物理学家用的几何代数 英文PDF电子书下载

1 Introduction 1

1.1 Vector(linear)spaces 2

1.2 The scalar product 4

1.3 Complex numbers 6

1.4 Quaternions 7

1.5 The cross product 10

1.6 The outer product 11

1.7 Notes 17

1.8 Exercises 18

2 Geometric algebra in two and three dimensions 20

2.1 A new product for vectors 21

2.2 An outline of geometric algebra 23

2.3 Geometric algebra of the plane 24

2.4 The geometric algebra of space 29

2.5 Conventions 38

2.6 Reflections 40

2.7 Rotations 43

2.8 Notes 51

2.9 Exercises 52

3 Classical mechanics 54

3.1 Elementary principles 55

3.2 Two-body central force interactions 59

3.3 Celestial mechanics and perturbations 64

3.4 Rotating systems and rigid-body motion 69

3.5 Notes 81

3.6 Exercises 82

4 Foundations of geometric algebra 84

4.1 Axiomatic development 85

4.2 Rotations and reflections 97

4.3 Bases,frames and components 100

4.4 Linear algebra 103

4.5 Tensors and components 115

4.6 Notes 122

4.7 Exercises 124

5 Relativity and spacetime 126

5.1 An algebra for spacetime 127

5.2 Observers,trajectories and frames 131

5.3 Lorentz transformations 138

5.4 The Lorentz group 143

5.5 Spacetime dynamics 150

5.6 Notes 163

5.7 Exercises 164

6 Geometric calculus 167

6.1 The vector derivative 168

6.2 Curvilinear coordinates 173

6.3 Analytic functions 178

6.4 Directed integration theory 183

6.5 Embedded surfaces and vector manifolds 202

6.6 Elasticity 220

6.7 Notes 224

6.8 Exercises 225

7 Classical electrodynamics 228

7.1 Maxwell's equations 229

7.2 Integral and conservation theorems 235

7.3 The electromagnetic field of a point charge 241

7.4 Electromagnetic waves 251

7.5 Scattering and diffraction 258

7.6 Scattering 261

7.7 Notes 264

7.8 Exercises 265

8 Quantum theory and spinors 267

8.1 Non-relativistic quantum spin 267

8.2 Relativistic quantum states 278

8.3 The Dirac equation 281

8.4 Central potentials 288

8.5 Scattering theory 297

8.6 Notes 305

8.7 Exercises 307

9 Multiparticle states and quantum entanglement 309

9.1 Many-body quantum theory 310

9.2 Multiparticle spacetime algebra 315

9.3 Systems of two particles 319

9.4 Relativistic states and operators 325

9.5 Two-spinor calculus 332

9.6 Notes 337

9.7 Exercises 337

10 Geometry 340

10.1 Projective geometry 341

10.2 Conformal geometry 351

10.3 Conformal transformations 355

10.4 Geometric primitives in conformal space 360

10.5 Intersection and reflection in conformal space 365

10.6 Non-Euclidean geometry 370

10.7 Spacetime conformal geometry 383

10.8 Notes 390

10.9 Exercises 391

11 Further topics in calculus and group theory 394

11.1 Multivector calculus 394

11.2 Grassmann calculus 399

11.3 Lie groups 401

11.4 Complex structures and unitary groups 408

11.5 The general linear group 412

11.6 Notes 416

11.7 Exercises 417

12 Lagrangian and Hamiltonian techniques 420

12.1 The Euler-Lagrange equations 421

12.2 Classical models for spin-1/2 particles 427

12.3 Hamiltonian techniques 432

12.4 Lagrangian field theory 439

12.5 Notes 444

12.6 Exercises 445

13 Symmetry and gauge theory 448

13.1 Conservation laws in field theory 449

13.2 Electromagnetism 453

13.3 Dirac theory 457

13.4 Gauge principles for gravitation 466

13.5 The gravitational field equations 474

13.6 The structure of the Riemann tensor 490

13.7 Notes 495

13.8 Exercises 495

14 Gravitation 497

14.1 Solving the field equations 498

14.2 Spherically-symmetric systems 500

14.3 Schwarzschild black holes 510

14.4 Quantum mechanics in a black hole background 524

14.5 Cosmology 535

14.6 Cylindrical systems 543

14.7 Axially-symmetric systems 551

14.8 Notes 564

14.9 Exercises 565

Bibliography 568

Index 575