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# Imaginary Numbers are not Real - the Geometric Algebra of Spacetime

**Stephen Gull (a), Anthony Lasenby (a) and
Chris Doran (b) **

(a) MRAO, Cavendish Laboratory, Madingley Road,
Cambridge CB3 0HE, UK

(b) DAMTP, Silver Street, Cambridge, CB3 9EW, UK

**February 9, 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.
**
*

*Mark Ashdown*