Geometry and physics with geometric algebra

The aim of this essay is to provide a brief mathematical account of the geometric algebra formalism and to revisit a few outstanding applications to geometry and physics. The main emphasis is the presentation of geometric algebra as a powerful enrichment of Grassmann’ exterior algebra (of metric linear spaces) by the geometric product (see ♢4), which in our view better connects with Clifford’s seminal ideas on this topic and provides a transparent intrinsic formalism in which to cast concepts, applications and effective computations. Hopefully it will be useful to mathematicians wishing to understand geometric algebra in their own terms and to people with other backgrounds to sharpen their awareness of its mathematical underpinnings.