Geometric algebra
In
mathematics,
geometric algebra is a term applied to the theory of Clifford algebras and related theory, following a book of the same title by
Emil Artin. This term has also had recent currency for treatments of the same area in the
physics literature.
David Hestenes et al.'s
geometric algebra is a reinterpretation of Clifford algebras over the reals (said to be a return to the original name and interpretation intended by
William Clifford). The
real numbers are used as scalars in a
vector space V. From now on, a
vector is something in
V itself.
The
outer product (the
exterior product, or the
wedge product) is defined such that the
graded algebra (
exterior algebra of
Hermann Grassmann) of multivectors is generated. The
geometric algebra is the
algebra generated by the
geometric product (which is to be thought of as more fundamental) with (for all multivectors )
- Associativity\n# Distributivity over the addition of multivectors: and \n# Contraction for any "vector" (a grade-one element) is a scalar (real number)
We call this algebra a
geometric algebra . The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the
associative algebra.
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 ).
The usual
dot product and
cross product of traditional vector algebra (on ) find their places in geometric algebra as the inner product
-
(which is symmetric) and the outer product
-
with
-
(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 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 .
A useful example is , and to generate , an instance of geometric algebra called
spacetime algebra by Hestenes. The electromagnetic field tensor, in this context, becomes just a bivector 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.
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