In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not by all elements) and addition of objects of different dimensions.
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"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternion algebra. Adding the dual of the Grassmann exterior product allows the use of the Grassmann–Cayley algebra. In the late 1990s, plane-based geometric algebra and conformal geometric algebra (CGA) respectively provided a framework for euclidean geometry and classical geometries. In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations. 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 David Hestenes, who advocated its importance to relativistic physics.
The scalars and vectors have their usual interpretation and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities of 3D vector calculus that are derived as a cross product, such as oriented area, oriented angle of rotation, torque, angular momentum and the magnetic field. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra 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). 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.
Definition and notation 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 vector space over a field with a symmetric bilinear form (the inner product, e.g., the Euclidean or Lorentzian metric) , the geometric algebra of the quadratic space is the Clifford algebra , an element of which is called a multivector. The Clifford algebra is commonly defined as a quotient of the tensor algebra of , though this definition is abstract, so the following definition is presented without requiring abstract algebra.
Definition A unital associative algebra with a nondegenerate symmetric bilinear form is the Clifford algebra of the quadratic space if it contains and as distinct subspaces for generates as an algebra is not generated by any proper subspace of .
To cover degenerate symmetric bilinear forms, the last condition must be modified. It can be shown that these conditions uniquely characterize the geometric product.
For the remainder of this article, only the real case, , will be considered. The notation (respectively ) will be used to denote a geometric algebra for which the bilinear form has the signature (respectively ).
The product in the geometric algebra is called the geometric product, and the product in the contained exterior algebra is called the exterior product (frequently called the wedge product or the outer product). It is standard to denote these respectively by juxtaposition (i.e., suppressing any explicit multiplication symbol) and the symbol . Their connection to the inner product will be elaborated on below.
The above definition of the geometric algebra is still somewhat abstract, so we summarize the properties of the geometric product here. For multivectors : (closure) , where is the identity element (existence of an identity element) (associativity) and (distributivity) for .
The exterior product has the same properties, except that the last property above is replaced by for .
Note that in the last property above, the real number need not be nonnegative if is not positive-definite. An important property of the geometric product is the existence of elements that have a multiplicative inverse. For a vector , if then exists and is equal to . A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if is a vector in such that , the element is both a nontrivial idempotent element and a nonzero zero divisor, and thus has no inverse.
It is usual to identify and with their images under the natural embeddings and . In this article, this identification is assumed. Throughout, the terms scalar and vector refer to elements of and respectively (and of their images under this embedding).
Geometric product
For vectors and , we may write the geometric product of any two vectors and as the sum of a symmetric product and an antisymmetric product:
Thus we can define the inner product of vectors as
so that the symmetric product can be written as
Conversely, is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the contained exterior algebra:
Then by addition: the restricted or vector form of the geometric product.
The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, and are parallel if their geometric product is equal to their inner product, whereas and are perpendicular if their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the dot product of standard vector algebra. The exterior product of two vectors can be identified with the signed area enclosed by a parallelogram the sides of which are the vectors. The cross product of two vectors in dimensions with positive-definite quadratic form is closely related to their exterior product.
Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fully degenerate, the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.
The exterior product is naturally extended as an associative bilinear binary operator between any two elements of the algebra, satisfying the identities
where the sum is over all permutations of the indices, with the sign of the permutation, and are vectors (not general elements of the algebra). Since every element of the algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the algebra. It follows from the definition that the exterior product forms an alternating algebra.
The equivalent structure equation for Clifford algebra is
where is the Pfaffian of and provides combinations, , of indices divided into and parts and is the parity of the combination.
The Pfaffian provides a metric for the exterior algebra and, as pointed out by Claude Chevalley, Clifford algebra reduces to the exterior algebra with a zero quadratic form. The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra from simplices. This derivation provides a better connection between Pascal's triangle and simplices because it provides an interpretation of the first column of ones.
Blades, grades, and basis A multivector that is the exterior product of linearly independent vectors is called a blade, and is said to be of grade . A multivector that is the sum of blades of grade is called a (homogeneous) multivector of grade . From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.
Consider a set of linearly independent vectors spanning an -dimensional subspace of the vector space. With these, we can define a real symmetric matrix (in the same way as a Gramian matrix)
By the spectral theorem, can be diagonalized to diagonal matrix by an orthogonal matrix via
Define a new set of vectors , known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:
Since orthogonal transformations preserve inner products, it follows that and thus the are perpendicular. In other words, the geometric product of two distinct vectors is completely specified by their exterior product, or more generally
Therefore, every blade of grade can be written as the exterior product of vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a block matrix that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace are normalized according to
then these normalized vectors must square to or . By Sylvester's law of inertia, the total number of and the total number of s along the diagonal matrix is invariant. By extension, the total number of these vectors that square to and the total number that square to is invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra . For example, models three-dimensional Euclidean space, relativistic spacetime and a conformal geometric algebra of a three-dimensional space.
The set of all possible products of orthogonal basis vectors with indices in increasing order, including as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem). For example, the following is a basis for the geometric algebra :
A basis formed this way is called a standard basis for the geometric algebra, and any other orthogonal basis for will produce another standard basis. Each standard basis consists of elements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements. If the standard basis elements are with being an index set, then the geometric product of any two multivectors is