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III. Geometric Algebra
The
tangent vector spaces ${T_x}\mathcal{M} = {V_4}({\mathbf{x}})$ of the manifold
${\mathcal{M}_4} = \{ {\mathbf{x}}\} $ can be extended to geometric tangent spaces
$G{T_x}\mathcal{M}$ at each point. The generalization involves transforming the tangent
space ${T_x}\mathcal{M}$ from a vector space into a linear space structured by the
Clifford algebra $\text{Cl}(1,3)$. 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 $\text{Cl(1,3)}$ 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 $G{T_x}\mathcal{M}$.
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.