Or said another way, forces carry the structure of G-torsors! The terminology is way too pompous for such a simple concept, though. John Baez has a really clean writeup on them, accessible to anyone interested: https://math.ucr.edu/home/baez/torsors.html.
Heck, even differences are usually non-physical; it's their ratio that matters. I.e. choosing feet or meters doesn't change the physics; the same goes for energy. So we have two free parameters: choice of origin and choice of units.
Baez only hints at this near the end of the above article, but we can actually fuse translations and scalings into a single group of elements that are combined translation + scale operations, an affine group. It turns out that this combined group is just a certain combination (semidirect product[0]) of the translation and scaling groups, as one would hope.
And once we are thinking about affine groups, it's natural to consider more complicated ones. Most famous is probably the Poincaré group[1]. That is, points in space are physically described by G-torsors over the Poincaré group!
Excellent comment. The only thing I can add is the obligatory "why stop there?", Poincaré true to form gives a beautiful description of the symmetries of flat spacetime, but we know that spacetime is only locally flat, the measurement symmetries at large scale are carried by the (unfortunately named) Killing fields[0].
Heck, even differences are usually non-physical; it's their ratio that matters. I.e. choosing feet or meters doesn't change the physics; the same goes for energy. So we have two free parameters: choice of origin and choice of units.
Baez only hints at this near the end of the above article, but we can actually fuse translations and scalings into a single group of elements that are combined translation + scale operations, an affine group. It turns out that this combined group is just a certain combination (semidirect product[0]) of the translation and scaling groups, as one would hope.
And once we are thinking about affine groups, it's natural to consider more complicated ones. Most famous is probably the Poincaré group[1]. That is, points in space are physically described by G-torsors over the Poincaré group!
[0]:https://ncatlab.org/nlab/show/semidirect+product+group
[1]:https://en.wikipedia.org/wiki/Poincar%C3%A9_group