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^{T }A 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.
External links:
All Wikipedia text
is available under the
terms of the GNU Free Documentation License