# David HestenesArizona State University | ASU · Department of Physics

David Hestenes

PhD, Physics

## About

118

Publications

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Introduction

From his Wikipedia page: David Hestenes, 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, and as founder of Modelling Instruction, a research-based program to reform K–12 Science, Technology, Engineering, and Mathematics (STEM) education.
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## Publications

Publications (118)

Alternative versions of the energy–momentum complex in general relativity are given compact new formulations with spacetime algebra. A new unitary form for Einstein’s equation greatly simplifies the derivation and analysis of gravitational superpotentials. Interpretation of Einstein’s equations as a gauge field theory on flat spacetime is shown to...

Geometric Calculus is developed for curved-space treatments of General Rel-ativity and comparison with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein's Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Rel-ativity as a Gauge Theory of Gravity on a cu...

The Dirac equation is reinterpreted as a constitutive equation for singularities in the electromagnetic vacuum, with the electron as a point singularity on a lightlike toroidal vortex. The diameter of the vortex is a Compton wavelength and its thickness is given by the electron's anomalous magnetic moment. The photon is modeled as an electron-posit...

Understanding the electron clock and the role of complex numbers in quantum mechanics is grounded in the geometry of spacetime, and best expressed with Spacetime Algebra (STA). The efficiency of STA is demonstrated with coordinate-free applications to both relativistic and non-relativistic QM. Insight into the structure of Dirac theory is provided...

The book Clifford Algebra to Geometric Calculus is the first and still the most complete exposition of Geometric Calculus (GC). But it is more of a reference book than a textbook, so can it be a difficult read for beginners. This tutorial is a guide for serious students who want to dig deeply into the subject. It presents helpful background and aim...

In a previous conference honouring Hermann Grassmann’s profound intellectual con tributions [Schubring 1996a], I cast him
as a central figure in the historical development of a universal geometric calculus for mathematics and physics [Hestenes 1996]. Six teen years later I am here to report that impressive new applications in
this tradition are rap...

The Modeling Instruction Program at Arizona State University has demonstrated the feasibility and effectiveness of a university-based graduate program dedicated to professional development of in-service physics teachers. The program culminates in a Master of Natural Science degree, although not all students in the program are degree candidates. The...

Geometric Algebra makes it possible to formulate simple spinor equations of motion for classical particles and rigid bodies.
In the Newtonian case, these equations have proven their value by simplifying orbital computations. The relativistic case
is not so well known, but it has new and surprising features worth exploiting, including close connecti...

Mathematics has been described as the science of patterns. Natural science can be characterized as the investigation of patterns
in nature. Central to both domains is the notion of model as a unit of coherently structured knowledge. Modeling Theory is
concerned with models as basic structures in cognition as well as scientific knowledge. It maintai...

Conformal Geometric Algebraic (CGA) provides ideal mathematical tools
for construction, analysis, and integration of classical Euclidean,
Inversive & Projective Geometries, with practical applications to
computer science, engineering, and physics. This paper is a
comprehensive introduction to a CGA tool kit. Synthetic statements in
classical geomet...

A long‐standing debate over the interpretation of quantum mechanics has centered on the meaning of Schroedinger’s wave function Ψ for an electron. Broadly speaking, there are two major opposing schools. On the one side, the Copenhagen school (led by Bohr, Heisenberg and Pauli) holds that Ψ provides a complete description of a single electron state;...

The discovery of Mathematical Viruses is announced here for the flrst time. Such viruses are a serious threat to the general mental health of the mathematical community. Several viruses inimical to the unity of mathematics are identifled, and their deleterious characteristics are described. A strong dose of geometric algebra and calculus is the bes...

Reformulation of the Dirac equation in terms of the real Spacetime Algebra (STA) reveals hidden geometric structure, including a geometric role for the unit imaginary as generator of rotations in a spacelike plane. The STA and the real Dirac equation play essential roles in a new Gauge Theory Gravity (GTG) version of General Relativity (GR). Beside...

If electron zitterbewegung is a real effect, it should generate an electric dipole field oscillating with the zitterbewegung frequency 2mc^2/hbar. The possibility of detecting it as a resonance in electron channeling is analyzed.

The possibility that zitterbewegung opens a window to particle substructure in quantum mechanics is explored by constructing a particle model with structural
features inherent in the Dirac equation. This paper develops a self-contained dynamical model of the electron as a lightlike
particle with helical zitterbewegung and electromagnetic interactio...

The authors describe a Modeling Instruction program that places an emphasis on the construction and application of conceptual models of physical phenomena as a central aspect of learning and doing science.

We present a complete formulation of the two-dimensional and three-dimensional crystallographic space groups in the conformal geometric algebra of Euclidean space. This enables a simple new representation of translational and orthogonal symmetries in a multiplicative group of versors. The generators of each group are constructed directly from a bas...

Modeling Theory provides common ground for interdisciplinary research in science education and the many branches of cognitive science, with implications for scientific practice, instructional design, and connections between science, mathematics and common sense.

A new gauge theory of gravity on flat spacetime has recently been developed by Lasenby, Doran, and Gull. Einstein’s principles of equivalence and general relativity are replaced by gauge principles asserting, respectively, local rotation and global displacement gauge invariance. A new unitary formulation of Einstein’s tensor illuminates long-standi...

This is an introduction to spacetime algebra (STA) as a unified
mathematical language for physics. STA simplifies, extends, and
integrates the mathematical methods of classical, relativistic, and
quantum physics while elucidating geometric structure of the theory. For
example, STA provides a single, matrix-free spinor method for rotational
dynamics...

The connection between physics teaching and research at its deepest level can be illuminated by physics education research (PER). For students and scientists alike, what they know and learn about physics is profoundly shaped by the conceptual tools at their command. Physicists employ a miscellaneous assortment of mathematical tools in ways that con...

The Dirac equation has a hidden geometric structure that is made manifest by reformulating it in terms of a real spacetime algebra. This reveals an essential connection between spin and complex numbers with profound implications for the interpretation of quantum mechanics. Among other things, it suggests that to achieve a complete interpretation of...

Geometric algebra provides the essential foundation for a new approach to symmetry,groups. Each of the 32 lattice point groups and 230 space groups in three dimensions is generated from a set of three symmetry vectors. This greatly facilitates representation, analysis and application of the groups to molecular modeling and crystal- lography.

Geometric algebra is used in an essential way to provide a coordinate- free approach to Euclidean geometry and rigid body mechanics that fully inte- grates rotational and translational dynamics. Euclidean points are given a homo- geneous representation that avoids designating one of them as an origin of coordi- nates and enables direct computation...

The study of relations among Euclidean, spherical and hyperbolic geometries dates back to the beginning of last century. The attempt to prove Euclid’s fifth postulate led C. F. Gauss to discover hyperbolic geometry in the 1820’s. Only a few years passed before this geometry was rediscovered independently by N. Lobachevski (1829) and J. Bolyai (1832...

The standard algebraic model for Euclidean space En
is an n-dimensional real vector space ℝn
or, equivalently, a set of real coordinates. One trouble with this model is that, algebraically, the origin is a distinguished element, whereas all the points of En
are identical. This deficiency in the vector space model was corrected early in the 19th cen...

Classical geometry has emerged from efforts to codify perception of space and motion. With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean, non-Euclidean and projective geometries were given precise mathematical formulations and the rich properties of geometric objects were explored. Though...

The well-known Force Concept Inventory (FCI) instrument has been in use over the last 15 years, and is now credited with stimulating reform of physics education. An instructor can give the FCI as both a pre-test and as a post-test to produce data that can be used in a continuous improvement manner to evaluate the effectiveness of various instructio...

The recorded study of spheres dates back to the first century in the book Sphaerica of Menelaus. Spherical trigonometry was thoroughly developed in modern form by Euler in his 1782 paper [75]. Spherical geometry in n-dimensions was first studied by Schläfli in his 1852 treatise, which was published posthumously in [202]. The most important transfor...

My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean space with all sorts of applications to computer-aided geometry, robotics, computer vision and the like. A detailed description and analysis of the model is soon to be published elsewhere, so I can concentrate on highlights here, although with a slightly...

Student views about knowing and learning physics have been probed with the Views About Sciences Survey (VASS) along six conceptual dimensions, and classified into four distinct profiles: expert, high transitional, low transitional, and folk. As an aid to interpreting VASS results, this article provides a qualitative analysis of student responses to...

Scientific practice involves the construction, validation and application of scientific models, so science instruction should be designed to engage students in making and using models. Scientific models are coherent units of structured knowledge. They are used to organize factual information into coherent wholes, often by the coordinated use of gen...

To meet National Standards recommended by the National Research Council for high school physics, inservice teachers must be integrated into the physics community. They must be empowered by access to resources of the physics community and by sustained support for their professional development. To that end, university and college physics departments...

The Dirac theory is completely reformulated in terms of Spacetime Algebra, a real Clifford Algebra char-acterizing the geometrical properties of spacetime. This eliminates redundancy in the conventional matrix formulation and reveals a hidden geometric structure in the theory. Among other things, it reveals that complex numbers in the Dirac equatio...

After nearly a century on the brink of obscurity, Hermann Grassmann is widely recognized as the originator of Grassmann algebra, an indispensable tool in modern mathematics. Still, in conception and applications, conventional renditions of his exterior algebra fall far short of Grassmann’s original vision. A fuller realization of his vision is foun...

The Force Concept Inventory (FCI)1 is a unique kind of "test" designed to assess student understanding of the most basic concepts in Newtonian physics. It can be used for several different purposes, but the most important one is to evaluate the effectiveness of instruction. For that purpose, the FCI is probably the most widely used instrument in ph...

The design and development of a new method for high school physics instruction is described. Students are actively engaged in understanding the physical world by constructing and using scientific models to describe, explain, predict, and to control physical phenomena. Course content is organized around a small set of basic models. Instruction is or...

The personal computer revolution confronts physics education with an unprecedented challenge and opportunity. Microcomputer hardware and software for data collection and management, for numerical calculation, symbolic manipulation and computer simulation are developing at a breathless pace. Processor power, network speed and capacity are no longer...

A new invariant formulation of 3D eye-head kinematics improves on the computational advantages of quaternions. This includes a new formulation of Listing's Law parametrized by gaze direction leading to an additive, rather than a multiplicative, saccadic error correction with a gaze vector difference control variable. A completely general formulatio...

Invariant methods for formulating and analyzing the mechanics of the skeleto-muscular system with geometric algebra are further developed and applied to reaching kinematics. This work is set in the context of a neurogeometry research program to develop a coherent mathematical theory of neural sensory-motor control systems.

It is shown that every Lie algebra can be represented as a bivector alge- bra; hence every Lie group can be represented as a spin group. Thus, the computa- tional power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thor- oughly analyzed, an...

Guidelines for constructing point particle models of the electron withzitterbewegung and other features of the Dirac theory are discussed. Such models may at least be useful approximations to the Dirac theory, but the more exciting possibility is that this approach may lead to a more fundamental reality.

Hamiltonian mechanics is given an invariant formulation in terms of Geometric Calculus, a general difierential and integral calculus with the structure of Clifiord algebra. Advantages over formulations in terms of difierential forms are explained.

Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system...

The basic principles of Newtonian mechanics can be interpreted as a
system of rules defining a medley of modeling games. The common
objective of these games is to develop validated models of physical
phenomena. This is the starting point for a promising new approach to
physics instruction in which students are taught from the beginning that
in scie...

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Abstract The Dirac wave function is represented in a form where all its compo- nents have obvious geometrical and physical interpretations. Six components compose a Lorentz transformation determining the electron velocity and spin directions. This provides the basis for a rigorous connection between relativis- tic rigid body dynamics and the time e...

Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebra into the foundations of linear algebra. There is a natural extension of linear transformations on a vector space to the associated Clifford algebra with...

Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra. This closes the gap between algebraic and synthetic approaches to projective geometry and facilitates connections with the rest of mathematics.

To cope with the explosion of information in mathematics and physics, we need a unifled mathematical language to integrate ideas and results from diverse flelds. Clifiord Algebra provides the key to a uni∞ed Geometric Calculus for expressing, developing, integrating and applying the large body of geometrical ideas running through mathematics and ph...

The purpose of this note is to record some of the early results obtained in the work in progress of the same title. A unipotent is a number other than plus or minus one which has square one. The unipodal numbers are formed by algebraically extending the complex numbers to include one such unipotent and its additive inverse. Historically complex num...

The zitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A refor-mulation of the Dirac theory shows that this interpretation can be sustained rigorously, with the complex phase factor in the wave function describing the local frequency and phase of the circulatory motion dir...

Fundamentos de mecánica clásica. Contenido: Orígenes del algebra geométrica; Desarrollo del algebra geométrica; Mecánica de una partícula simple; Fuerzas centrales en un sistema de dos partículas; Operadores y trasformaciones; Sistemas de muchas partículas; Mecánica de los cuerpos rígidos; Mecánica celeste; Mecánica relativista.

Thezitterbewegung is a local circulatory motion of the electron presumed to be the basis of the electron spin and magnetic moment. A reformulation of the Dirac theory shows that thezitterbewegung need not be attributed to interference between positive and negative energy states as originally proposed by Schroedinger. Rather, it provides a physical...

A means for separating subjective and objective aspects of the electron wave function is suggested, based on a reformulation of the Dirac Theory in terms of Spacetime Algebra. The reformulation admits a separa-tion of the Dirac wave function into a two parameter probability factor and a six parameter kinematical factor. The complex valuedness of th...

The claim that Clifford algebra should be regarded as a universal geometric algebra is strengthened by showing that the algebra is applicable to nonmetrical as well as metrical geometry. Clifford algebra is used to develop a coordinate-free algebraic formulation of projective geometry. Major theorems of projective geometry are reduced to algebraic...

An analysis of the conceptual structure of physics identifies essential factual and procedural knowledge which is not explicitly formulated and taught in physics courses. It leads to the conclusion that mathematical modeling of the physical world should be the central theme of physics instruction. There are reasons to believe that traditional metho...

Modeling theory was used in the design of a method to teach problem solving in introductory mechanics. A pedagogical experiment to evaluate the effectiveness of the method found positive results.

Geometric Calculus is primarily concerned with the theory and techniques for differentiating and integrating geometric functions, that is, functions whose domain and range are subsets of the Universal Geometric Algebra G. This chapter deals with the differentiation of functions defined on linear subspaces of G. Differentiation can be defined on mor...

This chapter defines Geometric Algebra by a set of axioms and develops a system of definitions and identities to make it a versatile and efficient computational tool. These results are used repeatedly in subsequent chapters. Many results are obtained in the form of algebraic identities, but they are seldom presented as theorems, because we wish to...

This chapter shows the advantages of developing the theory of linear and multilinear functions on finite dimensional spaces with Geometric Calculus. The theory is sufficiently well developed here to be readily applied to most problems of linear algebra.

The modern approach to calculus on manifolds, as typified by ref. [La], begins with the general notion of a topological space, from which spaces of increasing complexity are built up by introducing a succession of structures such as differentiable maps, fiber bundles, differential forms, connections and metrics. Without disputing that there are goo...

In spite of the enormous complexity of the human brain, there are good reasons to believe that only a few basic principles will be needed to understand how it processes sensory input and controls motor output. In fact, the most important principles may be known already! These principles provide the basis for a definite mathematical theory of learni...

This chapter continues the study of calculus on vector manifolds begun in Chapter 4. The emphasis here is on the central object of classical differential geometry, the curvature tensor. We have endeavored to supply simple and systematic derivations of all properties of the curvature tensor including relations to extrinsic geometry, behavior under t...

This chapter describes some basic contributions of Geometric Calculus to the theory of integration. The directed integral enables us to formulate and prove a few comprehensive theorems from which the main results of both real and complex variable theory are easily obtained.

This chapter develops an efficient method for expressing the intrinsic geometry of a manifold in terms of local properties of vector fields. The method is actually a special case of the theory in Section 5-6, but we develop it ab initio here to make its relation to the classical method of tensor analysis as direct and clear as possible. This chapte...

This report analyzes the eddy current induced in a solid conducting sphere by a sinusoidal current in a circular loop. Analytical expressions for the eddy currents are derived as a power series in the vectorial displacement of the center of the sphere from the axis of the loop. These are used for first order calculations of the power dissipated in...

A new method for calculating the curvature tensor is developed and applied to the Scharzschild case. The method employs Clifford algebra and has definite advantages over conventional methods using differential forms or tensor analysis.

The translational and rotational equations of motion for a small rigid body in a gravitational field are combined in a single spinor equation. Besides its computational advantages, this unifies the description of gravitational interaction in classical and quantum theory. Explicit expressions for gravitational precession rates are derived.

The Dirac theory has a hidden geometric structure. This talk traces the concep-tual steps taken to uncover that structure and points out significant implications for the interpre-tation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This impli...

Common sense beliefs of college students about motion and its causes are surveyed and analyzed. A taxonomy of common sense concepts which conflict with Newtonian theory is developed as a guide to instruction.

An instrument to assess the basic knowledge state of students taking a
first course in physics has been designed and validated. Measurements
with the instrument show that the student's initial qualitative, common
sense beliefs about motion and causes has a large effect on performance
in physics, but conventional instruction induces only a small cha...

1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5. Geometric Algebras of PseudoEuclidean Spaces.- 2 / Differentiation.- 2-1. Differentiation by Vectors.- 2-2. Multivector Derivative, Differential and Adjoints....

We explore the possibility thatzitterbewegung is the key to a complete understanding of the Dirac theory of electrons. We note that a literal interpretation of thezitterbewegung implies that the electron is the seat of an oscillating bound electromagnetic field similar to de Broglie's pilot wave. This opens up new possibilities for explaining two m...

Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.

A new spinor formulation of rotational dynamics is developed. A general theorem is established reducing the theory of the symmetric top to that of the spherical top. The classical problems of Lagrange and Poinsot are treated in detail, along with a modern application to the theory of magnetic resonance.

The Kustaanheimo theory of spinor regularization is given a new formulation in terms of geometric algebra. The Kustaanheimo-Stiefel matrix and its subsidiary condition are put in a spinor form directly related to the geometry of the orbit in physical space. A physically significant alternative to the KS subsidiary condition is discussed. Derivation...

Variational equations describing deviations from a reference orbit play a central role in space flight analysis. They are widely employed for error estimation in targeting problems, and they have many applications to optimal orbital transfer and spacecraft rendezvous problems. Since the variational equations have so many uses, it is important to ha...

The generator of electromagnetic gauge transformations in the Dirac equation has a unique geometric interpretation and a unique extension to the generators of the gauge group SU(2) U(1) for the Weinberg-Salam theory of weak and electromagnetic interactions. It follows that internal symmetries of the weak interactions can be interpreted as space-tim...

A rigorous derivation of the Schrödinger theory from the Pauli (or Dirac) theory implies that the Schrödinger equation describes an electron in an eigenstate of spin. Furthermore, the ground-state kinetic energy is completely determined by the electron spin density. This can be explained by interpreting the spin as an orbital angular momentum, whic...

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Properties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of...

The geometrical formulation of the Dirac theory with spacetime algebra is shown to be equivalent to the usual matrix formalism. Imaginary numbers in the Dirac theory are shown to be related to the spin tensor. The relation of observables to operators and the wavefunction is analyzed in detail and compared with some purportedly general principles of...

A spinor formulation of the classical Lorentz force is given which describes the presession of an electron's spin as well as its velocity. Solutions are worked out applicable to an electron in a

Spacetime algebra is employed to formulate classical relativis- tic mechanics without coordinates. Observers are treated on the same footing as other physical systems. The kinematics of a rigid body are expressed in spinor form and the Thomas precession is derived.

By a new method, the Dirac electron theory is completely reexpressed as a set of conservation laws and constitutive relations for local observables, describing the local distribution and flow of mechanical quantities. The coupling of the electromagnetic field to the electron is shown to be determined by the definitions of the observables rather tha...

The Pauli theory of electrons is formulated in the language of multivector calculus. The advantages of this approach are demonstrated in an analysis of local observables. Planck's constant is shown to enter the theory only through the magnitude of the spin. Further, it is shown that, when obtained as a limiting case of the Pauli theory, the Schrödi...