... for geometry, you know, is the gate of science, and the gate is so low and small that one can only enter it as a little child. William K. Clifford
This paper was commissioned to chronicle the impact that David Hestenes' work has had on physics. Sadly, it seems to us that his work has so far not really had the impact it deserves to have, and that what is needed in this volume is that his message be AMPLIFIED and stated in a language that ordinary physicists understand. With his background in philosophy and mathematics, David is certainly no ordinary physicist, and we have observed that his ideas are a source of great mystery and confusion to many[2]. David accurately described the typical response when he wrote[3] that `physicists quickly become impatient with any discussion of elementary concepts' - a phenomenon we have encountered ourselves.
We believe that there are two aspects of Hestenes' work which physicists should take particularly seriously. The first is that the geometric algebra of spacetime is the best available mathematical tool for theoretical physics, classical or quantum[4,5,6]. Related to this part of the programme is the claim that complex numbers arising in physical applications usually have a natural geometric interpretation that is hidden in conventional formulations[5,7,8,9]. David's second major idea is that the Dirac theory of the electron contains important geometric information[1,3,10,11], which is disguised in conventional matrix-based approaches. We hope that the importance and truth of this view will be made clear in this and the three following papers. As a further, more speculative, line of development, the hidden geometric content of the Dirac equation has led David to propose a more detailed model of the motion of an electron than is given by the conventional expositions of quantum mechanics. In this model[12,13], the electron has an electromagnetic field attached to it, oscillating at the `zitterbewegung' frequency, which acts as a physical version of the de Broglie pilot-wave[14].
David Hestenes' willingness to ask the sort of question that
Feynman specifically warned against
, and to engage in varying degrees of
speculation, has undoubtedly had the unfortunate effect of diminishing
the impact of his first idea, that geometric algebra can provide a
unified language for physics - a contention that we strongly
believe. In this paper, therefore, we will concentrate on the first
aspect of David's work, deferring to a companion
paper[16] any critical examination of his interpretation
of the Dirac equation.
In Section 2 we provide a gentle introduction to geometric algebra, emphasising the geometric meaning of the associative (Clifford) product of vectors. We illustrate this with the examples of 2- and 3-dimensional space, showing that it is possible to interpret the unit scalar imaginary number as arising from the geometry of real space. Section 3 introduces the powerful techniques by which geometric algebra deals with rotations. This leads to a discussion of the role of spinors in physics. In Section 4 we outline the vector calculus in geometric algebra and review the subject of monogenic functions; these are higher-dimensional generalisations of the analytic functions of two dimensions. Relativity is introduced in Section 5, where we show how Maxwell's equations can be combined into a single relation in geometric algebra, and give a simple general formula for the electromagnetic field of an accelerating charge. We conclude by comparing geometric algebra with alternative languages currently popular in physics. The paper is based on an lecture given by one of us (SFG) to an audience containing both students and professors. Thus, only a modest level of mathematical sophistication (though an open mind) is required to follow it. We nevertheless hope that physicists will find in it a number of surprises; indeed we hope that they will be surprised that there are so many surprises!