GEOMETRIC ALGEBRA

In mathematical physics, a 'geometric algebra' is a multilinear algebra described technically as a Clifford algebra over a real vector space equipped with a non-degenerate quadratic form. Informally, a geometric algebra is a Clifford algebra that includes a ''geometric product''. This allows the theory and properties of the algebra to be built up in an intuitive, geometric way. The term is also used in a more general sense to describe the study and application of these algebras: so 'Geometric algebra' is the study of ''geometric algebras''.
Geometric algebra is useful in physics problems that involve rotations, phases or imaginary numbers. Proponents of geometric algebra argue it provides a more compact and intuitive description of classical and quantum mechanics, electromagnetic theory and relativity. Current applications of geometric algebra include computer vision, biomechanics and robotics, and spaceflight dynamics.

Contents

The geometric product

Inverting a vector

The contraction rule

Inner and outer product

Applications of geometric algebra

History

See also

Notes

References

Further reading

External links

Research groups

Online reading

The geometric product

A geometric algebra $mathcal\{G\}\_n(mathcal\{V\}\_n)$ is an algebra constructed over a vector space $mathcal\; V\_n$ in which a ''geometric product'' is defined. The elements of geometric algebra are multivectors. The original vector space $mathcal\; V$ is constructed over the real numbers as scalars. From now on, a ''vector'' is something in $mathcal\; V$ itself. Vectors will be represented by boldface, small case letters (e.g. $mathbf\; a$), and multivectors by boldface, upper case letters (e.g. $mathbf\{A\}$).
The geometric product has the following properties, for all multivectors $mathbf\{A\},\; mathbf\{B\},\; mathbf\{C\}$:
# Closure
# Distributivity over the addition of multivectors:
#
★ $mathbf\{A\}(mathbf\{B\}\; +\; mathbf\{C\})\; =\; mathbf\{A\}mathbf\{B\}\; +\; mathbf\{A\}mathbf\{C\}$
#
★ $(mathbf\{A\}\; +\; mathbf\{B\})mathbf\{C\}\; =\; mathbf\{A\}mathbf\{C\}\; +\; mathbf\{B\}mathbf\{C\}$
# Associativity
# Unit (scalar) element:
#
★ $1\; ,\; mathbf\; A\; =\; mathbf\; A$
# Tensor contraction: for any "vector" (a grade-one element) $mathbf\{a\},\; mathbf\{a\}^2$ is a scalar (real number)
# Commutativity of the product by a scalar:
#
★ $lambda\; mathbf\; A\; =\; mathbf\; A\; lambda$
Properties (1) and (2) are among those needed for an algebra over a field. (3) and (4) mean that a geometric algebra is 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
:$mathbf\; a\; ,\; mathbf\; b\; =\; mathbf\; a\; cdot\; mathbf\; b\; +\; mathbf\; a\; wedge\; mathbf\; b$
The definition and the associativity of geometric product entails the concept of the inverse of a vector (or division by vector). Thus, one can easily set and solve vector algebra equations that otherwise would be cumbersome to handle. In addition, one gains a geometric meaning that would be difficult to retrieve, for instance, by using matrices. Although not all the elements of the algebra are invertible, the inversion concept can be extended to multivectors. Geometric algebra allows one to deal with subspaces directly, and manipulate them too. Furthermore, geometric algebra is a coordinate-free formalism.
Geometric objects like $mathbf\; a\; wedge\; mathbf\; b$ are called ''bivectors''. A bivector can be pictured as a plane segment (a parallelogram, a circle etc.) endowed with orientation. One bivector represents all planar segments with the same magnitude ''and'' direction, no matter where they are in the space that contains them. However, once either the vector $mathbf\; a$ or $mathbf\; b$ is meant to depart from some preferred point (e.g. in problems of Physics), the oriented plane $mathbf\; B\; =\; mathbf\; a\; wedge\; mathbf\; b$ is determined unambiguously.
The outer product (the exterior product, or the wedge product) $wedge$ is defined such that the graded algebra (exterior algebra of Hermann Grassmann) $wedge^nmathcal\{V\}\_n$ 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 $mathcal\; V$. Multivectors are represented here by boldface caps. Note that scalars and vectors become special cases of multivectors ("0-vectors" and "1-vectors", respectively).

Inverting a vector

As a meaningful result one can consider a fixed non-zero vector $mathbf\; v$, from a point chosen as the origin, in the usual 3-D Euclidean space, $mathbb\{R\}^3$. The set of all vectors $mathbf\; x$ such that $mathbf\; x\; wedge\; mathbf\; v\; =\; mathbf\; B$, $mathbf\; B$ denoting a given bivector containing $mathbf\; v$, determines a line $l$ parallel to $mathbf\; v$. Since $mathbf\; B$ is a ''directed'' area, $l$ is uniquely determined with respect to the chosen origin. The set of all vectors $mathbf\; x$ such that $mathbf\; x\; cdot\; mathbf\; v\; =\; s$, $s$ denoting a given (real) scalar, determines a plane P orthogonal to $mathbf\; v$. Again, P is uniquely determined with respect to the chosen origin. The two information pieces, $mathbf\; B$ and $s$, can be set independently of one another. Now, what is (if any) the vector $mathbf\; x$ that satisfies the system {$mathbf\; x\; wedge\; mathbf\; v\; =\; mathbf\; B$, $mathbf\; x\; cdot\; mathbf\; v\; =\; s$}? Geometrically, the answer is plain: it is the vector that departs from the origin and arrives at the intersection of $l$ and P. By geometric algebra, even the algebraic answer is simple:
:$mathbf\; x\; mathbf\; v\; =\; s\; +\; mathbf\; B\; =>\; mathbf\; x\; =\; (s\; +\; mathbf\; B\; )/\; mathbf\; v\; =\; (s\; +\; mathbf\; B\; )\; mathbf\; v$^-1,
where the inverse of a non-zero vector is expressed by
:$mathbf\; z$^-1 $=\; mathbf\; z\; /(mathbf\; z\; cdot\; mathbf\; z\; )$.
Note that the division by a vector transforms the multivector $s\; +\; mathbf\; B$ into the sum of two vectors. Namely, $s\; mathbf\; v$^-1 is the projection of $mathbf\; x$ on $mathbf\; v$, and $mathbf\; B\; mathbf\; v$^-1 is the rejection of $mathbf\; x$ from $mathbf\; v$ (i.e. the component of $mathbf\; x$ orthogonal to $mathbf\; v$). Note also that the structure of the solution does not depend on the chosen origin.

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 $langle\; x,\; x$
angle ge 0).
The contraction rule can be put in the form:
:$Q(mathbf\; a)\; =\; mathbf\; a^2\; =\; epsilon\_a\; \{Vert\; mathbf\; a\; Vert\}^2$
where $Vert\; mathbf\; a\; Vert$ is the modulus of vector 'a', and $epsilon\_a=0,\; ,\; pm1$ is called the ''signature'' of vector 'a'. This is especially useful in the construction of a Minkowski space (the spacetime of special relativity) through $mathbb\{R\}\_\{1,3\}$. 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 $mathbb\{R\}\_\{3,1\}$ instead).

Inner and outer product

The usual dot product and cross product of traditional vector algebra (on $mathbb\{R\}^3$) find their places in geometric algebra $mathcal\{G\}\_3$ as the inner product
:
(which is symmetric) and the outer product
:
with
:$mathbf\{a\}\; imesmathbf\{b\}\; =\; -i(mathbf\{a\}wedgemathbf\{b\})$
(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 $i$ here is the unit pseudoscalar of Euclidean 3-space, which establishes a duality between the vectors and the bivectors, and is named so because of the expected property $i^2\; =\; -1$.
While the 'cross' product can only be defined in a three-dimensional space, the 'inner' and 'outer' products can be generalized to any dimensional $mathcal\; G\_\{p,q,r\}$.
Let $mathbf\{a\},,\; mathbf\{A\}\_\{langle\; k$
angle} be a vector and a homogeneous multivector of grade ''k'', respectively. Their inner product is then
:$mathbf\; a\; cdot\; mathbf\; A\_\{langle\; k$
angle} = {1 over 2} , left ( mathbf a , mathbf A_{langle k
angle} + (-1)^{k+1} , mathbf{A}_{langle k
angle} , mathbf{a}
ight ) = (-1)^{k+1} mathbf A_{langle k
angle} cdot mathbf{a}
and the outer product is
:$mathbf\; a\; wedge\; mathbf\; A\_\{langle\; k$
angle} = {1 over 2} , left ( mathbf a , mathbf A_{langle k
angle} - (-1)^{k+1} , mathbf{A}_{langle k
angle} , mathbf{a}
ight ) = (-1)^{k} mathbf A_{langle k
angle} wedge mathbf{a}

Applications of geometric algebra

A useful example is $mathbb\{R\}\_\{3,\; 1\}$, and to generate $mathcal\{G\}\_\{3,\; 1\}$, an instance of geometric algebra sometimes called spacetime algebra.^[1] The electromagnetic field tensor, in this context, becomes just a bivector $mathbf\{E\}\; +\; imathbf\{B\}$ 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 as rotation in Euclidean space, where 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

The concinnity of geometry and algebra dates as far back at least to Euclid's ''Elements'' in the 3rd century B.C.^[2] It was not, however, until 1844 that algebra would be used in a ''systematic way'' to describe the geometrical properties and transformations of a space. In that year, Hermann Grassmann introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the propositional calculus) which encoded all of the geometrical information of a space. Grassmann's algebraic system could be applied to a number of different kinds of spaces: the chief among them being Euclidean space, affine space, and projective space. Following Grassmann, in 1878 William Kingdon Clifford examined Grassmann's algebraic system alongside the quaternions of William Rowan Hamilton. From his point of view, the quaternions described certain ''transformations'' (which he called ''rotors''), whereas Grassmann's algebra described certain ''properties'' (or ''Strecken'' such as length, area, and volume). His contribution was to define a new product — the 'geometric product' — on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently Rudolf Lipschitz in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in ''n'' dimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.
Nevertheless, another revolutionary development of the 19th-century would completely overshadow the geometric algebras: that of vector analysis, developed independently by Josiah Willard Gibbs and Oliver Heaviside. Vector analysis was motivated by James Clerk Maxwell's studies of electromagnetism, and specifically the need to express and manipulate conveniently certain differential equations. Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice. Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of abstract algebraists such as Hermann Weyl and Claude Chevalley.
The ''geometrical'' approach to geometric algebras has seen a number of 20th-century revivals. In mathematics, Emil Artin's ''Geometric Algebra'' discusses the algebra associated with each of a number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry. In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism.^[3] In computer graphics, they have been revived in order to represent efficiently rotations (and other transformations) on computer hardware^[4].

See also

★ Clifford algebra

★ Algebra of physical space

★ Spinor

★ Quaternion

★ Algebraic geometry

Notes

1. Cf. Hestenes (1966).
2. Euclid, Books II and VI.
3. Hestenes, ''et al'' (1984).
4. Dorst, ''et al'' (2007).

References

★ Geometric Algebra, , Emil, Artin, Interscience Publishers, 1957,

★ Applications of Grassmann's Extensive Algebra, Clifford, W., , , American Journal of Mathematics, 1878

★ Leo Dorst, Daniel Fontijne, Stephen Mann, "Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry" (The Morgan Kaufmann Series in Computer Graphics), Morgan Kaufmann (April 19, 2007), ISBN-10: 0123694655, ISBN-13: 978-0123694652.

★ Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik, , Hermann, Grassmann, , 1844, (The Linear Extension Theory - A new Branch of Mathematics)

★ Space-time Algebra, , David, Hestenes, Gordon and Breach, 1966,

★ David Hestenes and Sobczyk, G., 1984. ''Clifford Algebra to Geometric Calculus'', Springer Verlag ISBN 90-277-1673-0

Further reading

★ Baylis, W. E., ed., 1996. ''Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering''. Boston: Birkhäuser.

★ Baylis, W. E., 2002. ''Electrodynamics: A Modern Geometric Approach'', 2nd ed. Birkhäuser. ISBN 0-8176-4025-8

★ Nicolas Bourbaki, 1980. ''Eléments de Mathématique. Algèbre''. Chpt. 9, "Algèbres de Clifford". Paris: Hermann.

★ Chris Doran and Anthony Lasenby, 2003. ''Geometric Algebra for Physicists''. Cambridge Univ. Press.

★ Hestenes, D., 1999. ''New Foundations for Classical Mechanics'', 2nd ed. Springer Verlag ISBN 0-7923-5302-1

★ Lasenby, J., Lasenby, A. N., and Doran, C. J. L., 2000, "A Unified Mathematical Language for Physics and Engineering in the 21st Century," ''Philosophical Transactions of the Royal Society of London A 358'': 1-18.

External links

Research groups

★ Geometric Calculus International. Links to Research groups, Software, and Conferences, worldwide.

★ Cambridge Geometric Algebra group. Full-text online publications, and other material.

★ University of Amsterdam group

★ Geometric Calculus research & development (University of Arizona).

★ GA-Net blog and newsletter archive. Geometric Algebra/Clifford Algebra development news.

Online reading

★ Imaginary Numbers are not Real - the Geometric Algebra of Spacetime. Introduction (Cambridge GA group).

★ Physical Applications of Geometric Algebra. Final-year undergraduate course (Cambridge GA group; see also 1999 version).

★ Maths for (Games) Programmers: 5 - Multivector methods. Comprehensive introduction and reference for programmers, from Ian Bell.

★ A Geometric Algebra Primer, especially for computer scientists.

★

★ Exploring Hyperspace with the Geometric Product - Highlights applications to higher dimensions and cosmology that includes wormholes.