Geometric Product

For vectors, the Geometric Product is the sum of a Symmetric Product and an Anti-Symmetric Product.

ab = \dfrac{1}{2}(ab+ba) + \dfrac{1}{2}(ab - ba)

We found another definition that was called the vector form of the geometric product:

ab = a \cdot b + a \wedge b

You can calculate this vector form from the first definition if you have the two definitions shown below:

  • a \cdot b = \dfrac{1}{2}(ab + ba)
  • a \wedge b = \dfrac{1}{2}(ab - ba)

The geometric product is defined by the following rules:

  • Associative: (uv)w=u(vw)
  • Left Distributive: u(v+w)=uv+uw
  • Right Distributive: (v+w)u=vu+wu
  • Euclidean Metric: \hat a^2 = a^2
    • \hat a is the vector and a is the length of the vector
  • Contraction: v^2=\epsilon_v |v|^2

v^2=\epsilon_v |v|^2

\epsilon_v is the signature of v.

  • v is timelike if its signature is positive
  • v is spacelike if its signature is negative
  • The geometric product decomposes into a symmetric inner product.
  • The geometric product decomposes into an antisymmetric outer product.

The choice of c=1 makes it so spacelike intervals and timelike intervals are measured in the same unit.

The value i is a geometrical \sqrt{−1}, but it anticommutes with all spacetime vectors.

Reciprocal Frame:

\gamma_\mu=g_{\mu\nu} \gamma^\nu

\gamma_\mu dot \gamma^nu = \delta_{\mu}\{nu}