User:Maschen/Geometric algebra

In pure and applied mathematics, the field of geometric algebra is the application of Clifford algebra to geometric problems in physics and computation. The main algebraic structures of study are called geometric algebras which are Clifford algebras over the real numbers. Results from geometric algebra are applied in physics, graphics, robotics, and computational science, among other fields.

Geometric algebra primarily exploits the algebraic structure of a Clifford algebra to represent geometric objects and operations. Because of the correspondence (isomorphism) and the versatile algebraic structure of CA, this is a particularly powerful and convenient algebraic formalism of geometric concepts. Concepts such as vectors, surface elements, scaling, rotation, displacement and more have particularly simple representations in the suitably selected GA.

Technically, a geometric algebra for an n dimensional real vector space V is an algebra containing V whose multiplication operation encodes a bilinear form on V. Intuitively, the bilinear form determines a geometry for V, and the geometric algebra carries geometric information about V in its algebraic operations: for instance, things like distance, angles and length. A basis of n elements for V can be used to produce a basis of 2ⁿ elements for the algebra. The bilinear form dictates the multiplication of basis elements, and hence influences the character of the whole algebra. Many geometric operations in the vector space V (such as rotation, projection, reflection, projection etc.) can be translated into simple operations (addition, multiplication, coordinate projection) of the algebra.

A wide range of spaces can be studied using geometric algebra, including Euclidean and non-Euclidean spaces of any dimension. Two non-Euclidean examples include that of the spacetime algebra for Minkowski spacetime and the universal geometric algebra. Although the algebra is created using real number scalars, the complex numbers and quaternions can still appear as other elements of the geometric algebra. For example, there can exist elements that have square -1, as with imaginary numbers. In fact, the real numbers and the complex numbers can both be considered as special cases of geometric algebras. Other physically important structures appear in geometric algebras: for example, the gamma matrices introduced by the Dirac equation appear as elements in a certain geometric algebra.

In the late 19th century, William Kingdon Clifford originally called these types of algebras "geometric algebras." Later, they were called "Clifford algebras" in his honor, but now that term spans algebras over other fields than the real numbers. In the 1960s, David Hestenes repopularized the term "geometric algebra" for the real Clifford algebras with applications to geometry.