S. F. Gull, A. N. Lasenby and C. J. L. Doran.
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