Traditional vectors as an introduction to geometric algebra

Abstract

The 2002 Oersted Medal Lecture by David Hestenes concerns themany advantages for education in physics if geometric algebra were toreplace standard vector algebra. However, such a change has difficultiesfor those who have been taught traditionally. A new way of introducinggeometric algebra is presented here using a four-element array composed oftraditional vector and scalar products. This leads to an explicit 4 × 4matrix representation which contains key requirements for three-dimensionalgeometric algebra. The work can be extended to include Maxwell’s equationswhere it is found that curl and divergence appear naturally together.However, to obtain an explicit representation of space–time algebrawith the correct behaviour under Lorentz transformations, an 8 × 8matrix representation has to be formed. This leads to a Dirac representation ofMaxwell’s equations showing that space–time algebra has hidden within itsformalism the symmetry of ‘parity, charge conjugation and time reversal’.