SpaceTime Calculus

© D. Hestenes (1998)

Preface This is a book in progress, so it will be modified as expanded as time allows. It is a concise update of the book Space-Time Algebra (STA). The basic algebraic structure of STA remains the same, having passed the test of time. The treatment of electrodynamics has also survived but has expanded. The main changes are in quantum theory and gravitation, where substantial advances have been made, some quite recently.

Table of Contents

Introduction

PART I: Mathematical Fundamentals

      1. Spacetime Algebra
      2. Vector Derivatives and Differentials
      3. Linear Algebra
      4. Spacetime Splits
      5. Rigid Bodies and Charged Particles
      6. Electromagnetic Fields
      7. Transformations on Spacetime
      8. Directed Integrals and the Fundamental Theorem
      9. Integral Equations and Conservation Laws

PART II: Quantum Theory

      10. The Real Dirac Equation
      11. Observables and Conservation Laws
      12. Electron Trajectories
      13. The Zitterbewegung Interpretation
      14. 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

Related Papers


Proper Particle Mechanics.

Abstract: Spacetime algebra is employed to formulate classical relativistic mechanics without coordinates. Observers are treated on the same footing as other physical systems. The kinematics of a rigid body are expressed in spinor form and the Thomas precession is derived.

D. Hestenes, J. Math. Phys. 15, 1768-1777 (1974).
American Institute of Physics.

Proper Dynamics of a Rigid Point Particle.

Abstract: A spinor formulation of the classical Lorentz force is given which describes the presession of an electron's spin as well as its velocity. Solutions are worked out applicable to an electron in a uniform field, plane wave, and a Coulomb field.

D. Hestenes, J. Math. Phys. 15, 1778-1786 (1974).
American Institute of Physics.

Spinor Particle Mechanics.

Abstract: Geometric Algebra makes it possible to formulate simple spinor equations of motion for classical particles and rigid bodies. In the Newtonian case, these equations have proven their value by simplifying orbital computations. The relativistic case is not so well known, but it has new and surprising features worth exploiting, including close connections to quantum mechanical equations. The current status of spinor particle mechanics is reviewed, and directions for extention are pointed out.

D. Hestenes, Proceedings of the Fourth International Conference on Clifford Algebras and Their Applications to Mathematical Physics, Aachen/Germany (May 1996).
Kluwer Academic Publishers, Dordrecht..

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