>> Do you think it is all of GA that helps, or really just the concept of bivectors / multivectors?
Yes, definitely all of it. Especially in physics, where every well known equation comes with its own, not so well known algebra. There are dozens (maybe even hundreds?) of them, all with their own notations and shenanigans. The thing is, all of them can be covered by GA. And you only need three instances of GA to cover everything [0]: Classical mechanics, quantum mechanics and relativistic mechanics / space time.
>> I have a lot of skepticism about the geometric product as a useful object
The geometric product is kind of like: "Here is everything you could possibly want to do, all at once!". We usually pick out specific parts of it (called grade projections) to operate on sub-algebras matching the specific problem domain at hand. Yet, all of them are still connected in one über-algebra so to speak. So, we don't need to switch notations.
>> bivectors / multivectors
Multivectors, the vectors of GA, are like "normal" vectors, from linear algebra, just tuples. There is nothing special about them. The genius lies in the idea that unlike "normal" vector algebra, where you have one element in your tuple for every dimension, you also have elements for every combination of dimensions in a multivector. The multivector is usually sorted by the grade (number of dimensions an element combines). That way a multivector has a 0D scalar, 1D vectors, 2D bivectors, 3D trivectors, etc.
I'm very aware of GA and multivectors and the usual equations that are included in GA texts, like the one-term form of Maxwell's equations. I'm specifically interested in whether you know of a good reason to include the geometric product in all this. It seems to me that for the most part you can do all of the useful stuff without ever mentioning it, and the equations it produces tend to be very awkward (compute something, grade project, compute something else, project again...) -- it's very 'just so', and kinda feels like it's all using the wrong tool. But I haven't figured out what the better tool is yet.
Yes, definitely all of it. Especially in physics, where every well known equation comes with its own, not so well known algebra. There are dozens (maybe even hundreds?) of them, all with their own notations and shenanigans. The thing is, all of them can be covered by GA. And you only need three instances of GA to cover everything [0]: Classical mechanics, quantum mechanics and relativistic mechanics / space time.
>> I have a lot of skepticism about the geometric product as a useful object
The geometric product is kind of like: "Here is everything you could possibly want to do, all at once!". We usually pick out specific parts of it (called grade projections) to operate on sub-algebras matching the specific problem domain at hand. Yet, all of them are still connected in one über-algebra so to speak. So, we don't need to switch notations.
>> bivectors / multivectors
Multivectors, the vectors of GA, are like "normal" vectors, from linear algebra, just tuples. There is nothing special about them. The genius lies in the idea that unlike "normal" vector algebra, where you have one element in your tuple for every dimension, you also have elements for every combination of dimensions in a multivector. The multivector is usually sorted by the grade (number of dimensions an element combines). That way a multivector has a 0D scalar, 1D vectors, 2D bivectors, 3D trivectors, etc.
[0]: https://www.scirp.org/journal/PaperInformation.aspx?PaperID=...