Imaginary Numbers are not Real – the Geometric Algebra of Spacetime

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.

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)