**Undergraduate Lecture Notes in Physics**

<Physics from Symmetry>-by Jakob Schwichtenberg 2015

Contents

Part I Foundations

1 Introduction 3

1.1 What we Cannot Derive . . . . . . . . . . . . . . . . . . . 3
1.2 Book Overview . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Elementary Particles and Fundamental Forces . . . . . . 7

2 Special Relativity 11
2.1 The Invariant of Special Relativity . . . . . . . . . . . . . 12
2.2 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Upper Speed Limit . . . . . . . . . . . . . . . . . . . . . . 16
2.4 The Minkowski Notation . . . . . . . . . . . . . . . . . . 17
2.5 Lorentz Transformations . . . . . . . . . . . . . . . . . . . 19
2.6 Invariance, Symmetry and Covariance . . . . . . . . . . . 20

Part II Symmetry Tools

3 Lie Group Theory 25

3.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Rotations in two Dimensions . . . . . . . . . . . . . . . . 29
3.2.1 Rotations with Unit Complex Numbers . . . . . . 31

3.3 Rotations in three Dimensions . . . . . . . . . . . . . . . 33
3.3.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1 The Generators and Lie Algebra of SO(3) . . . . 41
3.4.2 The Abstract Definition of a Lie Algebra . . . . . 44
3.4.3 The Generators and Lie Algebra of SU(2) . . . . 45
3.4.4 The Abstract Definition of a Lie Group . . . . . . 47

3.5 Representation Theory . . . . . . . . . . . . . . . . . . . . 49

3.6 SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.1 The Finite-dimensional Irreducible Representations of SU(2) . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.2 The Casimir Operator of SU(2) . . . . . . . . . . . 56
3.6.3 The Representation of SU(2) in one Dimension . 57
3.6.4 The Representation of SU(2) in two Dimensions 57
3.6.5 The Representation of SU(2) in three Dimensions 58

3.7 The Lorentz Group O(1, 3) . . . . . . . . . . . . . . . . . . 58
3.7.1 One Representation of the Lorentz Group . . . . 61
3.7.2 Generators of the Other Components of the Lorentz Group . . . . . . . . . . . . . . . . . . . . 64
3.7.3 The Lie Algebra of the Proper Orthochronous Lorentz Group . . . . . . . . . . . . . . . . . . . . 66
3.7.4 The (0, 0) Representation . . . . . . . . . . . . . . 67
3.7.5 The (1/2, 0) Representation . . . . . . . . . . . . . . 68
3.7.6 The (0, 1/2) Representation . . . . . . . . . . . . . . 69
3.7.7 Van der Waerden Notation . . . . . . . . . . . . . 70
3.7.8 The (1/2, 1/2) Representation . . . . . . . . . . . . . . 75
3.7.9 Spinors and Parity . . . . . . . . . . . . . . . . . . 78
3.7.10 Spinors and Charge Conjugation . . . . . . . . . . 81
3.7.11 Infinite-Dimensional Representations . . . . . . . 82

3.8 The Poincare Group . . . . . . . . . . . . . . . . . . . . . 84

3.9 Elementary Particles . . . . . . . . . . . . . . . . . . . . . 85

3.10 Appendix: Rotations in a Complex Vector Space . . . . . 87

3.11 Appendix: Manifolds . . . . . . . . . . . . . . . . . . . . . 87

4 The Framework 91

4.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . 91
4.1.1 Fermat’s Principle . . . . . . . . . . . . . . . . . . 92
4.1.2 Variational Calculus - the Basic Idea . . . . . . . . 92

4.2 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3 Particle Theories vs. Field Theories . . . . . . . . . . . . . 94

4.4 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . 95

4.5 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . 97
4.5.1 Noether’s Theorem for Particle Theories . . . . . 97
4.5.2 Noether’s Theorem for Field Theories - Spacetime Symmetries . . . . . . . . . . . . . . . . . . . . . . 101
4.5.3 Rotations and Boosts . . . . . . . . . . . . . . . . . 104
4.5.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5.5 Noether’s Theorem for Field Theories - Internal Symmetries . . . . . . . . . . . . . . . . . . . . . . 106

4.6 Appendix: Conserved Quantity from Boost Invariance for Particle Theories . . . . . . . . . . . . . . . . . . . . . 108
4.7 Appendix: Conserved Quantity from Boost Invariance for Field Theories . . . . . . . . . . . . . . . . . . . . . . . 109

Part III The Equations of Nature

5 Measuring Nature 113

5.1 The Operators of Quantum Mechanics . . . . . . . . . . . 113
5.1.1 Spin and Angular Momentum . . . . . . . . . . . 114

5.2 The Operators of Quantum Field Theory . . . . . . . . . 115

6 Free Theory 117

6.1 Lorentz Covariance and Invariance . . . . . . . . . . . . . 117

6.2 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . 118
6.2.1 Complex Klein-Gordon Field . . . . . . . . . . . . 119

6.3 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4 Proca Equation . . . . . . . . . . . . . . . . . . . . . . . . 123

7 Interaction Theory 127

7.1 U(1) Interactions . . . . . . . . . . . . . . . . . . . . . . . 129
7.1.1 Internal Symmetry of Free Spin 1/2 Fields . . . . . 129
7.1.2 Internal Symmetry of Free Spin 1 Fields . . . . . 131
7.1.3 Putting the Puzzle Pieces Together . . . . . . . . . 132
7.1.4 Inhomogeneous Maxwell Equations and Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . 134
7.1.5 Charge Conjugation, Again . . . . . . . . . . . . . 135
7.1.6 Noether’s Theorem for Internal U(1) Symmetry . 136
7.1.7 Interaction of Massive Spin 0 Fields . . . . . . . . 138
7.1.8 Interaction of Massive Spin 1 Fields . . . . . . . . 138

7.2 SU(2) Interactions . . . . . . . . . . . . . . . . . . . . . . 139

7.3 Mass Terms and Unification of SU(2) and U(1) . . . . . 145

7.4 Parity Violation . . . . . . . . . . . . . . . . . . . . . . . . 152

7.5 Lepton Mass Terms . . . . . . . . . . . . . . . . . . . . . . 156

7.6 Quark Mass Terms . . . . . . . . . . . . . . . . . . . . . . 160

7.7 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.7.1 Labelling States . . . . . . . . . . . . . . . . . . . . 162

7.8 SU(3) Interactions . . . . . . . . . . . . . . . . . . . . . . 164
7.8.1 Color . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.8.2 Quark Description . . . . . . . . . . . . . . . . . . 167

7.9 The Interplay Between Fermions and Bosons . . . . . . . 169

Part IV Applications

8 Quantum Mechanics 173

8.1 Particle Theory Identifications . . . . . . . . . . . . . . . . 174

8.2 Relativistic Energy-Momentum Relation . . . . . . . . . . 174

8.3 The Quantum Formalism . . . . . . . . . . . . . . . . . . 175
8.3.1 Expectation Value . . . . . . . . . . . . . . . . . . 177

8.4 The Schrödinger Equation . . . . . . . . . . . . . . . . . . 178
8.4.1 Schrödinger Equation with External Field . . . . 180

8.5 From Wave Equations to Particle Motion . . . . . . . . . 180
8.5.1 Example: Free Particle . . . . . . . . . . . . . . . . 181
8.5.2 Example: Particle in a Box . . . . . . . . . . . . . . 181
8.5.3 Dirac Notation . . . . . . . . . . . . . . . . . . . . 185
8.5.4 Example: Particle in a Box, Again . . . . . . . . . 187
8.5.5 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.6 Heisenberg’s Uncertainty Principle . . . . . . . . . . . . . 191

8.7 Comments on Interpretations . . . . . . . . . . . . . . . . 192

8.8 Appendix: Interpretation of the Dirac Spinor Components . . . . . . . . . . . . . . . . . . . . . . . . . . 194

8.9 Appendix: Solving the Dirac Equation . . . . . . . . . . . 199

8.10 Appendix: Dirac Spinors in Different Bases . . . . . . . . 200
8.10.1 Solutions of the Dirac Equation in the Mass Basis 202

9 Quantum Field Theory 205

9.1 Field Theory Identifications . . . . . . . . . . . . . . . . . 206

9.2 Free Spin 0 Field Theory . . . . . . . . . . . . . . . . . . . 207

9.3 Free Spin 1/2 Theory . . . . . . . . . . . . . . . . . . . . . . 212

9.4 Free Spin 1 Theory . . . . . . . . . . . . . . . . . . . . . . 215

9.5 Interacting Field Theory . . . . . . . . . . . . . . . . . . . 215
9.5.1 Scatter Amplitudes . . . . . . . . . . . . . . . . . . 216
9.5.2 Time Evolution of States . . . . . . . . . . . . . . . 216
9.5.3 Dyson Series . . . . . . . . . . . . . . . . . . . . . 220
9.5.4 Evaluating the Series . . . . . . . . . . . . . . . . . 221

9.6 Appendix: Most General Solution of the Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

10 Classical Mechanics 227
10.1 Relativistic Mechanics . . . . . . . . . . . . . . . . . . . . 229
10.2 The Lagrangian of Non-Relativistic Mechanics . . . . . . 230

11 Electrodynamics 233
11.1 The Homogeneous Maxwell Equations . . . . . . . . . . 234
11.2 The Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . 235
11.3 Coulomb Potential . . . . . . . . . . . . . . . . . . . . . . 237

12 Gravity 239

13 Closing Words 245

Part V Appendices

A Vector calculus 249
A.1 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 250
A.2 Change of Coordinate Systems . . . . . . . . . . . . . . . 251
A.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . 253
A.4 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
A.5 Right-handed and Left-handed Coordinate Systems . . . 254

B Calculus 257
B.1 Product Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 257
B.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . 257
B.3 The Taylor Series . . . . . . . . . . . . . . . . . . . . . . . 258
B.4 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
B.4.1 Important Series . . . . . . . . . . . . . . . . . . . 260
B.4.2 Splitting Sums . . . . . . . . . . . . . . . . . . . . 262
B.4.3 Einstein’s Sum Convention . . . . . . . . . . . . . 262
B.5 Index Notation . . . . . . . . . . . . . . . . . . . . . . . . 263
B.5.1 Dummy Indices . . . . . . . . . . . . . . . . . . . . 263
B.5.2 Objects with more than One Index . . . . . . . . . 264
B.5.3 Symmetric and Antisymmetric Indices . . . . . . 264
B.5.4 Antisymmetric × Symmetric Sums . . . . . . . . 265
B.5.5 Two Important Symbols . . . . . . . . . . . . . . . 265

C Linear Algebra 267
C.1 Basic Transformations . . . . . . . . . . . . . . . . . . . . 267
C.2 Matrix Exponential Function . . . . . . . . . . . . . . . . 268
C.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 268
C.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . 269
C.5 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . 269

D Additional Mathematical Notions 271
D.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 271
D.2 Delta Distribution . . . . . . . . . . . . . . . . . . . . . . . 272

Bibliography 273
Index 277

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