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.
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References
D. Hestenes, “Spin and uncertainty in the interpretation of quantum mechanics,”Am. J. Phys. 47(5), 399 (1979).
E. T. Jaynes, “Scattering of light by free electrons,” in A. Weingartshofer and D. Hestenes, eds.,The Electron (Kluwer Academic, Dordrecht, 1991), p. 1.
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.
D. Hestenes,New Foundations for Classical Mechanics (Reidel, Dordrecht, 1985).
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.
D. Hestenes and G. Sobczyk,Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984).
D. Hesteness, “Vectors, spinors, and complex numbers in classical and quantum physics,”Am. J. Phys. 39, 1013 (1971).
D. Hestenes, “Observables, operators, and complex numbers in the Dirac theory,”J. Math. Phys. 16(3), 556 (1975).
D. Hestenes, “The design of linear algebra and geometry,”Acta Appl. Math. 23, 65 (1991).
D. Hestenes and R. Gurtler, “Local observables in quantum theory,”Am. J. Phys. 39, 1028 (1971).
D. Hestenes, “Local observables in the Dirac theory,”J. Math. Phys. 14(7), 893 (1973).
D. Hestenes, “Quantum mechanics from self-interaction,”Found. Phys. 15(1), 63 (1985).
D. Hestenes, “Thezitterbewegung interpretation of quantum mechanics,”Found. Phys. 20(10), 1213 (1990).
L. de Broglie,La Réinterprétation de la Mécanique Ondulatoire (Gauthier-Villars, Paris, 1971).
R. P. Feynman,The Character of Physical Law (British Broadcasting Corporation, London, 1965).
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).
D. Hestenes, “Entropy and indistinguishability,”Am. J. Phys. 38, 840 (1970).
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.
D. Hestenes, “Modelling games in the Newtonian world,”Am. J. Phys. 60(8), 732 (1992).
D. Hestenes,Space-Time Algebra (Gordon & Breach, New York, 1966).
H. Grassmann,Die Ausdehnungslehre (Berlin, 1862).
H. Grassmann, “Der Ort der Hamilton'schen Quaternionen in der Ausdehnungslehere,”Math. Ann. 12, 375 (1877).
W. K. Clifford, “Applications of Grassmann's extensive algebra,”Am. J. Math. 1, 350 (1878).
C. J. L. Doran, A. N. Lasenby, and S. F. Gull, “States and operators in the spacetime algebra,”Found. Phys. 23(9), 1239 (1993).
H. Halberstam and R. E. Ingram,The Mathematical Papers of Sir William Rowan Hamilton, Vol. III (Cambridge University Press, Cambridge, 1967).
I. W. Benn and R. W. Tucker,An Introduction to Spinors and Geometry (Adam Hilger, London, 1988).
A. N. Lasenby, C. J. L. Doran, and S. F. Gull, “A multivector derivative approach to Lagrangian field theory,”Found. Phys. 23(10) (1993).
J. A. Wheeler and R. P. Feynman, “Classical electrodynamics in terms of direct interparticle action,”Rev. Mod. Phys. 21(3), 425 (1949).
J. A. Wheeler and R. P. Feynman, “Interaction with the absorber as the mechanism of radiation,”Rev. Mod. Phys. 17, 157 (1945).
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).
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.
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.
W. Misner, K. S. Thorne, and J. A. Wheeler,Gravitation (Freeman, New York, 1973).
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The title of this paper is inspired by David Hestenes, who is known to have a fondness for deliberate ambiguity.(1)
Supported by a SERC studentship.
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Gull, S., Lasenby, A. & Doran, C. Imaginary numbers are not real—The geometric algebra of spacetime. Found Phys 23, 1175–1201 (1993). https://doi.org/10.1007/BF01883676
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DOI: https://doi.org/10.1007/BF01883676