Geometric algebra

In mathematics, the geometric algebra (GA) of a vector space with a quadratic form (usually the Euclidean metric or the Lorentz metric) is an algebra over a field, the Clifford algebra of a vector space with a quadratic form with its multiplication operation called the geometric product. The algebra elements are called multivectors, which contains both the scalars



F


{\displaystyle F}
and the vector space



V


{\displaystyle V}
.
Clifford's contribution was to define a new product, the geometric product, that unified the Grassmann and Hamilton algebras into a single structure. Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations.
The scalars and vectors have their usual interpretation, and make up distinct subspaces of a GA. Bivectors provide a more natural representation of the pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum, electromagnetic field and the Poynting vector. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of



V


{\displaystyle V}
and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike vector algebra, a GA naturally accommodates any number of dimensions and any quadratic form such as in relativity.
Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics.
The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.

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