I am interested in understanding the physical/geometric meaning of the cross-product between a vector $v$ and a bi-vector $b$.
To help me understand better I have investigated a physical example. Let us take an example using $Cl_{3,1}$, with $v$ and $b$ defined as:
$$ v=t\gamma_0+x\gamma_1+y\gamma_2+z\gamma_3\\ b=E_x\gamma_0\gamma_1+E_y\gamma_0\gamma_2+E_z\gamma_0\gamma_2+B_xi\gamma_0\gamma_1+B_yi\gamma_0\gamma_2+B_zi\gamma_0\gamma_3 $$
Now I will take the geometric product of $u=v+b$:
$$ u^2=v^2+b^2+vb+bv $$
We now find a physical/geometric interpretation for each term:
- $v^2=t^2-x^2-y^2-z^2$, which is the square of the interval of special relativity.
- $b^2=E_x^2+E_y^2+E_z^2-B_x^2-B_y^2-B_z^2+2i(E_xB_x+E_yB_y+E_zB_z)$, which are the invariants of electromagnetism.
- But what about the term $vb+bv$?
After a couple of pages of algebra, I identify the term $vb+bv$ as:
$$ vb+bv=2\gamma_0\gamma_1\gamma_2(yF_x-xF_y)+2\gamma_0\gamma_1\gamma_3(zF_x-F_xz)+2\gamma_0\gamma_2\gamma_3(yF_z-F_zy) $$
where $F_x=E_x+iB_x$, $F_y=E_y+iB_y$, $F_z=E_z+iB_z$.
Now if I flip $\gamma_0\gamma_1\gamma_3$ to $\gamma_0\gamma_3\gamma_1$ then I simply obtain the cross-product between the space components of $v$ and the electromagnetic field.
$$ u^2=v^2+b^2+i v(x,y,z)\times b $$
I was not able to find any references in the physics literature regarding the cross product between position and electromagnetic field. So I am at a loss of what it could mean physically or geometrically. Can someone help me understand the meaning of the cross-product between a vector and a bi-vector?