David Hestenes et al.
's geometric algebra
is a radical reinterpretation of seemingly harmless Clifford algebras over the reals (or, stated ironically, return to the original name and interpretation intended by William Clifford
). The key ingredient of this formulation is the (natural) correspondence between geometric entities and the elements of the associative algebra
. Contrary to the claims its proponents make, the "mixing" of quantities of different grades is helpful only by the virtue of the computational advantages offered by associativity (and inverse of vectors), not particularly in visualization or conceptualization. The applications in physics are valuable, and Hestenes' unusual stand in favor of using real numbers
only (expelling complex numbers
) as the underlying field, has given the reformulation a sort of a distinction, given the fact that elements having a negative square are ubiquitous in the algebra.
We take to be the field for a linear space (otherwise known simply as a vector space, but following Hestenes I will reserve the word for the space of first grade elements) of dimention . 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 )
- Distributivity over the addition of multivectors: and
- Contraction for any "vector" (a grade one element) is a scalar (real number)
We call this algebra a geometric algebra
The connection between Clifford algebras and quadratic forms come from the distinctive 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
(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 .
One more useful example to convince yourself is to consider , and to generate , one instance of geometric algebra specifically dubbed spacetime algebra by Hestenes (not without reason!). Electromagnetic field tensor, in this context, becomes just a bivector where the imaginary unit, not surprizingly, is the volume element, giving an example of the geometric reinterpretation of the traditional "tricks", making them meaningful. 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.