is the See also
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More about the above paper: Imaginary Numbers are not Real - the Geometric Algebra of Spacetime
Found. Phys. 23(9), 1175-1201 (1993)
Abstract: This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a `geometric product' of vectors in 2- and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analysed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics). Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained - results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.
J. Lasenby, A.N. Lasenby and C.J.L. Doran
A unified mathematical language for physics and engineering in the 21st century
Phil. Trans. R. Soc. Lond. A 358, 21-39 (2000)
Abstract: The late 18th and 19th centuries were times of great mathematical progress. Many new mathematical systems and languages were introduced by some of the millennium's greatest mathematicians. Amongst these were the algebras of Clifford (1878) and Grassmann (1877). While these algebras caused considerable interest at the time, they were largely abandoned with the introduction of what people saw as a more straightforward and more generally applicable algebra - the vector algebra of Gibbs. This was effectively the end of the search for a unifying mathematical language and the beginning of a proliferation of novel algebraic systems, created as and when they were needed; for example, spinor algebra, matrix and tensor algebra, differential forms etc.
In this paper we will chart the resurgence of the algebras of Clifford and Grassmann in the form of a framework known as Geometric Algebra (GA). GA was pioneered in the mid-1960's by the American physicist and mathematician, David Hestenes. It has taken the best part of 40 years but there are signs that his claims that GA is the universal language for physics and mathematics are now beginning to take a very real form. Throughout the world there are an increasing number of groups who apply GA to a range of problems from many scientific fields. While providing an immensely powerful mathematical framework in which the most advanced concepts of quantum mechanics, relativity, electromagnetism etc. can be expressed, it is claimed that GA is also simple enough to be taught to school children! In this paper we will review the development and recent progress of GA and discuss whether it is indeed the unifying language for the physics and mathematics of the 21st century. The examples we will use for illustration will be taken from a number of areas of physics and engineering
- GeometricAlgebraForComputerScience by Leo Dorst, Daniel Fontijne and Stephen Mann
- GeometricAlgebraForPhysicists by Chris Doran and Anthony Lasenby
- GeometricAlgebraForComputerGraphics by John Vince
- CliffordAlgebrasAndSpinors by Pertti Lounesto (2nd edition) Cambridge University Press ISBN 0521005515
- Clifford Algebra to Geometric Calculus : A Unified Language for Mathematics and Physics (Fundamental Theories of Physics) by D. Hestenes, 1987 ISBN 9027725616
- New Foundations for Classical Mechanics (Fundamental Theories of Physics) by D. Hestenes, 2nd ed, 1999, ISBN 0792355148
- an introduction to geometric algebra as a unified language for physics and mathematics
- Geometric Algebra by Eduardo Bayro Corrochano (ed.), 2001 ISBN 0817641998
- with applications to theorem proving, computer-aided geometry, robotics, computer vision
- first chapter by Hestenes
- Electrodynamics : A Modern Geometric Approach (Progress in Mathematical Physics) by William E. Baylis, 2004, ISBN 0817640258
- The natural appearance of the Minkowski spacetime metric in the paravector space of Clifford's geometric algebra is used to formulate a covariant treatment in special relativity that seamlessly connects spacetime concepts to the spatial vector treatments common in undergraduate texts.
- HistoryOfVectorAnalysis Michael J. Crowe (1967, so background to the early stuff)
- wrote symbolic algebra package for Maple that supports Clifford algebra symbolic computations
wrote a symbolic algebra and computation tool, CLICAL, that can be used with a variety of different algebras, including CliffordAlgebra
See also CliffordAlgebra
(lots of them)