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III. Geometric Algebra
The tangent vector spaces ${T_x}\mathcal{M}_4 = {V_4}({\mathbf{x}})$
of the manifold ${\mathcal{M}_4} = \{ {\mathbf{x}}\} $ can be
extended to geometric tangent spaces $G{T_x}\mathcal{M}_4$
at each point. The generalization involves transforming the tangent
space ${T_x}\mathcal{M}_4$ from a vector space into a linear space
structured by the Clifford algebra $\mathcal{Cl}(1,3)$. This geometric
algebra (GA) encompasses the algebra of differential forms
(exterior algebra) as a sub-algebra under the wedge product.
Specifically, geometric algebra constitutes a linear space built on
an inner product vector space, augmented by the operation of wedge
multiplication. The construction endows the triad base vectors
$\left\{ {{{\mathbf{e}}_a}({\mathbf{x}});a = 0,1,2,3} \right\}$ with
the inner product rule
\[ {{\mathbf{e}}_a} \cdot
{{\mathbf{e}}_b}: = \frac{1}{2}({{\mathbf{e}}_a}{{\mathbf{e}}_b} +
{{\mathbf{e}}_b}{{\mathbf{e}}_a}) = \frac{1}{2}\{
{{\mathbf{e}}_a}{\text{,}}{{\mathbf{e}}_b}\} = {\eta _{ab}} \]
This means that the Clifford algebra $\mathcal{Cl}(1,3)$ of the
base vectors is isomorphic to the algebra of Dirac matrices.
However, the triad base vectors in (3.1) are not matrices, but are
to be regarded as four separate vectors with a clear geometric
meaning in the tangent space $G{T_x}\mathcal{M}_4
:={\mathcal{G}_4}$.
In addition to (3.1), the wedge (outer) product is defined
by:
\[ {{\mathbf{e}}_a} \wedge
{{\mathbf{e}}_b}: = \frac{1}{2}({{\mathbf{e}}_a}{{\mathbf{e}}_b} -
{{\mathbf{e}}_b}{{\mathbf{e}}_a}) \]
The dot (inner) product of vectors is a scalar, whereas the
wedge (outer) product of two vectors is a new type of object called
a 2-vector or bi-vector.
Axioms Geometric Algebra
- The geometric product of vectors is associative:
\[{\mathbf{a}}({\mathbf{bc}}) = ({\mathbf{ab}}){\mathbf{c}} =
{\mathbf{abc}}\]
- The geometric product of vectors is distributive over addition:
\[{\mathbf{a(b}} + {\mathbf{c}}) = {\mathbf{ab}} +
{\mathbf{ac}}\]
- The square of any vector is a real scalar:
${{\mathbf{a}}^2} \in \Re $