Abstract
Even today mathematicians typically typecast Clifford Algebra as the “algebra of a quadratic form,” with no awareness of its grander role in unifying geometry and algebra as envisaged by Clifford himself when he named it Geometric Algebra. It has been my privilege to pick up where Clifford left off—to serve, so to speak, as principal architect of Geometric Algebra and Calculus as a comprehensive mathematical language for physics, engineering and computer science. This is an account of my personal journey in discovering, revitalizing and extending Geometric Algebra, with emphasis on the origin and influence of my book Space-Time Algebra. I discuss guiding ideas, significant results and where they came from—with recollection of important events and people along the way. Lastly, I offer some lessons learned about life and science.
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Arnold, V.I.: On Teaching Mathematics. Paris (1997). http://pauli.uni-muenster.de/~munsteg/arnold.html
Artin E.: Geometric Algebra. Wiley, New York (1957)
Boudet R.: Relativistic Transitions in the Hydrogenic Atoms. Springer, Berlin (2009)
Boudet R.: Quantum Mechanics in the Geometry of Space-Time. Springer, Berlin (2011)
Brackx F., Chisholm J.S.R., Souchek V.: Clifford Analysis and its Applications. Kluwer, Dordrecht (2001)
Chevalley C.: Algebraic Theory of Spinors. Columbia University Press, New York (1955)
Chisholm J.S.R., Common A.K.: Clifford Algebras and their Applications in Mathematical Physics. Reidel, Dordrecht (1986)
Clifford, W.K.: Mathematical Papers. Chelsea, New York (1968). Reprint of the 1882 edition edited by Robert Tucker
Dechant P.: Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups. Clifford Anal. Appl. 25, 1–12 (2015)
Delange R., Brackx F., Sommen F.: Clifford Analysis. Pitman, Boston (1982)
Delanghe R.: Clifford Analysis: History and Perspective. Comput. Methods Funct. Theory 1, 107–153 (2001)
Doran C., Lasenby A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)
Doran C., Hestenes D., Sommen F., Van Acker N.: Lie groups as spin groups. J. Math. Phys. 34, 3642–3669 (1993)
Fokas, A., Pinotsis, D.: Quaternions, evaluation of integrals and boundary value problems. Comput. Methods Funct. Theory. 7(2), 443–476 (2007)
Gladwell M.: The Tipping Point. Back Bay Books, New York (2000)
Gurtler R., Hestenes D.: Consistency in the formulation of the Dirac, Pauli and Schrödinger theories. J. Math. Phys. 16, 573–584 (1975)
Havel T.F., Doran C.: Geometric algebra in quantum information processing. Contemp. Math. 305, 81–100 (2002)
Hestenes, D.: Geometric Calculus and Elementary Particles. University of California at Los Angeles, doctoral thesis (1963)
Hestenes D.: Space Time Algebra. Gordon and Breach, New York (1966)
Hestenes, D.: Space Time Algebra, 2nd edn. Birkhäuser, New York (2015)
Hestenes M.R.: Calculus of Variations and Optimal Control. Wiley, New York (1966)
Hestenes D.: Real Spinor Fields. J. Math. Phys. 8, 798–808 (1967)
Hestenes D.: Multivector calculus. J. Math. Anal. Appl. 24, 313–325 (1968)
Hestenes D.: Multivector functions. J. Math. Anal. Appl. 24, 467–473 (1968)
Hestenes D., Gurtler R.: Local observables in quantum theory. J. Math. Phys. 39, 1028 (1971)
Hestenes D.: Local observables in the Dirac theory. J. Math. Phys. 14, 893–905 (1973)
Hestenes D.: Spin and uncertainty in the interpretation of quantum mechanics. Am. J. Phys. 47, 399–415 (1979)
Hestenes M.R.: Conjugate Direction Methods in Optimization. Springer, New York (1980)
Hestenes, D.: Geometry of the Dirac theory. In: Keller, J. (ed.) A Symposium on the Mathematics of Physical Space-Time, pp. 67–96. Facultad de Quimica, Universidad Nacional Autónoma de México, Mexico City (1981)
Hestenes D.: Space-time structure of weak and electromagnetic interactions. Found. Phys. 12, 153–168 (1982)
Hestenes, D.: Clifford algebra and the interpretation of quantum mechanics. In: Chisholm, J.S.R., Common, A.K. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 321–346. Reidel, Dordrecht/Boston (1986)
Hestenes D.: New Foundations for Classical Mechanics. Kluwer, Dordrecht/Boston (1986)
Hestenes, D.: A unified language for mathematics and physics. In: Chisholm, J.S.R., Common, A.K. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 1–23. Reidel, Dordrecht/Boston (1986)
Hestenes, D.: How the brain works: the next great scientific revolution. In: Smith, C.R., Erickson, G.J. (eds.) Maximum Entropy and Bayesian Spectral Analysis and Estimation Problems, pp. 173–205. Reidel, Dordrecht/Boston (1987)
Hestenes, D.: Old wine in new bottles: a new algebraic framework for computational geometry. In: Bayro-Corrochano, E., Sobczyk, G. (eds.) Advances in Geometric Algebra with Applications in Science and Engineering, pp. 1–14. Birkhäuser, Boston (2001)
Hestenes D.: Spacetime physics with geometric algebra. Am. J. Phys. 71, 691–704 (2003)
Hestenes D.: Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics. Am. J. Phys. 71, 104–121 (2003)
Hestenes D.: Gauge theory gravity with geometric calculus. Found. Phys. 36, 903–970 (2006)
Hestenes, D.: Modeling theory for math and science education. In: Lesh, R., Galbraith, P., Hines, C., Hurford, A. (eds.) Modeling Students’ Mathematical Competencies. Springer, New York (2008)
Hestenes, D.: Gauge gravity and electroweak theory. In: Kleinert, H., Jantzen, R.T., Ruffi, R. (eds.) Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity, pp. 629–647. World Scientific, Singapore (2008)
Hestenes, D.: Grassmann’s legacy. In: Petsche, H.-J., Lewis, A., Liesen, J., Russ, S. (eds.) From Past to Future: Grassmann’s Work in Context. Birkhäuser, Berlin (2011). http://geocalc.clas.asu.edu/html/Overview.html
Hestenes D., Jaynes E.T.: Papers on probability, statistics and statistical physics. Found. Phys. 14, 187–191 (1984)
Hestenes D., Lounesto P.: Geometry of spinor regularization. Celest. Mech. 30, 171–179 (1983)
Hestenes D., Sobczyk G.: Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics. Kluwer, Dordrecht/Boston (1984)
Jaynes E.T.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge (2003)
Lasenby, A., Doran, C., Gull, S.: Gravity, gauge theories and geometric algebra. Philos. Trans. R. Lond. A 355, 487–582 (1998)
Lipschitz, R.: Principes d’un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions. C. R. Acad. Sci. Paris 91, 619–621, 660–664 (1880)
Lounesto P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (1997)
Merton R.K.: Sociology of Science: Theoretical and Empirical Investigations. University of Chicago Press, Chicago (1973)
O’Neill, B.: Elementary Differential Geometry. Academic Press, New York (1966). Now free online
Oziewicz Z.: In Memoriam Jaime Keller (1936–2011). Adv. Appl. Clifford Algebras 25, 1–12 (2001)
Riesz M.: L’integral de Riemann–Liouville et le Problème de Cauchy. Acta Math. 81, 1–223 (1949)
Riesz, M.: Marcel Riesz: Collected Papers. Springer, Berlin (1988). Edited by L. Garding and L. Hörmander
Riesz, M.: Clifford Numbers and Spinors. Kluwer, Dordrecht/Boston (1993). Reprint of Riesz’s lectures at the University of Maryland. Edited by E. Folke Bolinder and Pertti Lounesto (1958)
Rocha-Chavez R., Shapiro M., Sommen F.: Integral theorems for functions and differential forms in C(m). Chapman & Hall, Boca Raton (2002)
Schroedinger E.: Spacetime Structure. Cambridge University Press, Cambridge (1950)
Sudbury A.: Quaternion analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)
Tits, J.L.: Review of “Clifford Numbers and Spinors” by Marcel Riesz. Math. Rev. 31, 6177 (1966)
Vrbik, J.: New Methods of Celestial Mechanics. Bentham ebooks (2010)
Whittaker, E.T., Watson, G.N.: A Course on Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1952)
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Hestenes, D. The Genesis of Geometric Algebra: A Personal Retrospective. Adv. Appl. Clifford Algebras 27, 351–379 (2017). https://doi.org/10.1007/s00006-016-0664-z
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DOI: https://doi.org/10.1007/s00006-016-0664-z