Table of Contents |
1. | Introduction | 1 |
1.1. | Symmetry: Argument, Principle, and Leitmotif | 1 |
1.2. | Operations and Invariants | 2 |
1.3. | "Symmetries" in "Fundamental Physics" | 4 |
1.3.1. | What is Meant by "Fundamental Physics"? | 4 |
1.3.2. | "Physics" on Which Level of Description? | 5 |
1.3.3. | Which Kind of "Symmetry"? | 9 |
1.4. | The Scope of Symmetries | 11 |
1.4.1. | Ontology of Symmetries | 11 |
1.4.2. | Symmetry Groups in Fundamental Physics | 13 |
1.4.3. | The Use of Symmetries | 14 |
1.5. | Bibliographical Notes | 16 |
2. | Classical Mechanics | 17 |
2.1. | Newtonian and Analytical Mechanics | 18 |
2.1.1. | Newtonian Mechanics | 18 |
2.1.2. | Lagrange Form of Mechanics | 20 |
2.1.3. | Hamiltonian Formulation | 22 |
2.1.4. | Principle of Stationary Action | 25 |
2.1.5. | *Classical Mechanics in Geometrical Terms | 30 |
2.2. | Symmetries and Conservation Laws | 34 |
2.2.1. | Conservation Laws | 34 |
2.2.2. | Noether Theorem-A First Glimpse | 40 |
2.2.3. | Symmetry and Canonical Transformations | 48 |
2.2.4. | Conservation Laws and Symmetries | 50 |
2.2.5. | *Noether-Geometrically | 60 |
2.3. | Galilei Group | 61 |
2.3.1. | Transformations and Invariants of Classical Mechanics | 61 |
2.3.2. | Structure of the Galilei Group | 62 |
2.3.3. | Lie Algebra of the Galilei Group | 63 |
2.4. | Concluding Remarks and Bibliographical Notes | 64 |
3. | Electrodynamics and Special Relativity | 67 |
3.1. | Electrodynamics a la Maxwell | 67 |
3.1.1. | Maxwell Equations | 67 |
3.1.2. | Lorentz Boosts | 68 |
3.2. | Special Relativity | 69 |
3.2.1. | "Deriving" Special Relativity | 69 |
3.2.2. | Minkowski Geometry | 73 |
3.2.3. | Relativistic Mechanics | 78 |
3.2.4. | Relativistic Field Theory | 81 |
3.3. | Noether Theorems | 87 |
3.3.1. | Variational Symmetries in Field Theories | 88 |
3.3.2. | Global Symmetries and 1st Noether Theorem | 91 |
3.3.3. | Local Symmetries and 2nd Noether Theorem | 96 |
3.3.4. | Further Topics Relating to Variational Symmetries | 101 |
3.4. | Poincare Transformations | 108 |
3.4.1. | Poincare and Lorentz Groups | 108 |
3.4.2. | Poincare Algebra | 110 |
3.4.3. | Galilei and Bargmann Algebra | 112 |
3.4.4. | Forms of Relativistic Dynamics | 113 |
3.4.5. | Kinematical Groups and Their Mutual Contractions | 114 |
3.5. | *Generalizations of Poincare Symmetry | 120 |
3.5.1. | Conformal Symmetry | 120 |
3.5.2. | de Sitter Group | 128 |
3.6. | On the Validity of Special Relativity | 131 |
3.7. | Concluding Remarks and Bibliographical Notes | 133 |
4. | Quantum Mechanics | 135 |
4.1. | Principles of Quantum Mechanics | 135 |
4.1.1. | Hilbert Space | 136 |
4.1.2. | Operators | 136 |
4.1.3. | States, Observables, and Measurements | 138 |
4.1.4. | Time Evolution | 140 |
4.2. | Symmetry Transformations in Quantum Mechanics | 142 |
4.2.1. | Wigner Theorem | 143 |
4.2.2. | Symmetry Transformations and Observables | 145 |
4.2.3. | "Noether Theorem of Quantum Mechanics" | 145 |
4.2.4. | Symmetries and Superselection Rules | 146 |
4.3. | Quantum Physics and Group Representation | 148 |
4.3.1. | Why Group Representation? | 148 |
4.3.2. | Galilei Operators | 148 |
4.3.3. | Bargmann Group | 152 |
4.3.4. | Symmetries of the Schrodinger Equation | 154 |
4.4. | Concluding Remarks and Bibliographical Notes | 159 |
5. | Relativistic Field Theory | 161 |
5.1. | Representations of the Poincare Group | 163 |
5.1.1. | Global Structure of ISO(3,1) | 164 |
5.1.2. | Transformation of the Generators | 164 |
5.1.3. | The "Little Group" | 165 |
5.1.4. | Classification of Particles | 167 |
5.2. | Symmetry and Quantum Field Theory | 173 |
5.2.1. | Lorentz Symmetry Rules Field Variants | 173 |
5.2.2. | Representations of SL(2, C) | 174 |
5.2.3. | Field Variants | 175 |
5.2.4. | Quantum-Field Theoretical Symmetry Transformations | 177 |
5.3. | Actions | 178 |
5.3.1. | Requirements on a QFT Action | 178 |
5.3.2. | Scalar Fields | 180 |
5.3.3. | Spinor Actions | 186 |
5.3.4. | Gauge Vector Fields | 194 |
5.3.5. | Higher-Spin Fields | 209 |
5.4. | Spontaneous Symmetry Breaking | 213 |
5.4.1. | Goldstone Bosons | 213 |
5.4.2. | Nambu-Goldstone Model | 216 |
5.4.3. | Higgs Mechanism | 218 |
5.5. | Discrete Symmetries | 222 |
5.5.1. | General Preliminary Remarks and Definition of Terms | 222 |
5.5.2. | Space Inversion P | 223 |
5.5.3. | Time Reversal T | 226 |
5.5.4. | Charge Conjugation C | 228 |
5.5.5. | CPT Theorem | 229 |
5.6. | Effective Field Theories | 231 |
5.6.1. | EFT: The Very Idea | 231 |
5.6.2. | Historical Examples | 232 |
5.6.3. | Renormalization (Group) | 234 |
5.6.4. | Chain of Effective Theories | 240 |
5.7. | Concluding Remarks and Bibliographical Notes | 242 |
6. | Particle Physics | 249 |
6.1. | Particles and Interactions | 249 |
6.1.1. | Standard Model Constituents | 249 |
6.1.2. | Quarks as Building Blocks of Hadrons | 252 |
6.1.3. | Interaction Processes | 260 |
6.1.4. | Lagrangian of the Standard Model | 262 |
6.2. | Strong Interactions | 264 |
6.2.1. | Lagrangian of Quantum Chromo Dynamics | 264 |
6.2.2. | Symmetries of QCD | 265 |
6.2.3. | Theoretical Consistency and Experimental Support | 266 |
6.3. | Weak and Electromagnetic Interaction | 268 |
6.3.1. | Fermi-Type Model of Weak Interactions | 269 |
6.3.2. | Current Algebra | 270 |
6.3.3. | Glashow-Salam-Weinberg Model | 274 |
6.3.4. | Theoretical Consistency and Experimental Support | 279 |
6.4. | Paralipomena on the Standard Model | 279 |
6.4.1. | Limits of the Standard Model | 280 |
6.4.2. | Massive Neutrinos | 282 |
6.4.3. | Anomalies | 285 |
6.4.4. | Strong CP Problem | 288 |
6.4.5. | Standard Model and Effective Field Theories | 290 |
6.5. | Concluding Remarks and Bibliographical Notes | 294 |
7. | General Relativity and Gravitation | 297 |
7.1. | Introductory Remarks | 297 |
7.2. | Equivalence Principle | 299 |
7.2.1. | Different Versions of the Equivalence Principle | 299 |
7.2.2. | Reference Systems and Gravitation | 302 |
7.2.3. | Geodesies | 303 |
7.2.4. | The "Principle" of General Covariance | 307 |
7.3. | Riemann-Cartan Geometry | 307 |
7.3.1. | Tensors | 308 |
7.3.2. | Affine Connection and Covariant Derivative | 309 |
7.3.3. | Torsion and Curvature | 312 |
7.3.4. | Metric | 313 |
7.3.5. | Tetrads and Spin Connections | 317 |
7.4. | Physics in Curved Spacetime | 322 |
7.4.1. | Mechanics, Hydrodynamics, Electrodynamics | 322 |
7.4.2. | Coupling Relativistic Fields to Gravity | 325 |
7.5. | Geometrodynamics | 327 |
7.5.1. | Field Equations | 327 |
7.5.2. | Action Functionals for General Relativity | 331 |
7.5.3. | Covariance, Invariance, and Symmetries | 343 |
7.5.4. | Noether Identities and Conservation Laws | 348 |
7.6. | *Modifications and Extensions of/to General Relativity | 361 |
7.6.1. | Interpreting GR as a Spin-2 Field Theory | 362 |
7.6.2. | Altering the Geometry | 364 |
7.6.3. | Gravitation as a Gauge Theory | 368 |
7.6.4. | Changing Structures and Modifying Principles | 378 |
7.7. | Concluding Remarks and Bibliographical Notes | 385 |
8. | *Unified Field Theories | 387 |
8.1. | Grand Unified Theories | 387 |
8.1.1. | Motivation and Basic Concepts | 387 |
8.1.2. | SU(5) Grand Unification | 390 |
8.1.3. | SO(10) Grand Unification | 394 |
8.1.4. | Instead of a Conclusion | 395 |
8.2. | Kaluza-Klein Theory | 395 |
8.2.1. | Kaluza's and Klein's Contributions to the KK Theory | 395 |
8.2.2. | The 5D Model | 398 |
8.2.3. | Beyond Five Dimensions: Einstein-Yang-Mills Theory | 405 |
8.2.4. | Instead of a Conclusion | 417 |
8.3. | Supersymmetry | 418 |
8.3.1. | Why Supersymmetry? | 419 |
8.3.2. | Compelling Consequences of Fermi-Bose Symmetry | 421 |
8.3.3. | Global Supersymmetry | 422 |
8.3.4. | Local Supersymmetry and Supergravity | 437 |
8.3.5. | Instead of a Conclusion | 444 |
8.4. | Further Speculations | 445 |
8.4.1. | Compositeness and Technicolor | 445 |
8.4.2. | Strings and Branes | 446 |
8.4.3. | Gauge/Gravity Duality Conjecture | 453 |
9. | Conclusion | 455 |
9.1. | Symmetries: The Road to Reality | 455 |
9.1.1. | Symmetry: The Golden Thread | 455 |
9.1.2. | The "Weltgesetze" and Their Symmetries | 459 |
9.1.3. | History of Symmetry Considerations | 462 |
9.2. | Are Symmetries a Principle of Nature? | 472 |
9.2.1. | ... and Other Philosophical Questions | 472 |
9.2.2. | Symmetries and the Unification of Physics | 474 |
9.2.3. | Laws of Nature and Principles of Physics | 479 |
9.2.4. | Origin of Symmetries | 486 |
9.3. | Physics Beyond Symmetries | 486 |
9.3.1. | Prominent Non-Symmetries | 487 |
9.3.2. | Other Notions of Fundamental Physics | 488 |
9.3.3. | Are we Biased, or Haughty, or Simply in a Specific World? | 491 |
Appendix A | Group Theory | 493 |
A.1. | Basics | 493 |
A.1.1. | Definitions: Algebraic Structures | 493 |
A.1.2. | Mapping of Groups | 497 |
A.1.3. | Simple Groups | 498 |
A.2. | Lie Groups | 501 |
A.2.1. | Definitions and Examples | 501 |
A.2.2. | Generators of a Lie Group | 505 |
A.2.3. | Lie Algebra Associated to a Lie Group | 506 |
A.2.4. | Inonu-Wigner Contraction of Lie Groups | 510 |
A.2.5. | Classification of Lie Groups | 512 |
A.2.6. | Infinite-Dimensional Lie Groups | 515 |
A.3. | Representation of Groups | 516 |
A.3.1. | Definitions and Examples | 516 |
A.3.2. | Representations of Finite Groups | 517 |
A.3.3. | Representation of Continuous Groups | 520 |
A.3.4. | Examples: Representations of SO(2), SO(3), SU(3) | 523 |
A.3.5. | Projective Representations and Central Charges | 530 |
Appendix B | Spinors, Z2-gradings, and Supergeometry | 535 |
B.1. | Spinors | 535 |
B.1.1. | Pauli and Dirac Matrices | 536 |
B.1.2. | Weyl Spinors | 538 |
B.1.3. | Spinors and Tensors | 541 |
B.1.4. | Dirac and Majorana Spinors | 542 |
B.2. | *Z2 Gradings | 544 |
B.2.1. | Definitions | 545 |
B.2.2. | Supertrace and Superdeterminant | 545 |
B.2.3. | Differentiation and Integration | 546 |
B.2.4. | Pseudo-Classical Mechanics | 549 |
B.3. | *Supergeometry | 550 |
B.3.1. | Superspace | 550 |
B.3.2. | Superfields | 552 |
B.3.3. | Superactions | 558 |
B.4. | *Supergroups | 563 |
B.4.1. | OSp(N/M) and the Super-Poincare Algebra | 563 |
B.4.2. | SU(N/M) and the Super-Conformal Algebra | 564 |
Appendix C | Symmetries and Constrained Dynamics | 565 |
C.1. | Constrained Dynamics | 565 |
C.1.1. | Singular Lagrangians | 565 |
C.1.2. | Constraints as a Consequence of Local Symmetries | 567 |
C.1.3. | Rosenfeld-Dirac-Bergmann Algorithm | 568 |
C.1.4. | First-Class Constraints and Symmetries | 574 |
C.1.5. | Second-Class Constraints and Gauge Conditions | 581 |
C.1.6. | Constraints in Field Theories: Some Remarks | 583 |
C.1.7. | Quantization of Constrained Systems | 584 |
C.2. | Yang-Mills Type Theories | 586 |
C.2.1. | Electrodynamics | 586 |
C.2.2. | Maxwell-Dirac Theory | 588 |
C.2.3. | Non-Abelian Gauge Theories | 590 |
C.3. | Reparametrization-Invariant Theories | 592 |
C.3.1. | Immediate Consequences of Reparametrization Invariance | 592 |
C.3.2. | Free Relativistic Particle | 594 |
C.3.3. | Metric Gravity | 604 |
C.3.4. | Tetrad Gravity | 619 |
C.3.5. | Einstein-Dirac-Yang-Mills-Higgs Theory | 622 |
C.4. | Alternative Approaches | 624 |
C.5. | Constraints and Presymplectic Geometry | 626 |
C.5.1. | Legendre Projectability | 626 |
C.5.2. | Symmetry Transformations in the Tangent and Cotangent Bundle | 628 |
C.5.3. | Constraint Stabilization | 631 |
Appendix D | *Symmetries in Path-Integral and BRST Quantization | 633 |
D.1. | Basics | 634 |
D.1.1. | Path-integral Formulation of Quantum Mechanics | 634 |
D.1.2. | Functional Integrals in Field Theory | 638 |
D.1.3. | Faddeev-Popov Ghost Fields in Theories with Local Symmetries | 641 |
D.2. | Noether and Functional Integrals | 645 |
D.2.1. | Noether Currents and Ward-Takahashi-Slavnov-Taylor Identities | 646 |
D.2.2. | Quantum Action and its Symmetries | 647 |
D.2.3. | BRST Symmetries | 649 |
D.2.4. | Fujikawa: Fermionic Path Integrals and Anomalies | 655 |
Appendix E | *Differential Geometry | 657 |
E.1. | Differentiable Manifolds | 658 |
E.1.1. | From Topological Spaces to Differentable Manifolds | 658 |
E.1.2. | Tensor Bundles | 659 |
E.1.3. | Flows and the Lie Derivative | 662 |
E.1.4. | Symplectic Manifolds | 663 |
E.2. | Cartan Calculus | 663 |
E.2.1. | Differential Forms | 663 |
E.2.2. | Differentiation with Respect to a Form | 665 |
E.2.3. | Hodge Duality* | 665 |
E.2.4. | Integration of Differential Forms and Stokes's Theorem | 667 |
E.2.5. | Poincare Lemma and de Rham Cohomology | 669 |
E.3. | Manifolds with Connection | 670 |
E.3.1. | Linear Connection on Tensor Fields | 671 |
E.3.2. | Covariant Derivative | 672 |
E.3.3. | Torsion and Curvature | 672 |
E.4. | Lie Groups | 674 |
E.4.1. | Lie Algebra | 675 |
E.4.2. | Group Covariant Derivative | 676 |
E.4.3. | Group Curvature | 677 |
E.4.4. | Isometries and Coset Manifolds | 678 |
E.5. | Fibre Bundles | 682 |
E.5.1. | Definition, Various Types, and Examples of Fibre Bundles | 682 |
E.5.2. | Connections in Fibre Bundles | 685 |
E.5.3. | Yang-Mills Gauge Field Theory in Fibre Bundle Language | 687 |
E.5.4. | Metric and Tetrad Gravity on a Bundle | 690 |
Appendix F | *Symmetries in Terms of Differential Forms | 695 |
F.1. | Actions and Field Equations | 695 |
F.1.1. | The "World" Action | 695 |
F.1.2. | Field Equations | 700 |
F.2. | Symmetries | 706 |
F.2.1. | Generic Variational Symmetries | 706 |
F.2.2. | Lorentz Transformations | 709 |
F.2.3. | Diffeomorphisms | 711 |
F.3. | Gravitational Theories | 718 |
F.3.1. | Energy-Momentum Conservation | 718 |
F.3.2. | Gravitational Theories Beyond Einstein | 723 |
F.3.3. | Topological Terms | 726 |
F.4. | Gauge Theories | 729 |
F.4.1. | Global Symmetry | 730 |
F.4.2. | Local Gauge Transformations | 730 |
| References | 735 |
| Index | 757 |