The n-Category Café

Jonathan wrote:

Tim Poston wrote:

So there is a theory of holomorphic quaternionic functions, in a way … but it is identical to the theory of linear quaternionic functions, and not separately interesting.

That’s true for one definition of ‘holomorphic quaternionic function’, but not the Fueter definition. That’s why people prefer the Fueter definition! The first three pages of Sudbery’s paper explain the situation quite clearly. Maybe you can show Poston this paper.

There is tradition to consider linear map $f(x)$ as product of number over variable $x$. Since product in quaternion algebra is noncommutative this view creates problem. But why not to consider that linear map has form

where $a$ and $b$ are quaternions. In such case we extend the set of linear maps and can consider derivatives of polynomial or exponent of quaternion argument.

I now think Frenkel and Sudbery are using the same notion of ‘quaternionic holomorphic function’ as the one the geometric algebra people are using — it’s basically a solution of a certain Dirac equation, a 4d analogue of the Cauchy-Riemann equation.

Hello:

I thought I’d share the story of where this proposal for a two limit definition of a quaternion derivative came from because it is a nice story with fun implications for physics, specifically the magic of the factors of 1/3 and 2/3 that pop up for holomorphic functions.

In 1999, I went to the only meeting devoted to quaternions and physics I had ever heard about (and have not heard of any since) in Rome. I traveled with Prof. Guido Sandri of BU. In the second talk of the conference by David Joyce of Oxford, he explained the problem of quaternion analysis well. He then described his own approach to hypercomplex geometry. I got angry at his proposal, thinking it was far too complicated and clucky (I know, clucky isn’t a technical word in math, but it accurately reflects my feelings of yucko). I decided that this was the problem to focus on during the conference: come up with a good definition of a quaternion derivative driven by the needs of physics that does analytical functions gracefully.

I was fortunate that Guido was happy to be my personal tutor in complex analysis. He taught me the importance of the manifold C¹ in the topic (and I apologize ahead of time for any poor phrasing I use since I have not continued my education since then, consumed mostly by the activities of daily living outside academia). The pair of variables z and z^* on C¹ can do everything that x and yi can do on R². Yet they can also do more because they have the properties of the complex division algebra built in.

For quaternions, it was clear that q and q^* was not enough, there being 4 degrees of freedom. Had I been a diligent student of the literature, I would have picked up Feuter’s approach. Yet in the outdoor cafes in Rome, I came up with a different idea. What the conjugate does is keep the sign of the scalar positive, while flipping the signs of the other three. Why not make a new widget, that would flip the sign of all but the first term in the 3-vector? I called this the first conjugate, q^*1 = (e₁ q e₁)^*. This could be classified as “cute”, not too tricky, yet does something unexpected, taking a quaternion such as G = (f, g, u, v) to (-f, g, -u, -v). There is also a second conjugate, q^*2 = (e₂ q e₂)^* that flips the sign of all but the e₂ term. They can be isolated by different combinations of these conjugates:

(f, 0, 0, 0) = (G + G^*)/2
(g, 0, 0, 0) = -e₁(G + G^*1)/2
(u, 0, 0, 0) = -e₂(G + G^*2)/2
(v, 0, 0, 0) = -e₃(G - (((G^*)^*1)^*2))/2

Back in my SPR days, I recall one comment from Toby Bartels that this little widget should be acknowledge in a paper he and John Baez were writing. What their paper was, its outcome, any one-to-one emails on the topic didn’t happen. Why don’t we just say whatever slim relationship between John and I had was tense. I was hoping someone would ask why I had developed the first and second conjugates: for a new approach to quaternion analysis!

Despite owning quaternions.com, my “quaternions [accounting] rule the universe” approach is considerably more subtle than such a label implies. If one uses standard quaternions in the standard way, there is zero new physics in such an exercise. Zero, none, nada - most of my site is devoted to nothing new, but it is great practice. New physics is exclusively the result of new math. This is why the majority of theoretical physics research money can and should go to work on strings. This is why we should encourage the new math being developed in loop quantum gravity and the spin foams. This is why we should be skeptical that someone can find anything actually new to do with quaternions since they have been around since Gauss’ time (in an overlooked notebook). Yet I will continue with this post since this definition of a quaternion derivative - nine years old - still looks new and unknown.

The flaw with Feuter’s approach to the quaternion derivative (and Sudbery’s technical update) is that a function as simple as the square of a quaternion cannot be shown to be analytical in q. If you don’t have the polynomials, you don’t have anything. Guido told me there were five equivalent ways to show a function was analytic. One was to use the limit definition, another the Cauchy-Riemann equations, a third was to apply the chain rule. I forget what the fourth one was, my bad. I was able to show q² was analytic in q, (q^*)² was analytic in q^*, (q^*1)² was analytic in q^*1, and (q^*2)² was analytic in q^*2. It was fun to work through all the details. It was the fifth one that was a stumper. I had to show that a derivative of a regular function vanishes. They do that in geometric algebra by a geometric derivative as Bruce Bartlett described, an approach that does not work with quaternions. I wanted this “dot derivative” to vanish:

scalar((d/dx₀, d/dx₁, d/dx₂, d/dx₃)(f, g, u, v))
= d f/dx₀ - d g/dx₁ - d u/dx₂ - d w/dx₃ = 0

The problem was this didn’t want to vanish. There was too much from the 3-vector. The idea I have seen adopted by Feuter and everyone else I have read looks at the 4 degrees of freedom as equals. If I used that approach, I was able to get four of five tests for an analytic function to work, and that is not good enough. My derivative definition does not look at the four as equals. Instead, it puts the scalar equal on equal footing with the 3-vector. So I decided to put in a factor of 1/3 for the 3-vector, and if done just right - the details get confusing - I was able to get the dot derivative to vanish. Five for five is good :-)

Now that I had my extra factor of 1/3 for the basis of the 3-vector, I had to go back to my other 4 tests for analytic functions, and redo them with this factor. It was amazing to see all these factors of +2/3 and -2/3 in the chain rule calculation work out. The only other place where I have heard of factors of 2/3 showing up in calculations has to do with quarks. Accident? I don’t know, but it was exciting. On the Air France flight back, I worked out the details for the cube of q being analytic in q. If you ever want to be left alone on an international flight, do a page of algebra. People will leave you alone and assume you are brilliant.

It is clear folks who chat here are absurdly busy, citing papers yet to be fully read and digested. There is no way someone like myself could prove this quaternion derivative definition works unless people do it themselves. I have met my responsibility by posting it here since the topic is spot on. My email box is open to discuss the new quaternion math. The other two new math bits I have developed is a way to animate any quaternion expression what-so-ever, and a way to write a 4D division algebra that commutes, a variation of Clyde Daver’s hypercomplex numbers.

Off to Brazil for the 8th International Conference on Clifford Algebras to talk about the new quaternion math and its implications for new physics.

Doug
sweetser@alum.mit.edu