The late 18th and 19th centuries were times of great mathematical progress. Many new mathematical systems and languages were introduced by some of the millenium’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.
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)