Geometric Algebra

A geometric algebra is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The spacetime algebra and the conformal geometric algebra are specific examples of such geometric algebras. The term is also used as a collective term for the approach to classical, computational and relativistic geometry that makes heavy use of such algebras.

In 1878, the year before his death, Clifford expanded upon Grassmann's Ausdehnungslehre to form what are now usually called Clifford algebras in his honor although Clifford himself chose to call them "geometric algebras" and this term was repopularized by Hestenes in the 1960s. Geometric algebra (GA) finds application in physics, in graphics and in robotics. A key feature of GA is its emphasis on geometric interpretations of certain elements of the algebra as geometric entities. Via this interpretation, geometric operations are realized as algebraic operations in the algebra.

Proponents argue that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. Others claim that in some cases the geometric algebra approach is able to sidestep a "proliferation of manifolds" that arises during the standard application of differential geometry.

The associated geometric calculus is a generalization of vector calculus that is an alternative to the use of differential forms and Hodge duality.