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A geometric algebra (GA) is a Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The Clifford multiplication that defines the GA as a unital ring is called the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general ''n''-vectors. The addition operation combines these into general multivectors, which are the elements of the ring. This includes, among other possibilities, a well-defined formal sum of a scalar and a vector. Geometric algebra is distinguished from Clifford algebra in general by its restriction to real numbers and its emphasis on its geometric interpretation and physical applications. Specific examples of geometric algebras applied in physics include the algebra of physical space, the spacetime algebra, 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, 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, which is the geometric algebra of the trivial quadratic form. 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 by Hestenes in the 1960s, who recognized its importance to relativistic physics. ==Definition and notation== Given a finite-dimensional real quadratic space with a quadratic form (e.g. the Euclidean or Lorentzian metric) , the geometric algebra for this quadratic space is the Clifford algebra ''C''ℓ(''V'',''g''). The algebra product is called the ''geometric product''. It is standard to denote the geometric product by juxtaposition (i.e., suppressing any explicit multiplication symbol). The above definition of the geometric algebra is abstract, so we summarize the properties of the geometric product by the following set of axioms. The geometric product has the following properties: :, where ''A'', ''B'' and ''C'' are any elements of the algebra (associativity) : and , where ''A'', ''B'' and ''C'' are any elements of the algebra (distributivity) :, where ''a'' is a vector. Note that in the final property above, the square need not be nonnegative if ''g'' is not positive definite. An important property of the geometric product is the existence of elements with multiplicative inverse, also known as units. If for some vector ''a'', then ''a''−1 exists and is equal to . Not every nonzero element of the algebra is necessarily a unit. For example, if ''u'' is a vector in ''V'' such that , the elements are zero divisors and thus have no inverse: . There may also exist nontrivial idempotent elements such as . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Geometric algebra」の詳細全文を読む スポンサード リンク
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