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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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[论文投稿] 有没有老师需要发表论文 +4 金老师论文助理- 2024-05-29 4/200 2024-05-29 16:51 by liuyupu132
[论文投稿] 核心初审被拒，理由是“选题的意义不明确，文章写得不像是科技论文”，怎么改 5+3 工藤雷花樱 2024-05-27 8/400 2024-05-29 10:09 by topedit
[基金申请] E10开始送了，希望有好运 +5 sail 2024-05-27 5/250 2024-05-28 18:36 by 芝小芝
[硕博家园] 我是很理想化一人 +6 hahamyid 2024-05-26 6/300 2024-05-27 18:13 by 大飞鱼鱼鱼
[基金申请] 感觉自然基金限制通过比例就是有点扯，学学B口，化学学部，不限制比例。 +10 wsjing 2024-05-26 14/700 2024-05-27 11:57 by kanmiaolucky
[硕博家园] 2024博士招生 +3 big 混子 2024-05-26 3/150 2024-05-26 20:47 by 宁多缺毋滥
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