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Contents Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I. The Euclidean Manifold as a Paradigm 1. The Euclidean Space E3 (Hilbert Space and Lie Algebra Structure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.1 A Glance at History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.2 Algebraic Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 1.2.1 Symmetrization and Antisymmetrization . . . . . . . . . . 72 1.2.2 Cramer’s Rule for Systems of Linear Equations . . . . 73 1.2.3 Determinants and the Inverse Matrix . . . . . . . . . . . . . 75 1.2.4 The Hilbert Space Structure . . . . . . . . . . . . . . . . . . . . . 78 1.2.5 Orthogonality and the Dirac Calculus . . . . . . . . . . . . 81 1.2.6 The Lie Algebra Structure . . . . . . . . . . . . . . . . . . . . . . 82 1.2.7 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 1.2.8 The Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 1.2.9 Grassmann’s Alternating Product . . . . . . . . . . . . . . . . 86 1.2.10 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 1.3 The Skew-Field H of Quaternions . . . . . . . . . . . . . . . . . . . . . . . 89 1.3.1 The Field C of Complex Numbers . . . . . . . . . . . . . . . . 90 1.3.2 The Galois Group Gal(C|R) and Galois Theory . . . . 91 1.3.3 A Glance at the History of Hamilton’s Quaternions . 94 1.3.4 Pauli’s Spin Matrices and the Lie Algebras su(2) and sl(2,C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 1.3.5 Cayley’s Matrix Approach to Quaternions . . . . . . . . . 101 1.3.6 The Unit Sphere U(1,H) and the Electroweak Gauge Group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 1.3.7 The Four-Dimensional Extension of the Euclidean Space E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 1.3.8 Hamilton’s Nabla Operator . . . . . . . . . . . . . . . . . . . . . . 104 1.3.9 The Indefinite Hilbert Space H and the Minkowski Space M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 1.4 Riesz Duality between Vectors and Covectors . . . . . . . . . . . . . 104 XV XVI Contents 1.5 The Heisenberg Group, the Heisenberg Algebra, and Quantum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 1.6 The Heisenberg Group Bundle and Gauge Transformations . 112 2. Algebras and Duality (Tensor Algebra, Grassmann Algebra, Clifford Algebra, Lie Algebra) . . . . . . . . . . . . . . . . . . 115 2.1 Multilinear Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.1.1 The Graded Algebra of Polynomials . . . . . . . . . . . . . . 115 2.1.2 Products of Multilinear Functionals . . . . . . . . . . . . . . 118 2.1.3 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.1.4 Grassmann Algebra (Alternating Algebra) . . . . . . . . 121 2.1.5 Symmetric Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . 121 2.1.6 The Universal Property of the Tensor Product . . . . . 122 2.1.7 Diagram Chasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.2 The Clifford Algebra (E1) of the One-Dimensional Euclidean Space E1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2.3 Algebras of the Two-Dimensional Euclidean Space E2 . . . . . 127 2.3.1 The Clifford Algebra (E2) and Quaternions . . . . . . 128 2.3.2 The Cauchy–Riemann Differential Equations in Complex Function Theory . . . . . . . . . . . . . . . . . . . . 129 2.3.3 The Grassmann Algebra (E2) . . . . . . . . . . . . . . . . . . 131 2.3.4 The Grassmann Algebra (Ed 2) . . . . . . . . . . . . . . . . . . 132 2.3.5 The Symplectic Structure of E2 . . . . . . . . . . . . . . . . . . 132 2.3.6 The Tensor Algebra (E2) . . . . . . . . . . . . . . . . . . . . . 133 2.3.7 The Tensor Algebra (Ed 2) . . . . . . . . . . . . . . . . . . . . . 133 2.4 Algebras of the Three-Dimensional Euclidean Space E3 . . . . 133 2.4.1 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 2.4.2 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 2.4.3 Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.4.4 Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.5 Algebras of the Dual Euclidean Space Ed 3 . . . . . . . . . . . . . . . . 135 2.5.1 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2.5.2 Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2.6 The Mixed Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.7 The Hilbert Space Structure of the Grassmann Algebra (Hodge Duality) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2.7.1 The Hilbert Space (E3) . . . . . . . . . . . . . . . . . . . . . . . 139 2.7.2 The Hilbert Space (Ed 3) . . . . . . . . . . . . . . . . . . . . . . . 140 2.7.3 Multivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 2.8 The Clifford Structure of the Grassmann Algebra (Exterior–Interior K¨ahler Algebra) . . . . . . . . . . . . . . . . . . . . . . 144 2.8.1 The K¨ahler Algebra (E3)∨ . . . . . . . . . . . . . . . . . . . . . 144 2.8.2 The K¨ahler Algebra (Ed 3 )∨ . . . . . . . . . . . . . . . . . . . . 145 2.9 The C∗-Algebra End(E3) of the Euclidean Space . . . . . . . . . 145 2.10 Linear Operator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Contents XVII 2.10.1 The Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 2.10.2 The Grassmann Theorem . . . . . . . . . . . . . . . . . . . . . . . 148 2.10.3 The Superposition Principle . . . . . . . . . . . . . . . . . . . . . 151 2.10.4 Duality and the Fredholm Alternative . . . . . . . . . . . . 153 2.10.5 The Language of Matrices . . . . . . . . . . . . . . . . . . . . . . . 157 2.10.6 The Gaussian Elimination Method . . . . . . . . . . . . . . . 163 2.11 Changing the Basis and the Cobasis . . . . . . . . . . . . . . . . . . . . . 164 2.11.1 Similarity of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 2.11.2 Volume Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 2.11.3 The Determinant of a Linear Operator . . . . . . . . . . . . 167 2.11.4 The Reciprocal Basis in Crystallography . . . . . . . . . . 168 2.11.5 Dual Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 2.11.6 The Trace of a Linear Operator . . . . . . . . . . . . . . . . . . 170 2.11.7 The Dirac Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 2.12 The Strategy of Quotient Algebras and Universal Properties 174 2.13 A Glance at Division Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 176 2.13.1 From Real Numbers to Cayley’s Octonions . . . . . . . . 176 2.13.2 Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 2.13.3 The Fundamental Dimension Theorem . . . . . . . . . . . . 178 3. Representations of Symmetries in Mathematics and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.1 The Symmetric Group as a Prototype . . . . . . . . . . . . . . . . . . . 181 3.2 Incredible Cancellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 3.3 The Symmetry Strategy in Mathematics and Physics . . . . . . 186 3.4 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.5 Basic Notions of Representation Theory . . . . . . . . . . . . . . . . . 189 3.5.1 Linear Representations of Groups . . . . . . . . . . . . . . . . 189 3.5.2 Linear Representations of Lie Algebras . . . . . . . . . . . . 193 3.6 The Reflection Group Z2 as a Prototype . . . . . . . . . . . . . . . . . 194 3.6.1 Representations of Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . 194 3.6.2 Parity of Elementary Particles . . . . . . . . . . . . . . . . . . . 195 3.6.3 Reflections and Chirality in Nature . . . . . . . . . . . . . . . 196 3.6.4 Parity Violation in Weak Interaction . . . . . . . . . . . . . 196 3.6.5 Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 3.7 Permutation of Elementary Particles . . . . . . . . . . . . . . . . . . . . 197 3.7.1 The Principle of Indistinguishability of Quantum Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 3.7.2 The Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . 197 3.7.3 Entangled Quantum States . . . . . . . . . . . . . . . . . . . . . . 198 3.8 The Diagonalization of Linear Operators . . . . . . . . . . . . . . . . . 199 3.8.1 The Theorem of Principal Axes in Geometry and in Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 3.8.2 The Schur Lemma in Linear Representation Theory 202 3.8.3 The Jordan Normal Form of Linear Operators . . . . . 202 XVIII Contents 3.8.4 The Standard Maximal Torus of the Lie Group SU(n) and the Standard Cartan Subalgebra of the Lie Algebra su(n) . . . . . . . . . . . . . . . . . . . . . . . . 204 3.8.5 Eigenvalues and the Operator Strategy for Lie Algebras (Adjoint Representation) . . . . . . . . . . . . . . . 204 3.9 The Action of a Group on a Physical State Space, Orbits, and Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 3.10 The Intrinsic Symmetry of a Group . . . . . . . . . . . . . . . . . . . . . 206 3.11 Linear Representations of Finite Groups and the Hilbert Space of Functions on the Group. . . . . . . . . . . . . . . . . . . . . . . . 207 3.12 The Tensor Product of Representations and Characters . . . . 211 3.13 Applications to the Symmetric Group Sym(n) . . . . . . . . . . . . 214 3.13.1 The Characters of the Symmetric Group Sym(2) . . . 214 3.13.2 The Characters of the Symmetric Group Sym(3) . . . 216 3.13.3 Partitions and Young Frames . . . . . . . . . . . . . . . . . . . . 217 3.13.4 Young Tableaux and the Construction of a Complete System of Irreducible Representations . . . . . . . . . . . . 222 3.14 Application to the Standard Model in Elementary Particle Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 3.14.1 Quarks and Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 3.14.2 Antiquarks and Mesons . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.14.3 The Method of Highest Weight for Composed Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 3.14.4 The Pauli Exclusion Principle and the Color of Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 3.15 The Complexification of Lie Algebras . . . . . . . . . . . . . . . . . . . . 244 3.15.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 3.15.2 The Complex Lie Algebra slC(3,C) and Root Functionals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 3.15.3 Representations of the Complex Lie Algebra slC(3,C) and Weight Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 252 3.16 Classification of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.16.1 Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.16.2 Direct Product and Semisimplicity . . . . . . . . . . . . . . . 255 3.16.3 Solvablity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 3.16.4 Semidirect Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 3.17 Classification of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 3.17.1 The Classification of Complex Simple Lie Algebras . 259 3.17.2 Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 261 3.17.3 Solvability and the Heisenberg Algebra in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 3.17.4 Semidirect Product and the Levi Decomposition . . . 264 3.17.5 The Casimir Operators . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.18 Symmetric and Antisymmetric Functions . . . . . . . . . . . . . . . . 267 Contents XIX 3.18.1 Symmetrization and Antisymmetrization . . . . . . . . . . 268 3.18.2 Elementary Symmetric Polynomials . . . . . . . . . . . . . . 270 3.18.3 Power Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 3.18.4 Completely Symmetric Polynomials . . . . . . . . . . . . . . 271 3.18.5 Symmetric Schur Polynomials . . . . . . . . . . . . . . . . . . . 272 3.18.6 Raising Operators and the Creation and Annihilation of Particles . . . . . . . . . . . . . . . . . . . . . . . . 274 3.19 Formal Power Series Expansions and Generating Functions . 275 3.19.1 The Fundamental Frobenius Character Formula . . . . 276 3.19.2 The Pfaffian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3.20 Frobenius Algebras and Frobenius Manifolds . . . . . . . . . . . . . 278 3.21 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 3.22 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 3.22.1 Graduation in Nature. . . . . . . . . . . . . . . . . . . . . . . . . . . 287 3.22.2 General Strategy in Mathematics . . . . . . . . . . . . . . . . 287 3.22.3 The Super Lie Algebra of the Euclidean Space . . . . . 288 3.23 Artin’s Braid Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 3.23.1 The Braid Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 3.23.2 The Yang–Baxter Equation . . . . . . . . . . . . . . . . . . . . . 291 3.23.3 The Geometric Meaning of the Braid Group . . . . . . . 292 3.23.4 The Topology of the State Space of n Indistinguishable Particles in the Plane . . . . . . . . . . . . . . . . . . . . . . 294 3.24 The HOMFLY Polynomials in Knot Theory . . . . . . . . . . . . . . 295 3.25 Quantum Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 3.25.1 Quantum Mechanics as a Deformation . . . . . . . . . . . . 297 3.25.2 Manin’s Quantum Planes R2q and C2q . . . . . . . . . . . . . 298 3.25.3 The Coordinate Algebra of the Lie Group SL(2,C) . 300 3.25.4 The Quantum Group SLq(2,C) . . . . . . . . . . . . . . . . . . 301 3.25.5 The Quantum Algebra slq(2,C) . . . . . . . . . . . . . . . . . . 302 3.25.6 The Coaction of the Quantum Group SLq(2,C) on the Quantum Plane C2q . . . . . . . . . . . . . . . . . . . . . . . 303 3.25.7 Noncommutative Euclidean Geometry and Quantum Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 3.26 Additive Groups, Betti Numbers, Torsion Coefficients, and Homological Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 3.27 Lattices and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 4. The Euclidean Manifold E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 4.1 Velocity Vectors and the Tangent Space . . . . . . . . . . . . . . . . . 321 4.2 Duality and Cotangent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 323 4.3 Parallel Transport and Acceleration . . . . . . . . . . . . . . . . . . . . . 323 4.4 Newton’s Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 4.5 Bundles Over the Euclidean Manifold . . . . . . . . . . . . . . . . . . . 324 4.5.1 The Tangent Bundle and Velocity Vector Fields . . . . 325 4.5.2 The Cotangent Bundle and Covector Fields . . . . . . . 325 XX Contents 4.5.3 Tensor Bundles and Tensor Fields . . . . . . . . . . . . . . . . 326 4.5.4 The Frame Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 4.6 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 4.6.1 Newton and Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 4.6.2 The Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 329 4.6.3 The Dirac Delta Function and Laurent Schwartz’s Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 4.6.4 The Algebraization of the Calculus . . . . . . . . . . . . . . . 330 4.6.5 Formal Power Series Expansions and the Ritt Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 4.6.6 Differential Rings and Derivations . . . . . . . . . . . . . . . . 331 4.6.7 The p-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 4.6.8 The Local–Global Principle in Mathematics . . . . . . . 336 4.6.9 The Global Adelic Ring . . . . . . . . . . . . . . . . . . . . . . . . . 337 4.6.10 Solenoids, Foliations, and Chaotic Dynamical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 4.6.11 Period Three Implies Chaos . . . . . . . . . . . . . . . . . . . . . 345 4.6.12 Differential Calculi, Noncommutative Geometry, and the Standard Model in Particle Physics . . . . . . . . . . . 346 4.6.13 BRST-Symmetry, Cohomology, and the Quantization of Gauge Theories. . . . . . . . . . . . . . . . . . . . . . . . . . 347 4.6.14 Itˆo’s Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . 348 5. The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 5.1 Linearization and the Lie Algebra u(1) . . . . . . . . . . . . . . . . . . 355 5.2 The Universal Covering Group of U(1) . . . . . . . . . . . . . . . . . . 356 5.3 Left-Invariant Velocity Vector Fields on U(1) . . . . . . . . . . . . . 356 5.3.1 The Maurer–Cartan Form of U(1) . . . . . . . . . . . . . . . . 357 5.3.2 The Maurer–Cartan Structural Equation . . . . . . . . . . 358 5.4 The Riemannian Manifold U(1) and the Haar Measure . . . . 358 5.5 The Discrete Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . 359 5.5.1 The Hilbert Space L2(U(1)) . . . . . . . . . . . . . . . . . . . . . 359 5.5.2 Pseudo–Differential Operators . . . . . . . . . . . . . . . . . . . 360 5.5.3 The Sobolev Space Wm 2 (U(1)) . . . . . . . . . . . . . . . . . . . 361 5.6 The Group of Motions on the Gaussian Plane . . . . . . . . . . . . 361 5.7 Rotations of the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . . 362 5.8 Pontryagin Duality for U(1) and Quantum Groups . . . . . . . . 369 6. Infinitesimal Rotations and Constraints in Physics . . . . . . . 371 6.1 The Group U(E3) of Unitary Transformations . . . . . . . . . . . . 371 6.2 Euler’s Rotation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 6.3 The Lie Algebra of Infinitesimal Rotations . . . . . . . . . . . . . . . 374 6.4 Constraints in Classical Physics. . . . . . . . . . . . . . . . . . . . . . . . . 375 6.4.1 Archimedes’ Lever Principle . . . . . . . . . . . . . . . . . . . . . 375 Contents XXI 6.4.2 d’Alembert’s Principle of Virtual Power . . . . . . . . . . . 377 6.4.3 d’Alembert’s Principle of Virtual Work . . . . . . . . . . . 378 6.4.4 The Gaussian Principle of Least Constraint and Constraining Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 6.4.5 Manifolds and Lagrange’s Variational Principle . . . . 383 6.4.6 The Method of Perturbation Theory . . . . . . . . . . . . . . 384 6.4.7 Further Reading on Perturbation Theory and its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 6.5 Application to the Motion of a Rigid Body . . . . . . . . . . . . . . . 388 6.5.1 The Center of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 389 6.5.2 Moving Orthonormal Frames and Infinitesimal Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 6.5.3 Kinetic Energy and the Inertia Tensor . . . . . . . . . . . . 391 6.5.4 The Equations of Motion – the Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 6.5.5 Euler’s Equation of the Spinning Top . . . . . . . . . . . . . 395 6.5.6 Equilibrium States and Torque . . . . . . . . . . . . . . . . . . 397 6.5.7 The Principal Bundle R3 × SO(3) – the Position Space of a Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . 397 6.6 A Glance at Constraints in Quantum Field Theory . . . . . . . . 398 6.6.1 Gauge Transformations and Virtual Degrees of Freedom in Gauge Theory . . . . . . . . . . . . . . . . . . . . 399 6.6.2 Elimination of Unphysical States (Ghosts). . . . . . . . . 400 6.6.3 Degenerate Minimum Problems . . . . . . . . . . . . . . . . . . 401 6.6.4 Variation of the Action Functional . . . . . . . . . . . . . . . 404 6.6.5 Degenerate Lagrangian and Constraints . . . . . . . . . . . 408 6.6.6 Degenerate Legendre Transformation . . . . . . . . . . . . . 408 6.6.7 Global and Local Symmetries . . . . . . . . . . . . . . . . . . . . 411 6.6.8 Quantum Symmetries and Anomalies . . . . . . . . . . . . . 414 6.7 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 6.7.1 Topological Constraints in Maxwell’s Theory of Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 6.7.2 Constraints in Einstein’s Theory of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 6.7.3 Hilbert’s Algebraic Theory of Relations (Syzygies) . 417 6.8 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 7. Rotations, Quaternions, the Universal Covering Group, and the Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 7.1 Quaternions and the Cayley–Hamilton Rotation Formula . . 425 7.2 The Universal Covering Group SU(2) . . . . . . . . . . . . . . . . . . . 426 7.3 Irreducible Unitary Representations of the Group SU(2) and the Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 7.3.1 The Spin Quantum Numbers . . . . . . . . . . . . . . . . . . . . 428 7.3.2 The Addition Theorem for the Spin . . . . . . . . . . . . . . 434 XXII Contents 7.3.3 The Model of Homogeneous Polynomials . . . . . . . . . . 435 7.3.4 The Clebsch–Gordan Coefficients. . . . . . . . . . . . . . . . . 436 7.4 Heisenberg’s Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 8. Changing Observers – A Glance at Invariant Theory Based on the Principle of the Correct Index Picture . . . . . 439 8.1 A Glance at the History of Invariant Theory . . . . . . . . . . . . . 439 8.2 The Basic Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 8.3 The Mnemonic Principle of the Correct Index Picture . . . . . 443 8.4 Real-Valued Physical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 8.4.1 The Chain Rule and the Key Duality Relation . . . . . 445 8.4.2 Linear Differential Operators . . . . . . . . . . . . . . . . . . . . 446 8.4.3 Duality and Differentials . . . . . . . . . . . . . . . . . . . . . . . . 447 8.4.4 Admissible Systems of Observers . . . . . . . . . . . . . . . . . 449 8.4.5 Tensorial Families and the Construction of Invariants via the Basic Trick of Index Killing . . . . . . . . . . . . . . . 452 8.4.6 Orientation, Pseudo-Tensorial Families, and the Levi-Civita Duality . . . . . . . . . . . . . . . . . . . . . . . . . 460 8.5 Differential Forms (Exterior Product) . . . . . . . . . . . . . . . . . . . 464 8.5.1 Cartan Families and the Cartan Differential . . . . . . . 464 8.5.2 Hodge Duality, the Hodge Codifferential, and the Laplacian (Hodge’s Star Operator) . . . . . . . . . . . . 469 8.6 The K¨ahler–Clifford Calculus and the Dirac Operator (Interior Product). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 8.6.1 The Exterior Differential Algebra . . . . . . . . . . . . . . . . 475 8.6.2 The Interior Differential Algebra . . . . . . . . . . . . . . . . . 477 8.6.3 K¨ahler Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 8.6.4 Applications to Fundamental Differential Equations in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 8.6.5 The Potential Equation and the Importance of the de Rham Cohomology . . . . . . . . . . . . . . . . . . . . 481 8.6.6 Tensorial Differential Forms . . . . . . . . . . . . . . . . . . . . . 482 8.7 Integrals over Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . 483 8.8 Derivatives of Tensorial Families . . . . . . . . . . . . . . . . . . . . . . . . 484 8.8.1 The Lie Algebra of Linear Differential Operators and the Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 8.8.2 The Inverse Index Principle . . . . . . . . . . . . . . . . . . . . . 493 8.8.3 The Covariant Derivative (Weyl’s Affine Connection) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 8.9 The Riemann–Weyl Curvature Tensor . . . . . . . . . . . . . . . . . . . 503 8.9.1 Second-Order Covariant Partial Derivatives . . . . . . . . 504 8.9.2 Local Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 8.9.3 The Method of Differential Forms (Cartan’s Structural Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 8.9.4 The Operator Method . . . . . . . . . . . . . . . . . . . . . . . . . . 510 Contents XXIII 8.10 The Riemann–Christoffel Curvature Tensor . . . . . . . . . . . . . . 511 8.10.1 The Levi-Civita Metric Connection . . . . . . . . . . . . . . . 512 8.10.2 Levi-Civita’s Parallel Transport . . . . . . . . . . . . . . . . . . 513 8.10.3 Symmetry Properties of the Riemann–Christoffel Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 8.10.4 The Ricci Curvature Tensor and the Einstein Tensor 516 8.10.5 The Conformal Weyl Curvature Tensor . . . . . . . . . . . 517 8.10.6 The Hodge Codifferential and the Covariant Partial Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 8.10.7 The Weitzenb¨ock Formula for the Hodge Laplacian . 519 8.10.8 The One-Dimensional σ-Model and Affine Geodesics 520 8.11 The Beauty of Connection-Free Derivatives . . . . . . . . . . . . . . . 522 8.11.1 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 8.11.2 The Cartan Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 523 8.11.3 The Weyl Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 8.12 Global Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 8.13 Summary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 8.14 Two Strategies in Invariant Theory . . . . . . . . . . . . . . . . . . . . . 529 8.15 Intrinsic Tangent Vectors and Derivations . . . . . . . . . . . . . . . . 529 8.16 Further Reading on Symmetry and Invariants . . . . . . . . . . . . 534 9. Applications of Invariant Theory to the Rotation Group . 557 9.1 The Method of Orthonormal Frames on the Euclidean Manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 9.1.1 Hamilton’s Quaternionic Analysis . . . . . . . . . . . . . . . . 557 9.1.2 Transformation of Orthonormal Frames . . . . . . . . . . . 559 9.1.3 The Coordinate-Dependent Approach (SO(3)-Tensor Calculus) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 9.1.4 The Coordinate-Free Approach . . . . . . . . . . . . . . . . . . 561 9.1.5 Hamilton’s Nabla Calculus . . . . . . . . . . . . . . . . . . . . . . 563 9.1.6 Rotations and Cauchy’s Invariant Functions . . . . . . . 565 9.2 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 9.2.1 Local Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 9.2.2 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 9.2.3 The Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 9.2.4 Special Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 9.3 The Index Principle of Mathematical Physics . . . . . . . . . . . . . 574 9.3.1 The Basic Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 9.3.2 Applications to Vector Analysis . . . . . . . . . . . . . . . . . . 575 9.4 The Euclidean Connection and Gauge Theory . . . . . . . . . . . . 576 9.4.1 Covariant Partial Derivative . . . . . . . . . . . . . . . . . . . . . 577 9.4.2 Curves of Least Kinectic Energy (Affine Geodesics) . 577 9.4.3 Curves of Minimal Length . . . . . . . . . . . . . . . . . . . . . . . 579 9.4.4 The Gauss Equations of Moving Frames . . . . . . . . . . 580 XXIV Contents 9.4.5 Parallel Transport of a Velocity Vector and Cartan’s Propagator Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 9.4.6 The Dual Cartan Equations of Moving Frames . . . . . 584 9.4.7 Global Parallel Transport on Lie Groups and the Maurer–Cartan Form . . . . . . . . . . . . . . . . . . . . . . . 585 9.4.8 Cartan’s Global Connection Form on the Frame Bundle of the Euclidean Manifold . . . . 587 9.4.9 The Relation to Gauge Theory . . . . . . . . . . . . . . . . . . 590 9.4.10 The Reduction of the Frame Bundle to the Orthonormal Frame Bundle . . . . . . . . . . . . . . . 593 9.5 The Sphere as a Paradigm in Riemannian Geometry and Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 9.5.1 The Newtonian Equation of Motion and Levi-Civita’s Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 9.5.2 Geodesic Triangles and the Gaussian Curvature . . . . 599 9.5.3 Geodesic Circles and the Gaussian Curvature . . . . . . 600 9.5.4 The Spherical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . 600 9.5.5 Geodesics and Gauge Transformations . . . . . . . . . . . . 603 9.5.6 The Local Hilbert Space Structure . . . . . . . . . . . . . . . 606 9.5.7 The Almost Complex Structure . . . . . . . . . . . . . . . . . . 607 9.5.8 The Levi-Civita Connection on the Tangent Bundle and the Riemann Curvature Tensor . . . . . . . . . . . . . . 608 9.5.9 The Components of the Riemann Curvature Tensor and Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 9.5.10 Computing the Riemann Curvature Operator via Parallel Transport Along Loops . . . . . . . . . . . . . . . . . . 619 9.5.11 The Connection on the Frame Bundle and Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 9.5.12 Poincar′e’s Topological No-Go Theorem for Velocity Vector Fields on a Sphere . . . . . . . . . . . . . . . . . . . . . . . 623 9.6 Gauss’ Theorema Egregium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 9.6.1 The Natural Basis and Cobasis . . . . . . . . . . . . . . . . . . 623 9.6.2 Intrinsic Metric Properties . . . . . . . . . . . . . . . . . . . . . . 627 9.6.3 The Extrinsic Definition of the Gaussian Curvature 628 9.6.4 The Gauss–Weingarten Equations for Moving Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 9.6.5 The Integrability Conditions and the Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 9.6.6 The Intrinsic Characterization of the Gaussian Curvature (Theorema Egregium) . . . . . . . . . . . . . . . . . 632 9.6.7 Differential Invariants and the Existence and Uniqueness Theorem of Classical Surface Theory . . . 633 9.6.8 Gauss’ Theorema Elegantissimum and the Gauss–Bonnet Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 634 Contents XXV 9.6.9 Gauss’ Total Curvature and Topological Charges . . . 635 9.6.10 Cartan’s Method of Moving Orthonormal Frames . . 636 9.7 Parallel Transport in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 9.8 Finsler Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 9.9 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 10. Temperature Fields on the Euclidean Manifold E3 . . . . . . . 645 10.1 The Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 10.2 The Lie Derivative of a Temperature Field along the Flow of Fluid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 10.2.1 The Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 10.2.2 The Linearized Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 650 10.2.3 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 10.2.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 10.3 Higher Variations of a Temperature Field and the Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 10.4 The Fr′echet Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 10.5 Global Linearization of Smooth Maps and the Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 10.6 The Global Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 10.7 The Transformation of Temperature Fields . . . . . . . . . . . . . . . 657 11. Velocity Vector Fields on the Euclidean Manifold E3 . . . . . 659 11.1 The Transformation of Velocity Vector Fields . . . . . . . . . . . . . 661 11.2 The Lie Derivative of an Electric Field along the Flow of Fluid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 11.2.1 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 11.2.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663 11.2.3 The Lie Algebra of Velocity Vector Fields . . . . . . . . . 664 12. Covector Fields and Cartan’s Exterior Differential – the Beauty of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . 665 12.1 Ariadne’s Thread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 12.1.1 One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 12.1.2 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670 12.1.3 Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 12.1.4 Integration over Manifolds . . . . . . . . . . . . . . . . . . . . . . 681 12.1.5 Integration over Singular Chains . . . . . . . . . . . . . . . . . 684 12.2 Applications to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 12.2.1 Single-Valued Potentials and Gauge Transformations 685 12.2.2 Multi-Valued Potentials and Riemann Surfaces . . . . . 687 12.2.3 The Electrostatic Coulomb Force and the Dirac Delta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 12.2.4 The Magic Green’s Function and the Dirac Delta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 XXVI Contents 12.2.5 Conservation of Heat Energy – the Paradigm of Conservation Laws in Physics . . . . . . . . . . . . . . . . . 695 12.2.6 The Classical Predecessors of the Yang–Mills Equations in Gauge Theory (Fluid Dynamics and Electrodynamics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 12.2.7 Thermodynamics and the Pfaff Problem . . . . . . . . . . 698 12.2.8 Classical Mechanics and Symplectic Geometry . . . . . 700 12.2.9 The Universality of Differential Forms . . . . . . . . . . . . 700 12.2.10 Cartan’s Covariant Differential and the Four Fundamental Interactions in Nature . . . . . . . . . . . . . . 700 12.3 Cartan’s Algebra of Alternating Differential Forms . . . . . . . . 701 12.3.1 The Geometric Approach . . . . . . . . . . . . . . . . . . . . . . . 701 12.3.2 The Grassmann Bundle . . . . . . . . . . . . . . . . . . . . . . . . . 704 12.3.3 The Tensor Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 12.3.4 The Transformation of Covector Fields . . . . . . . . . . . 705 12.4 Cartan’s Exterior Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 12.4.1 Invariant Definition via the Lie Algebra of Velocity Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 12.4.2 The Supersymmetric Leibniz Rule . . . . . . . . . . . . . . . . 709 12.4.3 The Poincar′e Cohomology Rule . . . . . . . . . . . . . . . . . . 710 12.4.4 The Axiomatic Approach . . . . . . . . . . . . . . . . . . . . . . . 710 12.5 The Lie Derivative of Differential Forms . . . . . . . . . . . . . . . . . 712 12.5.1 Invariant Definition via the Flow of Fluid Particles . 712 12.5.2 The Contraction Product between Velocity Vector Fields and Differential Forms . . . . . . . . . . . . . . . . . . . . 714 12.5.3 Cartan’s Magic Formula . . . . . . . . . . . . . . . . . . . . . . . . 714 12.5.4 The Lie Derivative of the Volume Form . . . . . . . . . . . 715 12.5.5 The Lie Derivative of the Metric Tensor Field . . . . . 715 12.5.6 The Lie Derivative of Linear Operator Fields . . . . . . 716 12.6 Diffeomorphisms and the Mechanics of Continua – the Prototype of an Effective Theory in Physics. . . . . . . . . . . 717 12.6.1 Linear Diffeomorphisms and Deformation Operators 718 12.6.2 Local Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 719 12.6.3 Proper Maps and Hadamard’s Theorem on Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 12.6.4 Monotone Operators and Diffeomorphisms . . . . . . . . 720 12.6.5 Sard’s Theorem on the Genericity of Regular Solution Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 12.6.6 The Strain Tensor and the Stress Tensor in Cauchy’s Theory of Elasticity . . . . . . . . . . . . . . . . . 722 12.6.7 The Rate-of-Strain Tensor and the Stress Tensor in the Hydrodynamics of Viscous Fluids . . . . . . . . . . 725 12.6.8 Vorticity Lines of a Fluid . . . . . . . . . . . . . . . . . . . . . . . 728 12.6.9 The Lie Derivative of the Covector Field . . . . . . . . . . 728 Contents XXVII 12.7 The Generalized Stokes Theorem (Main Theorem of Calculus) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 12.8 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 12.8.1 Infinitesimal Isometries (Metric Killing Vector Fields) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 12.8.2 Absolute Integral Invariants and Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734 12.8.3 Relative Integral Invariants and the Vorticity Theorems for Fluids due to Thomson and Helmholtz 735 12.8.4 The Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . 735 12.8.5 The Noether Theorem – Symmetry Implies Conservation Laws in the Calculus of Variations . . . 737 12.9 The Hamiltonian Flow on the Euclidean Manifold – a Paradigm of Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 744 12.9.1 Hamilton’s Principle of Critical Action . . . . . . . . . . . . 746 12.9.2 Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 12.9.3 The Poincar′e–Cartan Integral Invariant . . . . . . . . . . . 749 12.9.4 Energy Conservation and the Liouville Integral Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749 12.9.5 Jacobi’s Canonical Transformations, Lie’s Contact Geometry, and Symplectic Geometry . . . . . . . . . . . . . 750 12.9.6 Hilbert’s Invariant Integral . . . . . . . . . . . . . . . . . . . . . . 753 12.9.7 Jacobi’s Integration Method . . . . . . . . . . . . . . . . . . . . . 753 12.9.8 Legendre Transformation . . . . . . . . . . . . . . . . . . . . . . . . 754 12.9.9 Carath′eodory’s Royal Road to the Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 12.9.10 Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 12.10 The Main Theorems in Classical Gauge Theory (Existence of Potentials) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760 12.10.1 Contractible Manifolds (the Poincar′e–Volterra Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 12.10.2 Non-Contractible Manifolds and Betti Numbers (De Rham’s Theorem on Periods) . . . . . . . . . . . . . . . . 764 12.10.3 The Main Theorem for Velocity Vector Fields . . . . . . 766 12.11 Systems of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 12.11.1 Integrability Condition . . . . . . . . . . . . . . . . . . . . . . . . . 767 12.11.2 The Frobenius Theorem for Pfaff Systems . . . . . . . . . 769 12.11.3 The Dual Frobenius Theorem . . . . . . . . . . . . . . . . . . . . 770 12.11.4 The Pfaff Normal Form and the Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 770 12.12 Hodge Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 12.12.1 The Hodge Codifferential . . . . . . . . . . . . . . . . . . . . . . . 772 12.12.2 The Hodge Homology Rule . . . . . . . . . . . . . . . . . . . . . . 773 XXVIII Contents 12.12.3 The Relation between the Cartan–Hodge Calculus and Classical Vector Analysis via Riesz Duality . . . . 773 12.12.4 The Classical Prototype of the Yang–Mills Equation in Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 12.12.5 The Hodge–Laplace Operator and Harmonic Forms. 775 12.13 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775 12.14 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 Part II. Ariadne’s Thread in Gauge Theory 13. The Commutative Weyl U(1)-Gauge Theory and the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 13.1 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 13.2 The Fundamental Principle of Local Symmetry Invariance in Modern Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 13.2.1 The Free Meson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 13.2.2 Local Symmetry and the Charged Meson in an Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 818 13.3 The Vector Bundle M4×C, Covariant Directional Derivative, and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820 13.4 The Principal Bundle M4 ×U(1) and the Parallel Transport of the Local Phase Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 13.5 Parallel Transport of Physical Fields – the Propagator Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827 13.6 The Wilson Loop and Holonomy . . . . . . . . . . . . . . . . . . . . . . . . 829 14. Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 14.1 The Prototype in Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 14.2 The Goldstone-Particle Mechanism . . . . . . . . . . . . . . . . . . . . . . 832 14.3 The Higgs-Particle Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 834 14.4 Dimensional Reduction and the Kaluza–Klein Approach . . . 835 14.5 Superconductivity and the Ginzburg–Landau Equation . . . . 836 14.6 The Idea of Effective Theories in Physics . . . . . . . . . . . . . . . . 840 15. The Noncommutative Yang–Mills SU(N)-Gauge Theory 843 15.1 The Vector Bundle M4 × CN, Covariant Directional Derivative, and Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843 15.2 The Principal Bundle M4 ×G and the Parallel Transport of the Local Phase Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 15.3 Parallel Transport of Physical Fields – the Propagator Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 15.4 The Principle of Critical Action and the Yang–Mills Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854 15.5 The Universal Extension Strategy via the Leibniz Rule . . . . 858 Contents XXIX 15.6 Tensor Calculus on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . 859 15.6.1 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 860 15.6.2 Connection and Christoffel Symbols . . . . . . . . . . . . . . 863 15.6.3 Covariant Differential for Differential Forms of Tensor Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864 15.6.4 Application to the Riemann Curvature Operator . . . 867 16. Cocycles and Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 16.1 Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871 16.2 Physical Fields via the Cocycle Strategy . . . . . . . . . . . . . . . . . 872 16.3 Local Phase Factors via the Cocycle Strategy . . . . . . . . . . . . . 873 17. The Axiomatic Geometric Approach to Bundles . . . . . . . . . 875 17.1 Connection on a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . 875 17.2 Connection on a Principal Bundle . . . . . . . . . . . . . . . . . . . . . . . 879 17.3 The Philosophy of Parallel Transport . . . . . . . . . . . . . . . . . . . . 883 17.3.1 Vector Bundles Associated to a Principal Bundle . . . 884 17.3.2 Horizontal Vector Fields on a Principal Bundle . . . . 887 17.3.3 The Lifting of Curves in Fiber Bundles . . . . . . . . . . . 888 17.4 A Glance at the History of Gauge Theory . . . . . . . . . . . . . . . . 891 17.4.1 From Weyl’s Gauge Theory in Gravity to the Standard Model in Particle Physics . . . . . . . . . 891 17.4.2 From Gauss’ Theorema Egregium to Modern Differential Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . 896 17.4.3 The Work of Hermann Weyl . . . . . . . . . . . . . . . . . . . . . 900 Part III. Einstein’s Theory of Special Relativity 18. Inertial Systems and Einstein’s Principle of Special Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905 18.1 The Principle of Special Relativity . . . . . . . . . . . . . . . . . . . . . . 908 18.1.1 The Lorentz Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909 18.1.2 The Transformation of Velocities . . . . . . . . . . . . . . . . . 910 18.1.3 Time Dilatation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911 18.1.4 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911 18.1.5 The Synchronization of Clocks . . . . . . . . . . . . . . . . . . . 912 18.1.6 General Change of Inertial Systems in Terms of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912 18.2 Matrix Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914 18.2.1 The Group O(1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914 18.2.2 The Lorentz Group O(1,3) . . . . . . . . . . . . . . . . . . . . . . 916 18.3 Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 918 18.3.1 The Lie Algebra o(1, 3) of the Lorentz Group O(1, 3) 918 18.3.2 The Lie Algebra p(1, 3) of the Poincar′e Group P(1, 3) 921 XXX Contents 18.4 The Minkowski Space M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923 18.4.1 Pseudo-Orthonormal Systems and Inertial Systems . 923 18.4.2 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926 18.4.3 Proper Time and the Twin Paradox . . . . . . . . . . . . . . 926 18.4.4 The Free Relativistic Particle and the Energy-Mass Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927 18.4.5 The Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 18.5 The Minkowski Manifold M4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 18.5.1 Causality and the Maximal Signal Velocity . . . . . . . . 930 18.5.2 Hodge Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 18.5.3 Arbitrary Local Coordinates . . . . . . . . . . . . . . . . . . . . . 932 19. The Relativistic Invariance of the Maxwell Equations . . . . 935 19.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936 19.1.1 The Coulomb Force and the Gauss Law . . . . . . . . . . . 937 19.1.2 The Amp`ere Force and the Amp`ere Law . . . . . . . . . . 941 19.1.3 Joule’s Heat Energy Law . . . . . . . . . . . . . . . . . . . . . . . . 944 19.1.4 Faraday’s Induction Law . . . . . . . . . . . . . . . . . . . . . . . . 944 19.1.5 Electric Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945 19.1.6 Magnetic Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947 19.1.7 The Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948 19.1.8 The Dirac Magnetic Monopole . . . . . . . . . . . . . . . . . . . 951 19.1.9 Vacuum Polarization in Quantum Electrodynamics . 952 19.2 The Maxwell Equations in a Vacuum . . . . . . . . . . . . . . . . . . . . 954 19.2.1 The Global Maxwell Equations Based on Electric and Magnetic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955 19.2.2 The Local Maxwell Equations Formulated in Maxwell’s Language of Vector Calculus . . . . . . . . . 957 19.2.3 Discrete Symmetries and CPT . . . . . . . . . . . . . . . . . . . 958 19.3 Invariant Formulation of the Maxwell Equations in a Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 960 19.3.1 Einstein’s Language of Tensor Calculus . . . . . . . . . . . 960 19.3.2 The Language of Differential Forms and Hodge Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962 19.3.3 De Rham Cohomology and the Four-Potential of the Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . 964 19.3.4 The Language of Fiber Bundles . . . . . . . . . . . . . . . . . . 967 19.4 The Transformation Law for the Electromagnetic Field . . . . 967 19.5 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 19.6 Invariants of the Electromagnetic Field . . . . . . . . . . . . . . . . . . 969 19.6.1 The Motion of a Charged Particle and the Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970 19.6.2 The Energy Density and the Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971 19.6.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972 |
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2015-08-17 23:57
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五星好评 顶一下，感谢分享！
feiliping27楼
2015-09-24 17:03
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顶一下，感谢分享！
maketheway28楼
2015-11-09 15:30
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五星好评 顶一下，感谢分享！
ddcy29楼
2016-02-11 12:09
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五星好评 顶一下，感谢分享！
michael_jhy30楼
2016-08-03 20:41
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五星好评 顶一下，感谢分享！
zza461585931楼
2016-08-13 21:12
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五星好评 顶一下，感谢分享！
xiyeyoung32楼
2016-11-24 22:02
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五星好评 顶一下，感谢分享！
woodhole33楼
2016-11-25 11:25
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五星好评 顶一下，感谢分享！
霄河夜曾鸣34楼
2016-12-04 23:14
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五星好评 顶一下，感谢分享！
kamacite35楼
2018-05-12 15:39
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五星好评 顶一下，感谢分享！
wolterli36楼
2018-12-18 21:14
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五星好评 顶一下，感谢分享！
xjvoooo37楼
2019-04-24 06:46
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五星好评 顶一下，感谢分享！
1093138楼
2019-04-25 11:39
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五星好评 顶一下，感谢分享！
Charles_Wu40楼
2019-08-29 14:05
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五星好评 顶一下，感谢分享！
feiliping42楼
2024-10-11 22:33
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顶一下，感谢分享！