Wedge Products, Dot Products, Scalars, and Norms

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2021-11-25 20:00:07

I recently posted a couple of articles talking about what we should do when we take a wedge product or dot product and one or both of the operands is a scalar. Those posts were motivated by a quest for a fundamental “geometric property” shared by the meaningful objects in projective geometric algebra (PGA), and they were basically a brain dump of the various ideas running through my head. I think I have it all sorted out now, and this article lists the questions along with the answers I ultimately came up with. Some of the ideas I had turned out not to be the right way to go, so I have deleted the previous posts to avoid leading anybody in the wrong direction.

This question arose because the condition a ∧ ã = 0 was a candidate for the fundamental property shared by all geometrically meaningful objects a in PGA. A four-dimensional bivector L represents a line that could have been constructed from two points only if L ∧ L̃ = 0. The same formula applies to flectors as well, but when we try to apply it to motors or magnitudes, we get an extra scalar term because s ∧ t = st. One avenue I explored was the possibility that s ∧ t = 0 was a more natural definition of the wedge product between scalars, and it kind of made sense because the scalar unit would otherwise be the only basis element that didn’t square to zero under the wedge product. This fixed the problem with a ∧ ã = 0 by making it true for all the object types in PGA, but it ended up causing other problems. The wedge product lost associativity and a universal identity element, but those didn’t really cause any difficulties. The real problem arose when I looked at projections using the interior products (which depend on the wedge and antiwedge products). There were some specific cases that became zero when it didn’t make any sense for that to happen, and that meant the definition s ∧ t = 0 was almost certainly the wrong path to follow. My conclusions are that (a) we must keep s ∧ t = st, and (b) that the condition a ∧ ã = 0 simply does not generalize as a property satisfied by geometrically meaningful objects a.

This question arose for several reasons. There has historically been some disagreement about how dot products involving scalars should be defined, there is the question of whether the wedge product and dot product should be disjoint components of the geometric product, and there is a question about whether the dot product constitutes a valid inner product on which a canonical norm can be defined (see next question). I explored the possibility that for two scalars s and t, we have s ⋅ t = st, but for any non-scalar basis element b, we have s ⋅ b = 0. If it were the case that s ∧ t = 0, then the wedge product and dot product would be fully disjoint, but I’ve come to the conclusion that this is not a natural requirement. The geometric product naturally partitions into symmetric and antisymmetric components given by the commutators ½(a ⟑ b + b ⟑ a) and ½(a ⟑ b − b ⟑ a), and these are the only things we can expect to be disjoint. When one of a or b is a vector, then these parts correspond to the dot product and wedge product, respectively, but that relationship does not generalize at all. Defining s ⋅ b = 0 still makes it possible to derive a canonical norm from the dot product instead of using the full geometric product, but it also introduces an inconsistency with the interior products when scalars are involved, which I don’t think is correct. My opinion is that the dot product should continue to be defined as ⟨a ⟑ b⟩|g − h|, where g and h are the grades of a and b. However, in agreement with some other authors, I don’t think this dot product is a natural operation, and I believe we should be working with interior products (a.k.a. contractions) instead.

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