Geometric algebraDavid 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 Vn with an outer product "∧" (also called wedge product) defined on it, such that a graded algebra ∧Vn is generated. Then, we define a geometric product " " with the following properties, for all multivectors A, B, C in the graded algebra ∧Vn:
- Distributivity over the addition of multivectors: A (B + C) = A B + A C
- 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 Vn, a a is a scalar Q(a)
The contractive rule makes the difference with other associative algebras. In general, Q(x) is a quadratic form
- Q(x) = ∑aij xi xj = xT A x,
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)
- a·b = 1/2 (a b + b a)
- a∧b = 1/2 (a b - b a)
- a b = a∧b + a·b