Re:Why can't you multiply vectors? [video]
by nyssos on Monday October 30, 2023 @07:08PM
(#38077222)
Spacetime algebra is just the Clifford algebra Cl_1_3(R), so all of the above applies.
> They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts
Yes and no: the right way to unify rotations and boosts is to consider them as the orientation-preserving elements of the Lorentz group (sometimes called the proper Lorentz group). You can construct this from the corresponding Clifford algebra, but it's somewhat technical and not physically well-motivated until you start dealing with spinors. It's also the group of symmetries of spacetime that leave the origin unchanged, which is far, far more natural.
> They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts
Yes and no: the right way to unify rotations and boosts is to consider them as the orientation-preserving elements of the Lorentz group (sometimes called the proper Lorentz group). You can construct this from the corresponding Clifford algebra, but it's somewhat technical and not physically well-motivated until you start dealing with spinors. It's also the group of symmetries of spacetime that leave the origin unchanged, which is far, far more natural.