I'm a bit pressed for time, but one annoyance I've had with the classic "greek" physics notation is that they represent things from "both ends" of a graded vector space. So for example, they start with a scalar, then a vector, then ... pseudoscalar-1, and finally the pseudoscalar.
It's a shortcut useful only if you need to scribble on paper and your wrists hurt from writing too much, but it obscures the underlying physics.
The programming equivalent is putting abbreviations in identifiers where, sure, it's fewer characters, but then the reader needs to a track a mental lookup table to translate back to the intended meaning.
Pushing things like this too far results in meaningful aspects of the equations getting squeezed out entirely. For example, the generality of GA means that you have to (correctly) track negative signs and multiplications by pseudoscalars such that your formulas work in all dimensions. In traditional vector algebra it's all too tempting to eliminate certain products because in "your chosen dimension" they multiply to 1 or -1 or whatever and just... disappear due to traditional algebraic simplification conventions. But then if you need to work in 4D SR or curved spaces, you can't, because you threw away something essential while "optimising for characters on a page".
You have then "start over", typically reaching for a partial and incomplete subset of GA, reinventing that wheel over and over.
Hence the push for unification onto GA, to break this cycle.
Thanks. Claude tells me the essense of this example is "the GA formula for rotation works in 4D (vs quaternions), and to do something like rotation in kD you need a tuple of two objects of different grades because cross-product is a hack that only happens to work in 3D because the high-grade object there is degenerate, and to do 4D special relativity you need 4D rotations".
Is this more or less in the right direction to keep exploring?
It's a shortcut useful only if you need to scribble on paper and your wrists hurt from writing too much, but it obscures the underlying physics.
The programming equivalent is putting abbreviations in identifiers where, sure, it's fewer characters, but then the reader needs to a track a mental lookup table to translate back to the intended meaning.
Pushing things like this too far results in meaningful aspects of the equations getting squeezed out entirely. For example, the generality of GA means that you have to (correctly) track negative signs and multiplications by pseudoscalars such that your formulas work in all dimensions. In traditional vector algebra it's all too tempting to eliminate certain products because in "your chosen dimension" they multiply to 1 or -1 or whatever and just... disappear due to traditional algebraic simplification conventions. But then if you need to work in 4D SR or curved spaces, you can't, because you threw away something essential while "optimising for characters on a page".
You have then "start over", typically reaching for a partial and incomplete subset of GA, reinventing that wheel over and over.
Hence the push for unification onto GA, to break this cycle.