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 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.
Note that some figures are different to the published version.