Geometric Algebra for Physicists

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Cambridge University Press, May 29, 2003 - Mathematics - 578 pages
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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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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
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About the author (2003)

Chris Doran obtained his PhD from the University of Cambridge, having gained a distinction in Part II of his undergraduate degree. He was elected a Junior Research Fellow of Churchill College, Cambridge in 1993, was made a Lloyd's of London Fellow in 1996 and was the Schlumberger Interdisciplinary Research Fellow of Darwin College, Cambridge in 1997 and 2000. He is currently a Fellow of Sidney Sussex College, Cambridge and holds an EPSRC Advanced Fellowship. Dr Doran has published widely on aspects of mathematical physics and is currently researching applications of geometric algebra in engineering and computer science.

Anthony Lasenby is Professor of Astrophysics and Cosmology at the University of Cambridge, and is currently Head of the Astrophysics Group and the Mullard Radio Astronomy Observatory in the Cavendish Laboratory. He began his astronomical career with a PhD at Jodrell Bank, specialising in the Cosmic Microwave Background, which has been a major subject of his research ever since. After a brief period at the National Radio Astronomy Observatory in America, he moved from Manchester to Cambridge in 1984, and has been at the Cavendish since then. He is the author or coauthor of nearly 200 papers spanning a wide range of fields, from early universe cosmology to computer vision. His introduction to geometric algebra came in 1988, when he encountered the work of David Hestenes for the first time, and since then he has been developing geometric algebra techniques and employing them in his research in many areas.

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