David Hestenes

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David Orlin Hestenes
Born May 21, 1933 (1933-05-21) (age 85)
Known for Geometric algebra
Awards Oersted Medal (2002)
Scientific career
Fields Physics
Institutions Arizona State University

David Orlin Hestenes, Ph.D. (born May 21, 1933) is a theoretical physicist and science educator. He is best known as chief architect of geometric algebra as a unified language for mathematics and physics,[1] and as founder of Modelling Instruction, a research-based program to reform K–12 Science, Technology, Engineering, and Mathematics (STEM) education.[2]

For more than 30 years, he was employed in the Department of Physics and Astronomy of Arizona State University (ASU), where he retired with the rank of Research Professor and is now emeritus.

Life and career

Education and doctorate degree

David Orlin Hestenes (eldest son of mathematician Magnus Hestenes) was born 1933 in Chicago Illinois. Beginning college as a pre-medical major at UCLA from 1950 to 1952, he graduated from Pacific Lutheran University in 1954 with degrees in philosophy and speech. After serving in the U.S. Army from 1954 to 1956, he entered UCLA as an unclassified graduate student, completed a physics M.A. in 1958 and won a University Fellowship. His mentor at UCLA was the physicist Robert Finkelstein,[3] who was working on unified field theories at that time.[4] A serendipitous encounter with lecture notes by mathematician Marcel Riesz inspired Hestenes to study a geometric interpretation of Dirac matrices. He obtained his Ph.D. from UCLA with a thesis entitled Geometric Calculus and Elementary Particles.[4][5] Shortly thereafter he recognized that the Dirac algebras and Pauli matrices could be unified in matrix-free form by a device later called a spacetime split.[6] Then he revised his thesis and published it in 1966 as a book, Space Time Algebra,[7] now referred to as spacetime algebra (STA). This was the first major step in developing a unified, coordinate-free geometric algebra and calculus for all of physics.

Postdoctorate research and career

From 1964 to 1966, Hestenes was an NSF Postdoctoral Fellow at Princeton with John Archibald Wheeler. In 1966 he joined the physics department at Arizona State University, rising to full professor in 1976 and retiring in 2000 to Emeritus Professor of Physics.

In 1980 and 1981 as a NASA Faculty Fellow and in 1983 as a NASA Consultant he worked at Jet Propulsion Laboratory on orbital mechanics and attitude control, where he applied geometric algebra in development of new mathematical techniques published in a textbook/monograph New Foundations for Classical Mechanics.[8]

In 1983 he joined with entrepreneur Robert Hecht-Nielsen and psychologist Peter Richard Killeen in conducting the first ever conference devoted exclusively to neural network modeling of the brain. Hestenes followed this in 1987 with appointment as the first Visiting Scholar in the Department of Cognitive and Neural Systems (Boston University) and a period of neuroscience research.[9][10][11][12]

Hestenes has been a principal investigator for NSF grants seeking to teach physics through modeling and to measure student understanding of physics models at both the high school and university levels.


Hestenes has worked in mathematical and theoretical physics, geometric algebra, neural networks, and cognitive research in science education. He is the prime mover behind the contemporary resurgence of interest in geometric algebras and in other offshoots of Clifford algebras as ways of formalizing theoretical physics,[13][14]

Geometric algebra and calculus

Spacetime algebra provided the starting point for two main lines of research: on its implications for quantum mechanics specifically and for mathematical physics generally.

The first line began with the fact that reformulation of the Dirac equation in terms of spacetime algebra reveals hidden geometric structure[15] Among other things, it reveals that the complex factor in the equation is a geometric quantity (a bivector) identified with electron spin, where specifies the spin direction and is the spin magnitude. The implications of this insight have been studied in a long series of papers [16][17][18][19][20][21] with the most significant conclusion linking it to Schrödinger’s zitterbewegung and proposing a zitterbewegung interpretation of quantum mechanics.[22] Research in this direction is still active.

The second line of research was dedicated to extending geometric algebra to a self-contained geometric calculus for use in theoretical physics. Its culmination is the book Clifford Algebra to Geometric Calculus[23] which follows an approach to differential geometry that uses the shape tensor (second fundamental form). Innovations in the book include the concepts of vector manifold, differential outermorphism, vector derivative that enables coordinate-free calculus on manifolds, and an extension of the Cauchy integral theorem to higher dimensions.[23][24]

Hestenes emphasizes the important role of the mathematician Hermann Grassmann[25][26] for the development of geometric algebra, with William Kingdon Clifford building on Grassmann's work. Hestenes is adamant about calling this mathematical approach “geometric algebra” and its extension “geometric calculus,” rather than referring to it as “Clifford algebra”. He emphasizes the universality of this approach, the foundations of which were laid by both Grassmann and Clifford. He points out that contributions were made by many individuals, and Clifford himself used the term “geometric algebra” which reflects the fact that this approach can be understood as a mathematical formulation of geometry, whereas, so Hestenes asserts, the term “Clifford algebra” is often regarded as simply “just one more algebra among many other algebras”,[27] which withdraws attention from its role as a unified language for mathematics and physics.

Hestenes' work has been applied to Lagrangian field theory,[28] formulation of a gauge theory of gravity alternative to general relativity by Lasenby, Doran and Gull, which they call gauge theory gravity (GTG),[29][30] and it has been applied to spin representations of Lie groups.[31] Most recently, it led Hestenes to formulate conformal geometric algebra, a new approach to computational geometry.[32] This has found a rapidly increasing number of applications in engineering and computer science.[33][34][35][36][37][38]

Modeling theory and instruction

Since 1980, Hestenes has been developing a Modeling Theory of science and cognition, especially to guide the design of science instruction.[39][40][41][42][43][44][45] The theory distinguishes sharply between conceptual models that constitute the content core of science and the mental models that are essential to understand them. Modeling Instruction is designed to engage students in all aspects of modeling, broadly conceived as constructing, testing, analyzing and applying scientific models.[46] To assess the effectiveness of Modeling Instruction, Hestenes and his students developed the Force Concept Inventory,[47][48] a concept inventory tool for evaluating student understanding of introductory physics.[49]

After a decade of education research to develop and validate the approach, Hestenes was awarded grants from the National Science Foundation for another decade to spread the Modeling Instruction Program nationwide. As of 2011, more than 4000 teachers had participated in summer workshops on modeling, including nearly 10% of the United States' high school physics teachers. It is estimated that Modeling teachers reach more than 100,000 students each year.

One outcome of the program is that the teachers created their own non-profit organization, the American Modeling Teachers Association (AMTA),[50] to continue and expand the mission after government funding terminated. The AMTA has expanded to a nationwide community of teachers dedicated to addressing the nation's Science, Technology, Engineering, and Mathematics (STEM) education crisis. Another outcome of the Modeling Program was creation of a graduate program at Arizona State University for sustained professional development of STEM teachers.[51] This provides a validated model for similar programs at universities across the country.[52]

Awards and fellowships


  • D. Hestenes: New Foundations for Classical Mechanics, Foundamental Theories of Physics, 2nd ed., Springer Verlag, 1999, ISBN 978-0792355144
  • D. Hestenes, A. Weingartshofer (eds.): The Electron: New Theory and Experiment, Fundamental Theories of Physics, Springer, 1991, ISBN 978-0792313564
  • D. Hestenes, Garret Sobczyk: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Fundamental Theories of Physics, Springer, 1987, ISBN 978-9027725615
  • D. Hestenes: Space-Time Algebra, Routledge, 1966, ISBN 978-0677013909


  1. ^ D. Hestenes: A Unified Language for Mathematics and Physics. In: J.S.R. Chisholm/A.K. Common (eds.): Clifford Algebras and their Applications in Mathematical Physics (Reidel: Dordrecht/Boston, 1986), p. 1–23.
  2. ^ Home page on Modeling Instruction http://modeling.asu.edu/
  3. ^ Robert Finkelstein Archived 2012-02-04 at the Wayback Machine.
  4. ^ a b D. Hestenes:Clifford algebra and the interpretation of quantum mechanics Archived 2012-04-06 at the Wayback Machine.. In: J.S.R. Chisholm, A.K. Commons (eds.): Clifford Algebras and their Interpretations in Mathematical Physics, Reidel, 1986, pp. 321–346
  5. ^ D. Hestenes: Geometric Calculus and Elementary Particles,--~~~~ University of California, Los Angeles
  6. ^ D. Hestenes, Spacetime Physics with Geometric Algebra, American Journal of Physics 71: 691–714 (2003).
  7. ^ D. Hestenes, Space-Time Algebra (Gordon & Breach: New York, 1966).
  8. ^ D. Hestenes, New Foundations for Classical Mechanics (Kluwer: Dordrecht/Boston, 1986), Second Edition (1999).
  9. ^ D. Hestenes, How the Brain Works: the next great scientific revolution. In C.R. Smith and G.J. Erickson (eds.), Maximum Entropy and Bayesian Spectral Analysis and Estimation Problems (Reidel: Dordrecht/Boston, 1987). p. 173–205.
  10. ^ D. Hestenes, Invariant Body Kinematics: I. Saccadic and compensatory eye movements. Neural Networks 7: 65–77 (1994).
  11. ^ D. Hestenes, Invariant Body Kinematics: II. Reaching and neurogeometry. Neural Networks 7: 79–88 (1994).
  12. ^ D. Hestenes, Modulatory Mechanisms in Mental Disorders. In Neural Networks in Psychopathology, ed. D.J. Stein & J. Ludik (Cambridge University Press: Cambridge, 1998). pp. 132–164.
  13. ^ Abel Diek, R. Kantowski: Some Clifford algebra history, in: Rafal Ablamowicz, P. Lounesto (eds.): Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992), Mathematics and Its Applications, Kluwer Academic, 1995, ISBN 978-9048145256, pp. 3–12, p. 9
  14. ^ Chris J. L. Doran, Anthony Lasenby: Geometric Algebra for Physicists, Cambridge University Press, 2003, ISBN 978-0521480222, p. 123
  15. ^ D. Hestenes, Real Spinor Fields, Journal of Mathematical Physics 8: 798–808 (1967).
  16. ^ D. Hestenes and R. Gurtler, Local Observables in Quantum Theory, American Journal of Physics 39: 1028 (1971).
  17. ^ D. Hestenes, Local Observables in the Dirac Theory, Journal of Mathematical Physics 14: 893–905 (1973).
  18. ^ D. Hestenes, Observables, Operators and Complex Numbers in the Dirac Theory, Journal of Mathematical Physics. 16 556–572 (1975).
  19. ^ D. Hestenes (with R. Gurtler), Consistency in the Formulation of the Dirac, Pauli and Schroedinger Theories, Journal of Mathematical Physics 16: 573–583 (1975).
  20. ^ D. Hestenes, Spin and Uncertainty in the Interpretation of Quantum Mechanics, American Journal of Physics 47: 399–415 (1979).
  21. ^ D. Hestenes, Geometry of the Dirac Theory. Originally published in A Symposium on the Mathematics of Physical Space-Time, Facultad de Quimica, Universidad Nacional Autonoma de Mexico, Mexico City, Mexico (1981), p. 67–96.
  22. ^ D. Hestenes, The Zitterbewegung Interpretation of Quantum Mechanics, Foundations of Physics 20: 1213–1232 (1990).
  23. ^ a b D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, a unified language for mathematics and physics (Kluwer: Dordrecht/Boston, 1984).
  24. ^ D. Hestenes, Multivector Calculus, Journal of Mathematical Analysis and Applications 24: 313–325 (1968)
  25. ^ D. Hestenes, Grassmann's Vision. In G. Schubring (Ed.), Hermann Günther Grassmann (1809-1877) — Visionary Scientist and Neohumanist Scholar (Kluwer: Dordrecht/Boston, 1996), p. 191–201
  26. ^ D. Hestenes, Grassmann’s Legacy. In H-J. Petsche, A. Lewis, J. Liesen, S. Russ (eds.) From Past to Future: Grassmann’s Work in Context (Birkhäuser: Berlin, 2011)
  27. ^ D. Hestenes: Differential forms in geometric calculus. In: F. Brackx, R. Delanghe, H. Serras (eds.): Clifford Algebras and their Applications in Mathematical Physics: Proceedings of the Third Conference Held at Deinze, Belgium, 1993, Fundamental Theories of Physics, 1993, ISBN 978-0792323471, pp. 269–286, p. 270
  28. ^ A. Lasenby, C. Doran and S. Gull, A Multivector Derivative Approach to Lagrangian Field Theory, Foundations of Physics 23: 1295–12327 (1993)
  29. ^ A. Lasenby, C. Doran, & S. Gull, Gravity, gauge theories and geometric algebra, Philosophical Transactions of the Royal Society (London) A 356: 487–582 (1998)
  30. ^ C. Doran & A. Lasenby, Geometric Algebra for Physicists (Cambridge U Press: Cambridge, 2003)
  31. ^ C. Doran, D. Hestenes, F. Sommen & N. Van Acker, Lie Groups as Spin Groups, Journal of Mathematical Physics 34: 3642–3669 (1993)
  32. ^ D. Hestenes, Old Wine in New Bottles: A new algebraic framework for computational geometry. In E. Bayro-Corrochano and G. Sobczyk (eds), Advances in Geometric Algebra with Applications in Science and Engineering (Birkhauser: Boston, 2001). pp. 1–14
  33. ^ L. Dorst, C. Doran and J. Lasenby (Eds.), Applications of Geometric Algebra in Compute Science and Engineering, Birkhauser, Boston (2002)
  34. ^ L. Dorst, D. Fontjne and S. Mann, Geometric Algebra for Computer Science (Elsevier: Amsterdam, 2007)
  35. ^ D. Hestenes & J. Holt, The Crystallographic Space Groups in Geometric Algebra, Journal of Mathematical Physics 48: 023514 (2007)
  36. ^ H. Li, Invariant Algebras and Geometric Reasoning. (Beijing: World Scientific, 2008)
  37. ^ E. Bayro-Corrochano and G. Scheuermann (eds.), Geometric Algebra Computing for Engineering and Computer Science. (London: Springer Verlag, 2009)
  38. ^ L. Dorst and J. Lasenby, Guide to Geometric Algebra in Practice (Springer: London, 2011)
  39. ^ D. Hestenes, Wherefore a Science of Teaching? The Physics Teacher 17: 235–242 (1979)
  40. ^ D. Hestenes, Toward a Modeling Theory of Physics Instruction, American Journal of Physics 55: 440–454 (1987)
  41. ^ D. Hestenes, Modeling Games in the Newtonian World, American Journal of Physics 60: 732–748 (1992)
  42. ^ D. Hestenes, Modeling Software for learning and doing physics. In C. Bernardini, C. Tarsitani and M. Vincentini (Eds.), Thinking Physics for Teaching, Plenum, New York, pp. 25–66 (1996)
  43. ^ D. Hestenes (1997), Modeling Methodology for Physics Teachers. In E. Redish and J. Rigden (Eds.) The changing role of the physics department in modern universities, American Institute of Physics Part II. pp. 935–957
  44. ^ D. Hestenes, Notes for a Modeling Theory of Science, Cognition and Physics Education, In E. van den Berg, A. Ellermeijer and O. Slooten (Eds.) Modelling in Physics and Physics Education, (U. Amsterdam 2008)
  45. ^ D. Hestenes, Modeling Theory for Math and Science Education. In R. Lesh, P. Galbraith, Hines, A. Hurford (Eds.) Modeling Students’ Mathematical Competencies (New York: Springer, 2010)
  46. ^ M. Wells, D. Hestenes, and G. Swackhamer, A Modeling Method for High School Physics Instruction, American Journal of Physics 63: 606–619 (1995)
  47. ^ I. Halloun and D. Hestenes, The Initial Knowledge State of College Physics Students, American Journal of Physics 53: 1043–1055 (1985)
  48. ^ D. Hestenes, M. Wells, and G. Swackhamer, Force Concept Inventory, The Physics Teacher 30: 141–158 (1992)
  49. ^ R.R. Hake, "Interactive-engagement vs traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses," American Journal of Physics 66: 64– 74 (1998)
  50. ^ AMTA home page: http://modelinginstruction.org/
  51. ^ D. Hestenes, C. Megowan-Romanowicz, S.Osborn Popp, J. Jackson,and R. Culbertson, A graduate program for high school physics and physical science teachers, American Journal of Physics 79: 971–979 (2011)
  52. ^ D. Hestenes and J. Jackson (1997), Partnerships for Physics Teaching Reform ––a crucial role for universities and colleges. In E. Redish & J. Rigden (Eds.) The changing role of the physics department in modern universities, American Institute of Physics. Part I p. 449–459

External links

  • An interview with David Hestens: His life and achievements, M.F. Tasar et al., Eurasia Journal of Mathematics, Science and Technology Education, 2012, vol. 8, no. 2, pp. 139–153
  • Papers introducing geometric algebra: Research on Geometric Calculus
  • Oersted Medal Lecture “Reforming the Mathematical Language of Physics” on Geometric Algebra in Physics.
  • Writings on pedagogy: Papers on Modeling Instruction.
  • Imaginary numbers are not real – the geometric algebra of spacetime, a tutorial introduction to the ideas of geometric algebra, by S. Gull, A. Lasenby, C. Doran
  • Physical Applications of Geometric Algebra course-notes, see especially part 2.
  • Cambridge University Geometric Algebra group
  • Emeritus page at ASU, user page at ASU, ASU Modeling Instruction Program page
  • Hestenes' homepage on geometric calculus at ASU
  • A Critical Role for Physicists in K–12 Science Education Reform by David Hestenes and Jane Jackson
  • David Hestenes at the Mathematics Genealogy Project
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