Geometric algebra

David Hestenes' geometric algebra is a mathematical formalism that mixes quantities of different dimensionalities in a single value. This leads to apparently more natural treatments of several areas of physics without the use of complex numbers.

We start from a vectorial space V_n with an outer product "∧" (also called wedge product) defined on it, such that a graded algebra ∧V_n is generated. Then, we define a geometric product " " with the following properties, for all multivectors A, B, C in the graded algebra ∧V_n:

Closure
Distributivity over the addition of multivectors: A (B + C) = A B + A C
Associativity
Unit element: there is a scalar 1 such that 1 A = A
Commutativity with the product with a scalar a: a A = A a
Contractive rule: for any vector a in V_n, a a is a scalar Q(a)

Properties (1) and (2) converts the vector space V_n into an algebra. From (3) and (4) the algebra becomes an associative unitary algebra. We call this algebra a geometric algebra G_n.

The contractive rule makes the difference with other associative algebras. In general, Q(x) is a quadratic form

Q(x) = ∑a_ij x_i x_j = x^TA x,

where x is a vector, x^T its transposed vector, and A is a matrix. In this way, the contraction rule takes the form of a inner product. Usually the contraction rule is chosen so that Q(x) = ε ||x||², with ε = +1, -1, 0. ε is called the signature of the vector x. Given a vector space of dimension n, we can define a vector base such that p of the vector in the base have positive signature, q have negative signature and r have null signature (obviously, p + q + r = n). We call (p, q, r) the signature of the vector space, and write V_p,q,r, and by extension, of the geometric algebra generated from this.

When a metric is defined, the geometric algebra is called a Clifford algebra, otherwise is called exterior or Grassmann algebra.

The geometric product is not commutative, but the following epression is, for any vector a, b:

a b + b a = (a + b)(a + b) - a a - b b = Q(a + b) - Q(a) - Q(b)

It is also a scalar, which allow us to redefine the inner product "·" (also dot product or scalar product) in terms of the geometric product:

a·b = 1/2 (a b + b a)

This leaves the asymmetric part of the geometric product as

a∧b = 1/2 (a b'\ - b a')

As a consequence, the geometric product can be redefined as

a b = a∧b + a·b

Note that some authors define as the difference of outer and inner product instead.

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