Overview of Geometric Algebra in Physics

Oersted Medal Lecture 2002:
 Reforming the Mathematical Language of Physics
                In: D. Hestenes, Am. J. Phys. 71 (2), February 2003, pp. 104--121.


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 contribute to a fragmentation of knowledge. We can do better! Research on the design and use of mathematical systems provides a guide for designing a unified mathematical language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geometric Algebra, which I introduce with emphasis on how it integrates and simplifies classical, relativistic and quantum physics. Introducing research-based reform into a conservative physics curriculum is a challenge for the emerging PER community. Join the fun!
Oersted Medal Lecture (short version) in Power Point form: ppt version

A sequel to the Oersted paper:

Spacetime Physics with Geometric Algebra
                In: D. Hestenes, Am. J. Phys. 71 (7), July 2003, pp. 691--714.


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 with applications from classical rigid body mechanics to relativistic quantum theory -- thus significantly reducing the mathematical and conceptual barriers between classical and quantum mechanics. The entire physics curriculum can be unified and simplified by adopting STA as the standard mathematical language. This would enable early infusion of spacetime physics and give it the prominent place it deserves in the curriculum.

A third paper in the Oersted sequence:

Gauge Theory Gravity with Geometric Calculus
             D. Hestenes, Foundations of Physics, 35(6): 903-970 (2005).


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 leads to resolution of long-standing problems with energymomentum conservation in general relativity. Geometric calculus provides many simplifications and fresh insights in theoretical formulation and physical applications of the theory.

Energymomentum Complex in General Relativity and Gauge Theory


Alternative versions of the energymomentum 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 resolve ambiguities in energymomentum conservation laws and reveal intriguing new relations between superpotential, gauge connection and spin angular momentum with rich new possibilities for physical interpretation.

Spacetime Geometry with Geometric Calculus
               To be published.


Geometric Calculus is developed for curved-space treatments of General Relativity 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 Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides mathematical tools that streamline the formulation and simplify calculations. The formalism automatically includes spinors so the Dirac equation is incorporated in a geometrically natural way.

Gauge Gravity and Electroweak Theory
               In H. Kleinert, R. T. Jantzen & R. Ruffini (Eds.), Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity (World Scientific, Singapore, 2008) pp. 629-647.


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). Besides clarifying the conceptual foundations of GR and facilitating complex computations, GTG opens up new possibilities for a unified gauge theory of gravity and quantum mechanics, including spacetime geometry of electroweak interactions. The Weinberg-Salam model fits perfectly into this geometric framework, and a promising variant that replaces chiral states with Majorana states is formulated to incorporate zitterbewegung in electron states.

The Crystallographic Space Groups in Geometric Algebra.
               D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007)    (22 pages)


We present a complete formulation of the 2D and 3D 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 basis of lattice vectors that define its crystal class. A new system of space group symbols enables one to unambiguously write down all generators of a given space group directly from its symbol.

Hunting for Snarks in Quantum Mechanics.
               D. Hestenes, in P. Goggins and C-Y Chan (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering (American Institute of Physics, 2009).    (17 pages)


A long-standing debate over the interpretation of quantum mechanics has centered on the meaning of Schroedingerís 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; hence the probability interpretation of ψψ* expresses an irreducible uncertainty in electron behavior that is intrinsic in nature. On the other side, the realist school (led by Einstein, de Broglie, Bohm and Jaynes) holds that ψ represents a statistical ensemble of possible electron states; hence it is an incomplete description of a single electron state. I contend that the debaters have overlooked crucial facts about the electron revealed by Dirac theory. In particular, analysis of electron zitterbewegung (first noticed by Schroedinger) opens a window to particle substructure in quantum mechanics that explains the physical significance of the complex phase factor in ψ. This led to a testable model for particle substructure with surprising support by recent experimental evidence. If the explanation is upheld by further research, it will resolve the debate in favor of the realist school. I give details. The perils of research on the foundations of quantum mechanics have been foreseen by Lewis Carroll in The Hunting of the Snark!

Grassmannís Legacy.
               D. Hestenes    (16 pages)


In a previous conference honouring Hermann Grassmannís profound intellectual contributions (Schubring 1996), I cast him as a central figure in the historical development of a universal geometric calculus for mathematics and physics (Hestenes 1996). Sixteen years later I am here to report that impressive new applications in this tradition are rapidly developing in computer science and robotics as well as physics and mathematics. Especially noteworthy is the emergence of Conformal Geometric Algebra as an ideal tool for computational geometry, as it fulfils at last one of Grassmannís grandest goals and confirms the prescience of his mathematical insight. Geometric Calculus has finally reached sufficient maturity to serve as a comprehensive geometric language for the whole community of scientists, mathematicians and engineers. Moreover, its simplicity recommends it as a tool for reforming high school mathematics and physics, as Grassmann had envisioned.

Last modified on 7 May 2010.
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