|
The physical interpretation depends on context, but the concrete arithmetic is the same whether you write it using GA language or some other mathematical formalism (matrices, Gibbs–Heaviside style vectors, differential forms, tensors, complex numbers, ...), so if you have a problem with it your problem is with the physics. Like with any expressive language (e.g. English, C++, or matrix algebra), GA makes it possible to state a wide variety of nonsensical things. But that’s not the fault of the language. Where GA really shines IMO is in the ability to divide by vectors, something that is extremely useful but gets super cumbersome using other mathematical languages. The biggest “problem” I have with GA is that it takes a lot of practice to get familiarity with all of the powerful stuff it can do. There are pages and pages of (extremely powerful and useful) short identities which are impossible to memorize by just looking at them, and can only be learned as far as I can tell through years of hard-won experience. I feel like I still only really have a handle on the most basic stuff. I find regularly myself working on some complicated coordinate-based calculation for 3 pages of scratch paper full of mistakes and wrong turns, only to experience déjà vu, re-express the original thing in GA language, and end up with a clear and geometrically interpretable 4 lines of simple algebraic identities which solve the problem. But I’m not always sure if I’d be able to figure out which 4 lines to use right off the bat, without first going through the coordinate slog. |
Isn't that just what distinguishing scalar/vector/bivector/trivector/etc is for? I don't have problems with coordinates because I don't bother with coordinate-based calculations in the first place. What I don't get is how adding two multivectors of different [ranks? eg scalar + bivector] is supposed to simplify anything.
> > > the geometric product [...] is the sum of a scalar and a bivector
> any expressive language [...] makes it possible to state a wide variety of nonsensical things.
Yes, but if everything the language makes it possible to state is nonsensical, then what's the point?
I assume there's some sort of point to geometric algebra, but I have yet to encounter any convincing explanation of what that point is, and why I shouldn't keep my dot and wedge products properly separated.