This question is in the context of geometric algebra. I have read in several places that the highest order wedge product behaves as the imaginary unit $i$ and I would just like to see this proven somewhere. I am confused because if you are in 4D, it seems like this is not true (shown below), but does work for lower ordered products of the same space (also shown below).
Suppose we have a 4D space spanned by the orthonormal vectors $e_i$ where $i = 0,1,2,3$. The geometric product between any two of them, i.e. $e_ie_j$ where $i\ne j$ is given by:
$$e_ie_j = e_i\cdot e_j + e_i \wedge e_j = e_i \wedge e_j$$
I believe this also extends to more than 2 vectors, namely: $e_1e_2e_3 = e_1\wedge e_2\wedge e_3$ and so on. Since the wedge product is antisymmetric, swapping the vectors picks up a minus sign, i.e.:
$$e_ie_j = -e_je_i$$
If we start with squaring a bivector, using the rules above, we get:
$$(e_1e_2)^2 = e_1e_2e_1e_2 = -e_1e_2e_2e_1 = -e_1e_1 = -1$$
So right away, the statement of "the highest order wedge product behaves as $i$" is not the whole story because you can have lower ordered products also behaving like $i$. Let's move to 3 elements:
$$(e_1e_2e_3)^2 = e_1e_2e_3e_1e_2e_3 = -e_1e_1e_3e_2e_2e_3 = -e_3e_2e_2e_3 = -e_3e_3 = -1$$
Okay, that also squares to -1. If we now go to 4 we get:
$$(e_1e_2e_3e_4)^2 = e_1e_2e_3e_4e_1e_2e_3e_4 = -e_1e_1e_3e_4e_2e_2e_3e_4 = -e_3e_4e_3e_4 = e_3e_4e_4e_3 = +1$$
This was probably stated with the implicit assumption that 4D referred to an underlying space-time metric. In general, we have \begin{equation*} \begin{aligned} \left( e_1 e_2 e_3 e_4 \right)^2 &= e_1 e_2 e_3 e_4 e_1 e_2 e_3 e_4 \\ &= -e_1^2 e_2 e_3 e_4 e_2 e_3 e_4 \\ &= -e_1^2 e_2^2 e_3 e_4 e_3 e_4 \\ &= +e_1^2 e_2^2 e_3^2 e_4^2. \end{aligned} \end{equation*} However, for special relativistic physics, we have two choices of metric, one is: \begin{equation*} e_1^2 = e_2^2 = e_3^2 = -1 = -e_4^2, \end{equation*} or \begin{equation*} e_1^2 = e_2^2 = e_3^2 = 1 = -e_4^2, \end{equation*} In either case, we have:
$$e_1^2 e_2^2 e_3^2 e_4^2 = -1.$$