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SPACETIME CALCULUS David Hestenes Arizona State University Abstract:
This book provides a synopsis of spacetime calculus with applications to classical electrodynamics, quantum theory and gravitation. The calculus is a coordinate-free mathematical language enabling a unified treatment of all these topics and bringing new insights and methods to each of them.
CONTENTS PART I: Mathematical Fundamentals 1. 2. 3. 4. 5. 6. 7. 8. 9.
Spacetime Algebra Vector Derivatives and Differentials Linear Algebra Spacetime Splits Rigid Bodies and Charged Particles Electromagnetic Fields Transformations on Spacetime Directed Integrals and the Fundamental Theorem Integral Equations and Conservation Laws PART II: Quantum Theory
10. 11. 12. 13. 14.
The Real Dirac Equation Observables and Conservation Laws Electron Trajectories The Zitterbewegung Interpretation Electroweak Interactions Part III. Induced Geometry on Flat Spacetime
15. Gauge Tensor and Gauge Invariance 16. Covariant Derivatives and Curvature 17. Universal Laws for Spacetime Physics REFERENCES Appendix A. Tensors and their Classification Appendix B: Transformations and Invariants Appendix C: Lagrangian Formulation
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PART I: Mathematical Fundamentals 1. SPACETIME ALGEBRA. We represent Minkowski spacetime as a real 4-dimensional vector space M4 . The two properties of scalar multiplication and vector addition in M4 provide only a partial specification of spacetime geometry. To complete the specification we introduce an associative geometric product among vectors a, b, c, . . . with the property that the square of any vector is a (real) scalar. Thus for any vector a we can write (1.1) a2 = aa = ²| a |2 , where ² is the signature of a and | a | is a (real) positive scalar. As usual, we say that a is timelike, lightlike or spacelike if its signature is positive (² = 1), null (² = 0), or neg