## Geometric Algebra for PhysicistsGeometric 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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### 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 |

568 | |

575 | |

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### Common terms and phrases

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