[ajkjk]

I completely agree, incidentally... And I have come to see this more clearly since writing the article, in part due to talking to you. Maybe I should update it accordingly. I've been meaning to write a followup where I explain this viewpoint but have had trouble finding the mental energy for it.

[hamish_todd]

Ah! If you do it would probably would need to be a followup rather than addendum

[ajkjk]

Well, I think my points in the parent article here do stand (I'm aware that you do not). The fact that there is a good interpretation of the geometric product in some cases does not obviate the fact that everyone's writing crappy intuitive things about it. Anyway it has always been my stance that there is a _good_ version of GA, and people need to figure out what it is and write about that instead of bloviating about how good the current version is. Treating the GP as a composition of operators is a start, but it's not the whole picture --- why do operators compose in that particular way? why are these your operators in the first place? My hunch is that GA is really a subset of a larger and more intuitive algebra that has very obvious answers to these questions, probably from a representation-theory perspective.

(I meant to do this followup soon after the first article but I've been been having a lot of difficulty focusing / constant brain fog for the past few years so it's been on the to-do list instead; part of the problem is that to do it right I need to go read and digest everyone's different treatises / expositions on GA ... and that has felt taxing, to say the least.)

[hamish_todd]

Very sorry to hear about the brain fog, that's rough. Best of luck getting through it.

> in some cases

What cases do you have in mind?

> why are these your operators in the first place?

This is a perfectly reasonable question but I want to point out that it's a bit philosophical and not the kind of thing physics undergrads tend to enjoy hearing about. For example Clifford algebras like matrix algebras are associative algebras over commutative fields (real or complex numbers) - why? Why does the universe like that? I can hazard guesses but it's mostly above my paygrade, possibly above anyone's paygrade. But if it's to be asked of GA it should be asked of matrices too, I'm sure you can agree.

There's something that distinguishes Clifford algebras from matrix algebras. They start from the assumption that vectors anticommute when they're orthogonal. That's easy to explain. It says that if A and B are orthogonal vectors then A->B is the opposite (negation) of B->A.

Aside from those the last thing is that you say your basis vectors have a certain "metric" on them. There's deep philosophical questions about why the universe cares so much about metrics, but they're not at all specific to Clifford Algebra.

Personally I find that very intuitive, much more intuitive than any other tensor algebra I'm aware of.