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Foundations of Physics

, Volume 23, Issue 9, pp 1175–1201 | Cite as

Imaginary numbers are not real—The geometric algebra of spacetime

  • Stephen Gull
  • Anthony Lasenby
  • Chris Doran
Part I. Invited Papers Dedicated To David Hestenes

Abstract

This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a “geometric product” of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.

Keywords

Reflection Analytic Function Mathematical Physic Conventional Method Physical Application 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    D. Hestenes, “Spin and uncertainty in the interpretation of quantum mechanics,”Am. J. Phys. 47(5), 399 (1979).Google Scholar
  2. 2.
    E. T. Jaynes, “Scattering of light by free electrons,” in A. Weingartshofer and D. Hestenes, eds.,The Electron (Kluwer Academic, Dordrecht, 1991), p. 1.Google Scholar
  3. 3.
    D. Hestenes, “Clifford algebra and the interpretation of quantum mechanics,” in J. S. R. Chisholm and A. K. Common, eds.,Clifford Algebras and Their Applications in Mathematical Physics (Reidel, Dordrecht, 1986), p. 321.Google Scholar
  4. 4.
    D. Hestenes,New Foundations for Classical Mechanics (Reidel, Dordrecht, 1985).Google Scholar
  5. 5.
    D. Hestenes, “A unified language for mathematics and physics,” in J. S. R. Chisholm and A. K. Common, eds.,Clifford Algebras and Their Applications in Mathematical Physics (Reidel, Dordrecht, 1986), p. 1.Google Scholar
  6. 6.
    D. Hestenes and G. Sobczyk,Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984).Google Scholar
  7. 7.
    D. Hesteness, “Vectors, spinors, and complex numbers in classical and quantum physics,”Am. J. Phys. 39, 1013 (1971).Google Scholar
  8. 8.
    D. Hestenes, “Observables, operators, and complex numbers in the Dirac theory,”J. Math. Phys. 16(3), 556 (1975).Google Scholar
  9. 9.
    D. Hestenes, “The design of linear algebra and geometry,”Acta Appl. Math. 23, 65 (1991).Google Scholar
  10. 10.
    D. Hestenes and R. Gurtler, “Local observables in quantum theory,”Am. J. Phys. 39, 1028 (1971).Google Scholar
  11. 11.
    D. Hestenes, “Local observables in the Dirac theory,”J. Math. Phys. 14(7), 893 (1973).Google Scholar
  12. 12.
    D. Hestenes, “Quantum mechanics from self-interaction,”Found. Phys. 15(1), 63 (1985).Google Scholar
  13. 13.
    D. Hestenes, “Thezitterbewegung interpretation of quantum mechanics,”Found. Phys. 20(10), 1213 (1990).Google Scholar
  14. 14.
    L. de Broglie,La Réinterprétation de la Mécanique Ondulatoire (Gauthier-Villars, Paris, 1971).Google Scholar
  15. 15.
    R. P. Feynman,The Character of Physical Law (British Broadcasting Corporation, London, 1965).Google Scholar
  16. 16.
    S. F. Gull, A. N. Lasenby, and C. J. L. Doran, “Electron paths, tunneling and diffraction in the spacetime algebra,”Found. Phys. 23(10) (1993).Google Scholar
  17. 17.
    D. Hestenes, “Entropy and indistinguishability,”Am. J. Phys. 38, 840 (1970).Google Scholar
  18. 18.
    D. Hestenes, “Inductive inference by neural networks,” in G. J. Erickson and C. R. Smith, eds.,Maximum Entropy and Bayesian Methods in Science and Engineering (Vol. 2), (Reidel, Dordrecht, 1988), p. 19.Google Scholar
  19. 19.
    D. Hestenes, “Modelling games in the Newtonian world,”Am. J. Phys. 60(8), 732 (1992).Google Scholar
  20. 20.
    D. Hestenes,Space-Time Algebra (Gordon & Breach, New York, 1966).Google Scholar
  21. 21.
    H. Grassmann,Die Ausdehnungslehre (Berlin, 1862).Google Scholar
  22. 22.
    H. Grassmann, “Der Ort der Hamilton'schen Quaternionen in der Ausdehnungslehere,”Math. Ann. 12, 375 (1877).Google Scholar
  23. 23.
    W. K. Clifford, “Applications of Grassmann's extensive algebra,”Am. J. Math. 1, 350 (1878).Google Scholar
  24. 24.
    C. J. L. Doran, A. N. Lasenby, and S. F. Gull, “States and operators in the spacetime algebra,”Found. Phys. 23(9), 1239 (1993).Google Scholar
  25. 25.
    H. Halberstam and R. E. Ingram,The Mathematical Papers of Sir William Rowan Hamilton, Vol. III (Cambridge University Press, Cambridge, 1967).Google Scholar
  26. 26.
    I. W. Benn and R. W. Tucker,An Introduction to Spinors and Geometry (Adam Hilger, London, 1988).Google Scholar
  27. 27.
    A. N. Lasenby, C. J. L. Doran, and S. F. Gull, “A multivector derivative approach to Lagrangian field theory,”Found. Phys. 23(10) (1993).Google Scholar
  28. 28.
    J. A. Wheeler and R. P. Feynman, “Classical electrodynamics in terms of direct interparticle action,”Rev. Mod. Phys. 21(3), 425 (1949).Google Scholar
  29. 29.
    J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,”Rev. Mod. Phys. 17, 157 (1945).Google Scholar
  30. 30.
    A. N. Lasenby, C. J. L. Doran, and S. F. Gull, “Grassmann calculus, pseudoclassical mechanics and geometric algebra,”J. Math. Phys. 34(8), 3683 (1993).Google Scholar
  31. 31.
    C. J. L. Doran, A. N. Lasenby, and S. F. Gull, “Grassmann mechanics, multivector derivatives and geometric algebra” in Z. Oziewicz, A. Borowiec, and B. Jancewicz, eds.,Spinors, Twistors and Clifford Algebras (Kluwer Academic, Dordrecht, 1993), p. 215.Google Scholar
  32. 32.
    A. N. Lasenby, C. J. L. Doran, and S. F. Gull, “2-Spinors, twistors and supersymmetry in the spacetime algebra,” in Z. Oziewicz, A. Borowiec, and B. Jancewicz, eds.,Spinors, Twistors and Clifford Algebras (Kluwer Academic, Dordrecht, 1993), p. 233.Google Scholar
  33. 33.
    W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, New York, 1973).Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Stephen Gull
    • 1
  • Anthony Lasenby
    • 1
  • Chris Doran
    • 2
  1. 1.MRAO, Cavendish LaboratoryCambridgeUnited Kingdom
  2. 2.DAMTPCambridgeUnited Kingdom

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