Geometric algebra

Abstract: 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.

Abstract: Preface to the Second Edition.- Introduction.- Part I:Geometric Algebra.- 1.Intrepretation of Clifford Algebra.- 2.Definition of Clifford Algebra.- 3.Inner and Outer Products.- 4.Structure of Clifford Algebra.- 5.Reversion, Scalar Product.- 6.The Algebra of Space.- 7.The Algebra of Space-Time.- Part II:Electrodynamics.- 8.Maxwell's Equation.- 9.Stress-Energy Vectors.- 10.Invariants .- 11. Free Fields.- Part III:Dirac Fields.- 12.Spinors.- 13.Dirac's Equation.- 14.Conserved Currents.- 15.C, P, T.- Part IV:Lorentz Transformations.- 16.Reflections and Rotations.- 17.Coordinate Transformations.- 18.Timelike Rotations.- 19.Scalar Product.- Part V:Geometric Calculus.- 20.Differentiation.- 21.Coordinate Transformations.- 22.Integration.- 23.Global and Local Relativity.- 24.Gauge Transformation and Spinor Derivatives.- Conclusion.- Appendices.- A.Bases and Pseudoscalars.- B.Some Theorems.- C.Composition of Spacial Rotations.- D.Matrix Representation of the Pauli Algebra.

Abstract: 1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5. Geometric Algebras of PseudoEuclidean Spaces.- 2 / Differentiation.- 2-1. Differentiation by Vectors.- 2-2. Multivector Derivative, Differential and Adjoints.- 2-3. Factorization and Simplicial Derivatives.- 3 / Linear and Multilinear Functions.- 3-1. Linear Transformations and Outermorphisms.- 3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem.- 3-3. Eigenblades and Invariant Spaces.- 3-4. Symmetric and Skew-symmetric Transformations.- 3-5. Normal and Orthogonal Transformations.- 3-6. Canonical Forms for General Linear Transformations.- 3-7. Metric Tensors and Isometries.- 3-8. Isometries and Spinors of PseudoEuclidean Spaces.- 3-9. Linear Multivector Functions.- 3-10. Tensors.- 4 / Calculus on Vector Manifolds.- 4-1. Vector Manifolds.- 4-2. Projection, Shape and Curl.- 4-3. Intrinsic Derivatives and Lie Brackets.- 4-4. Curl and Pseudoscalar.- 4-5. Transformations of Vector Manifolds.- 4-6. Computation of Induced Transformations.- 4-7. Complex Numbers and Conformal Transformations.- 5 / Differential Geometry of Vector Manifolds.- 5-1. Curl and Curvature.- 5-2. Hypersurfaces in Euclidean Space.- 5-3. Related Geometries.- 5-4. Parallelism and Projectively Related Geometries.- 5-5. Conformally Related Geometries.- 5-6. Induced Geometries.- 6 / The Method of Mobiles.- 6-1. Frames and Coordinates.- 6-2. Mobiles and Curvature 230.- 6-3. Curves and Comoving Frames.- 6-4. The Calculus of Differential Forms.- 7 / Directed Integration Theory.- 7-1. Directed Integrals.- 7-2. Derivatives from Integrals.- 7-3. The Fundamental Theorem of Calculus.- 7-4. Antiderivatives, Analytic Functions and Complex Variables.- 7-5. Changing Integration Variables.- 7-6. Inverse and Implicit Functions.- 7-7. Winding Numbers.- 7-8. The Gauss-Bonnet Theorem.- 8 / Lie Groups and Lie Algebras.- 8-1. General Theory.- 8-2. Computation.- 8-3. Classification.- References.