Geometric algebra

The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which is a superset of both the scalars F {displaystyle F} and the vector space V {displaystyle V} . Mathematically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form. Clifford's contribution was to define a new product, the geometric product, that united 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. D ⋅ A = 0 {displaystyle Dcdot A=0} The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which is a superset of both the scalars F {displaystyle F} and the vector space V {displaystyle V} . Mathematically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form. Clifford's contribution was to define a new product, the geometric product, that united 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 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. Specific examples of geometric algebras applied in physics include the spacetime algebra (or the less common alternative formulation, the 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, 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. There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic, 'full of geometric significance' and equivalent to the universal Clifford algebra.Given a finite-dimensional quadratic space V {displaystyle V} over a field F {displaystyle F} with a symmetric bilinear form (the inner product, e.g. the Euclidean or Lorentzian metric) g : V × V → F {displaystyle g:V imes V ightarrow F} , the geometric algebra for this quadratic space is the Clifford algebra Cl ⁡ ( V , g ) {displaystyle operatorname {Cl} (V,g)} . As usual in this domain, for the remainder of this article, only the real case, F = R {displaystyle F=mathbf {R} } , will be considered. The notation G ( p , q ) {displaystyle {mathcal {G}}(p,q)} (respectively G ( p , q , r ) {displaystyle {mathcal {G}}(p,q,r)} ) will be used to denote a geometric algebra for which the bilinear form g {displaystyle g} has the signature ( p , q ) {displaystyle (p,q)} (respectively ( p , q , r ) {displaystyle (p,q,r)} ). The essential product in the algebra is called the geometric product, and the product in the contained exterior algebra is called the exterior product (frequently called the outer product and less often the wedge product). It is standard to denote these respectively by juxtaposition (i.e., suppressing any explicit multiplication symbol) and the symbol ∧ {displaystyle wedge } . 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, for A , B , C ∈ G ( p , q ) {displaystyle A,B,Cin {mathcal {G}}(p,q)} : The exterior product has the same properties, except that the last property above is replaced by a ∧ a = 0 {displaystyle awedge a=0} for a ∈ V {displaystyle ain V} . Note that in the last property above, the real number g ( a , a ) {displaystyle g(a,a)} need not be nonnegative if g {displaystyle g} is not positive-definite. An important property of the geometric product is the existence of elements having a multiplicative inverse. If a 2 ≠ 0 {displaystyle a^{2} eq 0} for some vector a {displaystyle a} , then a − 1 {displaystyle a^{-1}} exists and is equal to g ( a , a ) − 1 a {displaystyle g(a,a)^{-1}a} . A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if u {displaystyle u} is a vector in V {displaystyle V} such that u 2 = 1 {displaystyle u^{2}=1} , the element 1 2 ( 1 + u ) {displaystyle extstyle {frac {1}{2}}(1+u)} is both a nontrivial idempotent element and a nonzero zero divisor, and thus has no inverse. It is usual to identify 1 ∈ R {displaystyle 1in mathbf {R} } with 1 ∈ G ( p , q ) {displaystyle 1in {mathcal {G}}(p,q)} , with associated natural embeddings R → G ( p , q ) {displaystyle mathbf {R} o {mathcal {G}}(p,q)} and V → G ( p , q ) {displaystyle V o {mathcal {G}}(p,q)} . In this article, this identification is assumed. Throughout, the term vector refers to an element of V {displaystyle V} (and its image under this embedding).