Hamilton and his school pushed quaternions very hard for half a century. They re-formulated much of existing mathematics in quaternion form. Yet now, quaternions are all but forgotten in physics. Many physicists take this as an indication quaternions are unuseful. For others, the fact that they have been deemed uncool is grounds enough to snub them.
I propose that the modern absence of quaternions in physics is an accident of history and fashion, and try to clarify (for myself, at least) how to view the peculiar properties of this algebra as something useful and meaningful in physics.
One will hear it said that quaternions are useless on account of their non-commutativity.
Is non-commutativity a good thing or a bad thing?
Of course, this depends on what you’re trying to model. There are applications that dictate non-commutative algebras; for these, the quaternions may be just the ticket.
Note that the quaternions contain the complex numbers as a proper sub-algebra. So they do everything the complexes do, and something more.
Of course, if you don’t need it, to invoke such a complicated structure is silly.
Why are complex numbers so useful in physics?
Various metaphysical replies to this question are in circulation, some of which are silly. It's a phase in many physicist's career to attempt to purify physics of complex numbers, and write everything in terms of reals. But there are branches of physics where the can’t easily be disposed of: for example, electrodynamics, wave theory, and quantum mechanics.
I am unsatisfied with my understanding of quantum mechanics, but I have seen papers (by Jauch) that discuss the question in detail, and am aware of others.
In the case of electromagnetism however, complex numbers are much more than a notational convenience. When the equations for electrodynamics are written in complex form, the complex-valued functions involved are tacitly assumed to be complex-differentiable. That is, that their components satisfy a potential equation. The use of complex functions therefore amounts to factoring out a symmetry of the equations, thereby effectively reducing the number of independent variables. So this application of complex analysis (not just writing a pair of reals as a complex number) is a mathematical statement of a deep physical notion of symmetry, with a crucial practical effect.
Another curious bit of schoolyard wisdom is that complex numbers are somehow fictitious because only one number can be measured at a time. I have heard an adherent to this belief in the same breath proclaim that a position in space, or in the plane, is an observable!
Of course, mathematical entities are our conceptual tools, and their main virtue is their usefulness. It is instructive to remember that a past generation of mystics wanted to consider only ratios of integers as being meaningful quantities; this position is no longer regarded as rational.
Given that perspective of complex numbers in physics, would there be a place in physics for quaternion-differentiable functions of a quaternion variable? It seems very likely. One would expect their use to factor out further common symmetries, and indeed they do.
One common complaint one hears about quaternions is that they are less general than some other algebra, such as an algebra of matrices, or Clifford algebras. The opposite complaint is also levelled, that they are too complicated, that one can make do with good old real or complex numbers.
Is the fact that quaternions are not commutative a loss of something? It is surely a loss of one general statement, but is it a loss of structure? Well, of course not. The quaternions have more structure than the complex numbers. What is lost in general statements is gained in structure. There are ultimately more things to say about the algebra of the quaternions.
Are the quaternions unnecessarily complicated? This depends on what they’re used for. For many physical systems, the structure of the quaternions is precisely what is needed. That is, their complication precisely reflects the complication of the physical system. Whatever you can understand about one, sheds light on the other.
During the early 1800s Hamilton and his followers invented quaternions and applied them to much of the math and physics known at the time; they continued to push quaternions very hard for half a century. For a time, advanced mathematics consisted of the study of quaternions.
There was no such discipline as abstract algebra at the time. Indeed this time was the flowering of what is now known as algebra. A significant example includes the matrix theory developed by Grassmann which can be viewed as a later offshoot of the same body of thought that gave rise to the quaternions. This matrix theory, now known as linear algebra, has become a mainstay of all aspects of engineering and physics. (Curiously, it entered physics much later via quantum mechanics.)
Between 1880 and 1887, Heaviside developed the (·, ×) notation for operations on 3-dimensional coordinates as a means of simplifying Maxwell’s equations. In the late 1800s, a battle between quaternions and the coordinate notation took place. Thomson, Heaviside and Gibbs lead the new school. Harsh words were exchanged. By 1894, even Cayley, an early contributor to the theory of quaternions, argued with Tait that an algebra of coordinate systems was of more general practical applicability in physics than quaternions. In the end, the new school prevailed, to such a degree that quaternions were expunged from the study of physics.
In addition to having been over-sold, the quaternions were at a technical disadvantage. At that time, very little was known about the analysis of quaternions. By contrast, the foundations of complex analysis had been completed, polished and well publicized (largely by Cauchy).
It was not until the mid 1930s when Fueter and his school really broke ground in quaternion analysis. This work has never become well known, and while it is much better understood now, there remain big puzzles to be solved.
So at the time of the battle over notations for electrodynamics, the field of quaternions were not ready for the fight. They could not have been used to their full effect at this point.
Therefore, I think it remains to be seen whether quaternions can regain a place in basic physics studies.
It is important to keep in mind that these ideas they emerged contemporarily. Nobody knew how far any of them would go. Nobody had learned abstract algebra in school—it hadn’t yet been abstracted! I can see why in particular Hamilton and his school came to believe they had conquered a new continent: I wonder how badly they twist in their graves, seeing it was just a peninsula.
Then, the usual admonishment against hubris in our own science…
Several algebras are commonly used in physics.
Maybe each algebra tells us something about the world.
The Geometry of Quaternions: Geometry of non-commutativity
Finkelstein, Jauch, Schiminovich and Spiser “Foundations of Quaternion Quantum Mechanics”, J. Math. Phys. 3 (1962) 207-220
Wikipedia on Cayley and quaternions