and Curvature 345 7.3.4 Metric 346 7.3.5 Tetrads and Spin Connections 350 7.4 Physics in Curved Spacetime 355 7.4.1 Mechanics, Hydrodynamics, Electrodynamics 355 7.4.2 Coupling Relativistic Fields to Gravity 358 7.5 Geometrodynamics 360 7.5.1 Field Equations 360 7.5.2 Action Functionals for General Relativity 364 7.5.3 Covariance, Invariance, and Symmetries 377 7.5.4 Noether Identities and Conservation Laws 382 7.6 Modifications and Extensions of/to General Relativity 396 7.6.1 Interpreting GR as a Spin-2 Field Theory 396 7.6.2 Altering the Geometry 398 7.6.3 Gravitation as a Gauge Theory 402 7.6.4 Changing Structures and Modifying Principles 413 7.7 Concluding Remarks and Bibliographical Notes 420 8 *Unified Field Theories 422 8.1 Grand Unified Theories 422 8.1.1 Motivation and Basic Concepts 422 8.1.2 SU(5) Grand Unification 425 8.1.3 SO(10) Grand Unification 429 8.1.4 Instead of a Conclusion 430 8.2 Kaluza-Klein Theory 430 8.2.1 Kaluza's and Klein's Contributions to the KK Theory 430 8.2.2 The 5D Model 433 8.2.3 Beyond Five Dimensions: Einstein-Yang-Mills Theory 440 8.2.4 Instead of a Conclusion 452 8.3 Supersymmetry 453 8.3.1 Why Supersymmetry? 453 8.3.2 Compelling Consequences of Fermi-Bose Symmetry 455 8.3.3 Global Supersymmetry 457 8.3.4 Local Supersymmetry and Supergravity 472 8.3.5 Instead of a Conclusion 479 8.4 Further Speculations 480 8.4.1 Compositeness and Technicolor 480 8.4.2 Strings and Branes 481 8.4.3 Gauge/Gravity Duality Conjecture 488 9 Conclusion 490 9.1 Symmetries: The Road to Reality 490 9.1.1 Symmetry: The Golden Thread 490 9.1.2 The ``Weltgesetze'' and Their Symmetries 494 9.1.3 History of Symmetry Considerations 497 9.2 Are Symmetries a Principle of Nature? 507 9.2.1 ƒ and Other Philosophical Questions 507 9.2.2 Symmetries and the Unification of Physics 509 9.2.3 Laws of Nature and Principles of Physics 515 9.2.4 Origin of Symmetries 521 9.3 Physics Beyond Symmetries 522 9.3.1 Prominent Non-Symmetries 522 9.3.2 Other Notions of Fundamental Physics 523 9.3.3 Are we Biased, or Haughty, or Simply in a Specific World? 527 Appendix A Group Theory 529 Appendix B Spinors, Z2-gradings, and Supergeometry 571 Appendix C Symmetries and Constrained Dynamics 602 Appendix D *Symmetries in Path-Integral and BRSTQuantization 671 Appendix E*Differential Geometry 697 Appendix F*Symmetries in Terms of Differential Forms 737 References 779 Index 801 Front Matter....Pages i-xxviii Introduction....Pages 1-17 Classical Mechanics....Pages 19-68 Electrodynamics and Special Relativity....Pages 69-141 Quantum Mechanics....Pages 143-168 Relativistic Field Theory....Pages 169-257 Particle Physics....Pages 259-307 General Relativity and Gravitation....Pages 309-400 *Unified Field Theories....Pages 401-468 Conclusion....Pages 469-507 Back Matter....Pages 509-788