Geometric algebra

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Geometric algebra is a Clifford algebra given a geometric interpretation which makes it useful in an exceptionally wide range of physics problems, particularly those that involve rotations, phases or imaginary numbers. Proponents of geometric algebra say that it more compactly and intuitively describes classical mechanics, quantum mechanics, electromagnetic theory and relativity than standard methods do.

In mathematics, a geometric algebra <math>\mathcal{G}_n(\mathcal{V}_n)<math> is an algebra constructed over a vector space <math>\mathcal V_n<math> in which a geometric product is defined. For all multivectors (the elements of the algebra) <math>\mathbf{A}, \mathbf{B}, \mathbf{C}<math>, the geometric product has the following properties:

  1. Closure
  2. (left and right) Distributivity over the addition of multivectors:
    • <math>\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}<math>
    • <math>(\mathbf{A} + \mathbf{B})\mathbf{C} = \mathbf{A}\mathbf{C} + \mathbf{B}\mathbf{C}<math>
  3. Associativity
  4. Unit (scalar) element:
    • <math> 1 \, \mathbf A = \mathbf A <math>
  5. Tensor contraction: for any "vector" (a grade-one element) <math>\mathbf{a}, \mathbf{a}^2<math> is a scalar (real number)
  6. Commutativity of the product by an scalar:
    • <math> \lambda \mathbf A = \mathbf A \lambda <math>

Note that the first two properties are needed to be an algebra. Next two make it an associative, unital algebra.

The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebra. This comes from the fact that the geometric product is defined in terms of the dot product and the wedge product of vectors as

<math> \mathbf a \, \mathbf b = \mathbf a \cdot \mathbf b + \mathbf a \wedge \mathbf b <math>

The original vector space <math>\mathcal V<math> is constructed over the real numbers as scalars. From now on, a vector is something in <math>\mathcal V<math> itself. Vectors will be represented by boldface, small case letters.

The outer product (the exterior product, or the wedge product) <math>\wedge<math> is defined such that the graded algebra (exterior algebra of Hermann Grassmann) <math>\wedge^n\mathcal{V}_n<math> of multivectors is generated. Multivectors are thus the direct sum of grade k elements (k-vectors), where k ranges from 0 (scalars) to n, the dimension of the original vector space <math>\mathcal V<math>. Multivectors are represented here by boldface caps. Note that scalars and vectors become special cases of multivectors ("0-vectors" and "1-vectors", respectively).

Contents

The contraction rule

The connection between Clifford algebras and quadratic forms come from the contraction property. This rule also gives the space a metric defined by the naturally derived inner product. It is to be noted that in geometric algebra in all its generality there is no restriction whatsoever on the value of the scalar, it can very well be negative, even zero (in that case, the possibility of an inner product is ruled out if you require <math>\langle x, x \rangle \ge 0<math>).

The contraction rule can be put in the form:

<math>Q(\mathbf a) = \mathbf a^2 = \epsilon_a {\Vert \mathbf a \Vert}^2<math>

where <math>\Vert \mathbf a \Vert<math> is the modulus of vector a, and <math>\epsilon_a=0, \, \pm1<math> is called the signature of vector a. This is especially useful in the construction of a Minkowski space (the relativity spacetime) through <math> \mathbb{R}_{1,3}<math>. In that context, null-vectors are called "lightlike vectors", vectors with negative signature are called "spacelike vectors" and vectors with positive signature are called "timelike vectors" (these last two denominations are exchanged when using <math>\mathbb{R}_{3,1}<math> instead).

Inner and outer product

The usual dot product and cross product of traditional vector algebra (on <math>\mathbb{R}^3<math>) find their places in geometric algebra <math>\mathcal{G}_3<math> as the inner product

<math>\mathbf{a}\cdot\mathbf{b} = \frac{1}{2}(\mathbf{a}\mathbf{b} + \mathbf{b}\mathbf{a})<math>

(which is symmetric) and the outer product

<math>\mathbf{a}\wedge\mathbf{b} = \frac{1}{2}(\mathbf{a}\mathbf{b} - \mathbf{b}\mathbf{a})<math>

with

<math>\mathbf{a}\times\mathbf{b} = -i(\mathbf{a}\wedge\mathbf{b})<math>

(which is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the mere distinction between vectors and bivectors (elements of grade two). The <math>i<math> here is the unit pseudoscalar of Euclidean 3-space, with establishes a duality between the vectors and the bivectors, and is named so because of the expected property <math>i^2 = -1<math>.

The inner and outer product can be generalized to any dimensional <math>\mathcal G_{p,q,r}<math>; however the cross product is only defined in a 3-dimension space.

Let <math>\mathbf{a},\, \mathbf{A}_{\langle k \rangle}<math> be a vector and a homogenous multivector of grade k, respectively. Their inner product is then

<math> \mathbf a \cdot \mathbf A_{\langle k \rangle} = {1 \over 2} \, \left ( \mathbf a \, \mathbf A_{\langle k \rangle} + (-1)^{k+1} \, \mathbf{A}_{\langle k \rangle} \, \mathbf{a} \right ) = (-1)^{k+1} \mathbf A_{\langle k \rangle} \cdot \mathbf{a}<math>

and the outer product is

<math> \mathbf a \wedge \mathbf A_{\langle k \rangle} = {1 \over 2} \, \left ( \mathbf a \, \mathbf A_{\langle k \rangle} - (-1)^{k+1} \, \mathbf{A}_{\langle k \rangle} \, \mathbf{a} \right ) = (-1)^{k} \mathbf A_{\langle k \rangle} \wedge \mathbf{a}<math>

Applications of geometric algebra

A useful example is <math>\mathbb{R}_{3, 1}<math>, and to generate <math>\mathcal{G}_{3, 1}<math>, an instance of geometric algebra called spacetime algebra by Hestenes. The electromagnetic field tensor, in this context, becomes just a bivector <math>\mathbf{E} + i\mathbf{B}<math> where the imaginary unit is the volume element, giving an example of the geometric reinterpretation of the traditional "tricks".

Boosts in this Lorenzian metric space have the same expression <math>e^{\mathbf{\beta}}<math> as rotation in Euclidean space, where <math>\mathbf{\beta}<math> is of course the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.

History

David Hestenes et al.'s geometric algebra [H1999] is a reinterpretation of Clifford algebras over the reals (said to be a return to the original name and interpretation intended by William Clifford). A book of the same title by Emil Artin covers the algebra associated with many different "geometries," including affine, projective, symplectic, and orthogonal.

References

  • [H1999] David Hestenes: New Foundations for Classical Mechanics (Second Edition). ISBN 0792355148, Kluver Academic Publishers (1999)

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