David Hestenes

 ( 21st-century American Physicists ) 

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.D. Hestenes, New Foundations for Classical Mechanics (Kluwer: Dordrecht/Boston, 1986), Second Edition (1999).

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. In 1987, he became the first visiting scholar in the Department of Cognitive and Neural Systems (Boston University) and worked on neuroscience research for a period.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). pp. 173–205.D. Hestenes, Invariant Body Kinematics: I. Saccadic and compensatory eye movements. Neural Networks 7: 65–77 (1994).D. Hestenes, Invariant Body Kinematics: II. Reaching and neurogeometry. Neural Networks 7: 79–88 (1994).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.

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.

The first line began with the fact that reformulation of the Dirac equation in terms of spacetime algebra reveals hidden geometric structure.D. Hestenes, Real Spinor Fields, Journal of Mathematical Physics 8: 798–808 (1967). Among other things, it reveals that the complex factor $i\; \backslash hbar$ in the equation is a geometric quantity (a bivector) identified with electron spin, where $i$ specifies the spin direction and $\backslash hbar\; /2$ is the spin magnitude. The implications of this insight have been studied in a long series of papersD. Hestenes and R. Gurtler, Local Observables in Quantum Theory, American Journal of Physics 39: 1028 (1971).D. Hestenes, Local Observables in the Dirac Theory, Journal of Mathematical Physics 14: 893–905 (1973).D. Hestenes, Observables, Operators and Complex Numbers in the Dirac Theory, Journal of Mathematical Physics. 16 556–572 (1975).D. Hestenes (with R. Gurtler), Consistency in the Formulation of the Dirac, Pauli and Schroedinger Theories, Journal of Mathematical Physics 16: 573–583 (1975).D. Hestenes, Spin and Uncertainty in the Interpretation of Quantum Mechanics, American Journal of Physics 47: 399–415 (1979).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), pp. 67–96. with the most significant conclusion linking it to Schrödinger's zitterbewegung and proposing a zitterbewegung interpretation of quantum mechanics.D. Hestenes, The Zitterbewegung Interpretation of Quantum Mechanics, Foundations of Physics 20: 1213–1232 (1990). 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 CalculusD. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, a unified language for mathematics and physics (Kluwer: Dordrecht/Boston, 1984). 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 , and an extension of the Cauchy integral theorem to higher dimensions.D. Hestenes, Multivector Calculus, Journal of Mathematical Analysis and Applications 24: 313–325 (1968)

Hestenes emphasizes the important role of the mathematician Hermann GrassmannD. Hestenes, Grassmann's Vision. In G. Schubring (Ed.), Hermann Günther Grassmann (1809–1877) — Visionary Scientist and Neohumanist Scholar (Kluwer: Dordrecht/Boston, 1996), pp. 191–201D. 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) 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”,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, , pp. 269–286, p. 270 which withdraws attention from its role as a unified language for mathematics and physics.

Hestenes' work has been applied to Lagrangian field theory,A. Lasenby, C. Doran and S. Gull, A Multivector Derivative Approach to Lagrangian Field Theory, Foundations of Physics 23: 1295–12327 (1993) formulation of a gauge theory of gravity alternative to general relativity by Lasenby, Doran and Gull, which they call gauge theory gravity (GTG),A. Lasenby, C. Doran, & S. Gull, Gravity, gauge theories and geometric algebra, Philosophical Transactions of the Royal Society (London) A 356: 487–582 (1998)C. Doran & A. Lasenby, Geometric Algebra for Physicists (Cambridge U Press: Cambridge, 2003) and it has been applied to spin representations of .C. Doran, D. Hestenes, F. Sommen & N. Van Acker, Lie Groups as Spin Groups, Journal of Mathematical Physics 34: 3642–3669 (1993) Most recently, it led Hestenes to formulate conformal geometric algebra, a new approach to computational geometry.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 This has found a rapidly increasing number of applications in engineering and computer science.L. Dorst, C. Doran and Joan Lasenby (Eds.), Applications of Geometric Algebra in Computer Science and Engineering, Birkhauser, Boston (2002)L. Dorst, D. Fontjne and S. Mann, Geometric Algebra for Computer Science (Elsevier: Amsterdam, 2007)D. Hestenes & J. Holt, The Crystallographic Space Groups in Geometric Algebra, Journal of Mathematical Physics 48: 023514 (2007)H. Li, Invariant Algebras and Geometric Reasoning. (Beijing: World Scientific, 2008)E. Bayro-Corrochano and G. Scheuermann (eds.), Geometric Algebra Computing for Engineering and Computer Science. (London: Springer Verlag, 2009)L. Dorst and Joan Lasenby, Guide to Geometric Algebra in Practice (Springer: London, 2011)

He has contributed to the main conferences in this field, the International Conference on Clifford Algebras (ICCA) and the Applications of Geometric Algebra in Computer Science and Engineering (AGACSE) series.

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),AMTA home page: http://modelinginstruction.org/ 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.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) This provides a validated model for similar programs at universities across the country.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 pp. 449–459

• 2014 Excellence in Physics Education Award from the American Physical Society
• 2003 Award for excellence in educational research by the Council of Scientific Society Presidents
• 2002 Oersted Medal, awarded by the American Association of Physics Teachers for notable contributions to the teaching of physics
• Fellow of the American Physical Society
• Overseas Fellow of Churchill College, Cambridge
• Foundations of Physics Honoree (Sept.–Nov. issues, 1993)
• Fulbright Research Scholar (England) 1987–1988
• NASA Faculty Fellow (Jet Propulsion Laboratory) 1980, 1981
• NSF Postdoctoral Fellow (Princeton) 1964–1966
• University Fellow (UCLA) 1958–1959

Books

• D. Hestenes: Space-Time Algebra, 2nd ed., Birkhäuser, 2015,
• D. Hestenes: New Foundations for Classical Mechanics, Fundamental Theories of Physics, 2nd ed., Springer Verlag, 1999,
• D. Hestenes, A. Weingartshofer (eds.): The Electron: New Theory and Experiment, Fundamental Theories of Physics, Springer, 1991,
• D. Hestenes, Garret Sobczyk: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, Fundamental Theories of Physics, Springer, 1987,

• Archive of Hestenes's papers
• Archive of Hestenes's homepage on geometric calculus at ASU
• An interview with David Hestenes: 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
• The Genesis of Geometric Algebra: A Personal Retrospective, Adv. Appl. Clifford Algebras 27, 351–379 (2017)
• Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics
• David Hestenes at the Mathematics Genealogy Project
• Cambridge University Geometric Algebra group
• 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.