Translating vector equations into geometric algebra

1

I’ve been reading about geometric algebra and I want to apply it in my research to get a bit more intuition. However, as stated (here)[Introduction to Clifford Algebra], it seems much easier to move from geometric algebra to vector algebra equations, rather than the reverse.

Equations with multiple vector cross products resist my translation. For example, electron spin accumulations (with magnetic moment s) cause a damping like torque in ferromagnets (with moment m) proportional to \vec{\tau} = \vec{m} \times (\vec{s} \times \vec{m})

To translate this to geometric algebra, I’ve tried the following:
Vector hats above were psuedovectors, so changing them out for the appropriate bivevectors (no hats), we get
(im) \times ((is) \times (im))
Where i is the unit psuedoscalar in 3 dimensions i=xyz.

Then using \mathbf{a} \wedge \mathbf{b}=i(\mathbf{a} \times \mathbf{b})=i \mathbf{a} \times \mathbf{b} (from new foundations for classical mechanics), we get
i \tau = (m) \wedge ((s) \wedge (im))
and then by associativity
i \tau = (m) \wedge (s) \wedge (im)
i \tau = - s \wedge m \wedge (im)
\tau = i (s \wedge m \wedge (im))

And here’s where I feel a bit stuck.
This equation should not be chiral, so I should be able to get rid of all pseudo scalars. Also, it’s still non interpretable at an intuitive level.

Any help?

2

Clearly, you lack the concept of Grassmann complement and what a cross product is.

The cross product is complement(wedge(a,b)).

As I explained a million times before, your Grassmann complement operation is wrong, as it is for most people because they have no clue about Grassmann algebra and how it actually works.

For now, I will just give this as a hint to help out. of course I can give you exact instructions, but you only ask for help, not for a full answer.

That formula from new foundations to classical mechanics seems to be incorrect also, so I wouldnt be relying on all that. You gotta think for yourself with geometric algebra, cant rely on all those papers.