(This post is also in the “quaternions” category. If that’s what you’re looking for, just get the rest of the post and page down to the section heading.)
Let me begin by saying that the 2nd group theory post was not quite as popular as the 1st one… the blog “only” got 436 hits Monday, and the 2nd group theory post only got 102 hits itself. On Tuesday, as it had the week before, the blog as a whole got more than 300 hits.
I assure you I’m not disappointed in these kinds of numbers.
I hope to put out another post on group theory – about symmetric groups, permutations and cycles, and the quaternion group. On the one hand, the material is not complicated….
On the other hand, that post is at stage II. I’m not sure I’ve ever talked before about a specific post being at stage II. That means that I haven’t even done the mathematics for it! All I have is an outline in my head. We’ll see what happens.
As for the week gone by….
I’m still looking at electric circuits… I keep using the wrong sign for the initial voltage on the capacitor.
I’ve been looking at group theory in chemistry… some things are clearer in these books, and some are quite confusing… but I’m sure I can get it.
I’ve also been looking at rings and ideals. Yes, that was part of my undergraduate education… but I never really understood them. Oh, the shame.
Oh, out on the non-research math staff exchange, I found a fascinating discussion of how Laplace transforms make rigorous ideas originally propounded by Heaviside. Roughly speaking, what he did was work out a set of rules for working with the “inverse derivative” operator.
Interestingly, I had seen a presentation of Heaviside’s approach way back as a freshman taking calculus. Then as a first-year graduate student, I saw the Laplace transform. Somewhere along the line I learned that the 2 approaches were related… but I no longer had Heaviside’s approach available to me. I’m delighted to have seen this explanation now.
Finally, thinking about the quaternion group for this coming Monday, it suddenly occurred to me that
- Quaternion multiplication is associative
- the vector cross product is not associative
- but the vector cross product comes from the product of 2 quaternions
So how the hell did we lose associativity?
BTW, we know that the vector cross product is not associative if we remember that we must distinguish the two triple cross products
(That they are not the same means precisely that the product is not associative.) To show that they are different, use the BAC-CAB rule:
(which is true because we pick up two negative signs when we switch orders.)
why is the quaternion product associative while the vector cross product is not?
The short answer is that the vector cross product is the quaternion part of the product of two pure quaternions (no real part), but the product of two pure quaternions is not pure – so we have to drop the real part… and that’s where we lose associativity; roughly speaking, the function of taking the pure part does not commute with the quaternion product.
You might want to look at the beginning of my first post on quaternions.
Start by invoking Mathematica’s Quaternion package:
Here are two pure quaternions – they have no real parts – P and Q… and their product P**Q:
The product of pure quaternions is not a pure quaternion; the terms in i^2, j^2, and k^2 are all real.
Let me define two functions which will act on quaternions: “IJK” to make a vector out of the pure part of a quaternion… and “pure” to drop the real part of a quaternion. The difference is that while IJK gives me a vector, pure gives me a pure quaternion.
Let’s start gently, by comparing the quaternion product to the vector cross product. The pure part of the quaternion product P**Q is the vector
The two vectors corresponding to the quaternions P and Q are… and their vector cross product is
Is that the same as our pure quaternion? Yes:
So the vector cross product really is the quaternion part of the quaternion product of two pure quaternions.
Now we need a third quaternion; call it R.
Let’s continue gently. I compute the quaternion product (P**Q)**R… then the quaternion product P**(Q**R)… and we see that they are the same:
That is, we have just shown that the quaternion product (admittedly all pure quaternions) is associative. I assure you, it’s true for arbitrary quaternions, not just for pure quaternions.
Now let’s use quaternions to see how we get two different vector cross products. First using (P**Q)**R: we take the pure part of P**Q, then post-multiply by R… and then extract the vector (don’t worry about how messy it is):
For comparison, here’s the triple vector product corresponding to (P**Q)**R:
Are the results the same? Yes:
Once again we have matched the pure part of the quaternion product to the vector cross product.
Now we want to use P**(Q**R) instead. We start with Q**R … take the pure part of it… premultiply by P… and extract the vector:
For comparison, we now compute the vector cross product corresponding to P**(Q**R). Get the two vectors for Q and R… take the cross product… then cross (the vector corresponding to) P into the result:
Are they the same? That is, is the quaternion part equal to the final vector? Yes:
But, of course, the two vector cross products are not the same:
All I’ve really done is display the calculations and see that we lost associativity. But where?
The triple vector product – now I’m abusing notation horribly, using P for both the vector and the quaternion – is the following sequence of functions:
pure ( pure (P**Q) ** R )
while the other triple product is
pure( P** pure(Q**R) ).
The difference between them is the first (inside) application of “pure”:
pure (P**Q) ** R
It isn’t quite right to say that we lost associativity because “pure” doesn’t commute with **… but it’s a suggestive phrase.
And that’s how the associative quaternion product gives rise to a non-associative cross product.
And I begin to understand why some people were so vehemently opposed to vector analysis. I knew that they considered vector analysis a destruction of quaternions – but now perhaps I see an example, the loss of associativity.
That said, I’ll keep both quaternions and cross products. Each is useful somewhere.