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.
Question #1. What do we get when we take the wedge product s ∧ t between two scalars s and t?
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.
Question #2. What’s the right way to define dot products involving scalars?
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.
Question #3. What is the best way to define norms?
When the dot product between a scalar s and a non-scalar basis element b was defined to be zero, it allowed us to define norms based on the dot product instead of the full geometric product. I don’t think that went as far as it should have, and I concluded for the previous question that we couldn’t define s ⋅ b = 0 anyway. The inner product in PGA can simply be equated with what’s known as the scalar product, which can be defined as the scalar component of the geometric product. The inner product between two basis elements a and b is nonzero only when a = ±b, and it’s zero for everything else, so it is very basic. Using the fat dot • for the inner product and an open dot ∘ for its dual, we can define the bulk and weight norms as follows.
‖a‖● = (a • ã)½ (bulk norm)
‖a‖○ = (a ∘ a̰)½ (weight norm)
These produce exactly what we need for every type of object in PGA: magnitudes, points, lines, planes, motors, and flectors. In my opinion, there is no need to make it any more complicated than this.
Question #4. What is the most natural property shared by geometrically meaningful objects?
Finally, we answer the question that started all of this by stating a condition that must be satisfied by an object a in PGA for it to be geometrically meaningful:
a ⟑ ã = a • ã
a ⟇ ã = a ∘ a̰
This says that the geometric product of a with its own reverse must not contain any terms other than the square of its bulk norm (a scalar), and that the geometric antiproduct of a with its own antireverse must not contain any terms other than the square of its weight norm (an antiscalar). This leads to the formulas listed on the geometric property wiki page for the various types of objects in PGA.
The conclusions I’ve reached are summarized as follows:
- The wedge product of two scalars remains s ∧ t = st.
- The dot product between a scalar s and a non-scalar basis element b remains s ⋅ b = sb.
- This dot product is not natural, and it’s better to use interior products / contractions.
- The inner product should simply be defined as the scalar product.
- The bulk and weight norms should be defined canonically with the inner product and its dual.
- The natural geometric property is that a ⟑ ã is a scalar and a ⟇ a̰ is an antiscalar.