# Geometric Algebra for Physicists

Cambridge University Press, May 29, 2003 - Mathematics - 578 pages
Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.

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With the aids of the geometric algebra, it also adds to the level of covering the applications such as black hole physics and quantum computing. Thanks to Chris Doran.

### Contents

 Introduction 1 11 Vector linear spaces 2 12 The scalar product 4 13 Complex numbers 6 14 Quaternions 7 15 The cross product 10 16 The outer product 11 17 Notes 17
 Quantum theory and spinors 267 82 Relativistic quantum states 278 83 The Dirac equation 281 84 Central potentials 288 85 Scattering theory 297 86 Notes 305 87 Exercises 307 Multiparticle states and quantum entanglement 309

 18 Exercises 18 Geometric algebra in two and three dimensions 20 21 A new product for vectors 21 22 An outline of geometric algebra 23 23 Geometric algebra of the plane 24 24 The geometric algebra of space 29 25 Conventions 38 26 Reflections 40 27 Rotations 43 28 Notes 51 29 Exercises 52 Classical mechanics 54 31 Elementary principles 55 32 Twobody central force interactions 59 33 Celestial mechanics and perturbations 64 34 Rotating systems and rigidbody motion 69 35 Notes 81 36 Exercises 82 Foundations of geometric algebra 84 41 Axiomatic development 85 42 Rotations and reflections 97 43 Bases frames and components 100 44 Linear algebra 103 45 Tensors and components 115 46 Notes 122 47 Exercises 124 Relativity and spacetime 126 51 An algebra for spacetime 127 52 Observers trajectories and frames 131 53 Lorentz transformations 138 54 The Lorentz group 143 55 Spacetime dynamics 150 56 Notes 163 57 Exercises 164 Geometric calculus 167 61 The vector derivative 168 62 Curvilinear coordinates 173 63 Analytic functions 178 64 Directed integration theory 183 65 Embedded surfaces and vector manifolds 202 66 Elasticity 220 67 Notes 224 68 Exercises 225 Classical electrodynamics 228 71 Maxwells equations 229 72 Integral and conservation theorems 235 73 The electromagnetic field of a point charge 241 74 Electromagnetic waves 251 75 Scattering and diffraction 258 76 Scattering 261 77 Notes 264 78 Exercises 265
 91 Manybody quantum theory 310 92 Multiparticle spacetime algebra 315 93 Systems of two particles 319 94 Relativistic states and operators 325 95 Twospinor calculus 332 96 Notes 337 Geometry 340 101 Projective geometry 341 102 Conformal geometry 351 103 Conformal transformations 355 104 Geometric primitives in conformal space 360 105 Intersection and reflection in conformal space 365 106 NonEuclidean geometry 370 107 Spacetime conformal geometry 383 108 Notes 390 109 Exercises 391 Further topics in calculus and group theory 394 112 Grassmann calculus 399 113 Lie groups 401 114 Complex structures and unitary groups 408 115 The general linear group 412 116 Notes 416 117 Exercises 417 Lagrangian and Hamiltonian techniques 420 121 The EulerLagrange equations 421 122 Classical models for spin12 particles 427 123 Hamiltonian techniques 432 124 Lagrangian field theory 439 125 Notes 444 126 Exercises 445 Symmetry and gauge theory 448 131 Conservation laws in field theory 449 132 Electromagnetism 453 133 Dirac theory 457 134 Gauge principles for gravitation 466 135 The gravitational field equations 474 136 The structure of the Riemann tensor 490 137 Notes 495 Gravitation 497 141 Solving the field equations 498 142 Sphericallysymmetric systems 500 143 Schwarzschild black holes 510 144 Quantum mechanics in a black hole background 524 145 Cosmology 535 146 Cylindrical systems 543 147 Axiallysymmetric systems 551 148 Notes 564 149 Exercises 565 Bibliography 568 Index 575 Copyright