Whatever happened to quaternions?

Question

Quaternions were made up by Hamilton. They are an extension of complex numbers. It is said that he first introduced "3d tertions". He was thinking what the relation between $\bf i$ and $\bf j$ had to be (in $t = a + b {\bf i} + c {\bf j}$), walking on a bridge in his hometown. On the bridge it struck him: ${\bf i} \times {\bf j} = {\bf k}$ (and ${\bf j} \times {\bf i} = -{\bf k}$ as holds for all pairs of different bases). So the quaternion was born

$$ q = a + b {\bf i} + c {\bf j} + d {\bf k}$$

It didn't get a grip on physics though. Hamilton invented them for the sake of rotation in classical mechanics. Rotation in two dimensions can be described by complex numbers (which are two dimensional). That's why Hamilton thought to extend the complex number to a tertion. A tertion could be represented by a vector in the complex 3D space. It turned out that he needed one more dimension. To incorporate k. This was because although the square of i and j are both -1, the product ij could not be accounted for in a tertion. The multiplication has to be closed. Which requires a k.

Whatever happened to these guys? Are they just mathematical curiosities instead of the heroes they were supposed to be by Hamilton? I read that the whole of special relativity could be defined by them. Only to give rise to a positive metric of Minkowski space. Is that all? Whatever happened to them heroes?

Answer 1

"Are they mathematical curiosities instead of the heroes they were supposed to be by Hamilton?"

Neither. It is true that there was a great hype around them, after their discovery. The hype passed. And they took their modest place among the rest of mathematics. Quaternions indeed give a very convenient way to calculate with rotations, and widely used in applied mathematics, wherever calculations with rotations are necessary (for example in computer graphics, robotics, aircraft dynamics). They give a convenient rational parametrization of the orthogonal groups in dimensions 3 and 4 (actually guessed by Euler, without proof).

There were hopes that the monumental 19 century building called Functions of Complex Variables will be generalized to quaternions, but these hopes waned. Functions of a complex variable remain one of the central research areas of mathematics, while quaternions occupy their small place, as a useful tool.

Answer 2

Your question reminds me of another question on this site here, which was in a different direction: why quaternions are more popular than another algebraic system. I gave an answer there, which I will edit and include here to address your question of "what happened" to quaternions.

Hamilton's quaternions are far from being just a "mathematical curiosity". They are the simplest case of a very general construction that people who didn't study abstract algebra far enough may never have heard about: a quaternion algebra. This is a really significant idea within the areas of (mainstream) mathematics that make use of it.

The Hamilton quaternions $\mathbf H$ and the $2 \times 2$ real matrices ${\rm M}_2(\mathbf R)$, are both 4-dimensional noncommutative rings, with their center equal to the real numbers. In both of them there is no 2-sided ideal other than $(0)$ and the whole ring. A ring that is 4-dimensional over a field $K$, has $K$ as its center, and has no 2-sided ideals other than $(0)$ and the whole ring is called a quaternion algebra over $K$. It turns out that the matrix ring ${\rm M}_2(K)$ is always a quaternion algebra over $K$ and every other example (if it exists) is a division ring: all nonzero elements have a left and right multiplicative inverse, just like in Hamilton's quaternions.

There are two quaternion algebras over $\mathbf R$: ${\rm M}_2(\mathbf R)$ and $\mathbf H$. There is only one quaternion algebra over $K$ when $K$ is $\mathbf C$ or a finite field, namely ${\rm M}_2(K)$. If you have heard of the $p$-adic numbers $\mathbf Q_p$ (if you haven't, skip the rest of this paragraph) then you'll be amused to find out there are two quaternion algebras over $\mathbf Q_p$ for each prime $p$, just like over the real numbers. For $p = 2$ the two examples are the $2 \times 2$ matrices over $\mathbf Q_2$ and the Hamilton quaternions with $\mathbf Q_2$-coefficients, but for odd primes $p$ you need to do something different to get the quaternion algebra over $\mathbf Q_p$ that is a division ring because the Hamilton quaternions with $\mathbf Q_p$-coefficients is isomorphic to ${\rm M}_2(\mathbf Q_p)$ rather than being a division ring.

Things get really interesting for quaternion algebras over fields like $K = \mathbf Q$, because there are infinitely many different (that is, nonisomorphic) quaternion algebras over $\mathbf Q$. The ring ${\rm M}_2(\mathbf Q)$ and the rational Hamilton quaternions are merely two examples out of a huge list of quaternion algebras over $\mathbf Q$. The other rational quaternion algebras show up in various places in mathematics: (supersingular) elliptic curves, Brauer groups, Shimura curves, and hyperbolic geometry in 2 and 3 dimensions. Take a look at John Voight's book in order to appreciate that quaternion algebras other than Hamilton's one example have a lot of interesting math associated to them.

Answer 3

Hamilton introduced the terms "vector", "scalar" and many others. The Wikipedia article Classical Hamiltonion quaternions states

A quaternion can be represented as the sum of a scalar and a vector.

By the middle of the nineteenth century the concept of vector spaces was not yet developed fully although linear algebra with its systems of equations goes back to antiquity. Thus, Hamilton and his followers, used quaternions as a substitute for three dimensional vector analytic geometry. That is, the three quaternion units i, j, k represented the three unit vector basis of ordinary space. After the development of vector spaces, the three unit vectors lost their association with quaternions and stood on their own. Same with the real scalars. The multiplication of two vector quaternions is given by $\,u v = -u\cdot v+u\times v.\,$ Later, the dot product and the cross product became independent and also lost their association with quaternions.

Almost all of the machinery, notation and terminology that Hamilton developed for quaternions as mentioned in the Wikipedia article, is now obsolete, or else repurposed.

More about the history of this development is in the Wikipedia article History of Quaternions For example, this excerpt

Since quaternions demand their readers to imagine four dimensions, there is a metaphysical aspect to their invocation. Quaternions are a philosophical object. Setting quaternions before freshmen students of engineering asks too much. Yet the utility of dot products and cross products in three-dimensional space, for illustration of processes, calls for the uses of these operations which are cut out of the quaternion product. Thus Willard Gibbs and Oliver Heaviside made this accommodation, for pragmatism, to avoid the distracting superstructure.

In answer to the question

Whatever happened to quaternions?

the Wikipedia article also states

Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions and the sedenions; they are also a useful tool in number theory, particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group.

Answer 4

From Numbers by Ebbinghaus, et al.:

Hamilton believed that his quaternions would play a key role in physics. With missionary zeal he strove to get them accepted by the mathematical world. [...] In Ireland and England, Hamilton became the figurehead of a school of "quaternionists" who "outdid their master in intolerance and rigidity." At the center stood a mystic formalism treated with due reverence by the initiated. One dreamt of a quaternionistic theory of functions and expected to gain new and profound insights into the whole realm of mathematics. To promote these utopian aims there was even founded, in 1895, an "International Association for promoting the study of quaternions," at Yale University in New Haven, Connecticut.

However, such enthusiasm was not universal even in Britain. Thomson (Lord Kelvin), commented caustically:

"Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way."

So what happened to quaternions? Their significance was greatly diminished when it was realised that the quaternion algebra was only a particular algebra of complex $2 \times 2$ matrices.

A similar fate awaited the flood of hypercomplex systems which inundated the whole of algebra in the second half of the nineteenth century. The realisation that there are far fewer interesting $\mathbb R$-algebras than one might expect began with Frobenius' isomorphism theorem, which showed that there are only three isomorphically distinct real finite-dimensional associative division algebras - namely $\mathbb R$ itself, $\mathbb C$ and $\mathbb H$. This was followed by the isomorphism theorem of Mazur and Gelfand (building on the work of Hopf) which showed that $\mathbb R$, $\mathbb C$, and $\mathbb H$ are the only Banach division algebras.

Answer 5

As a side note I want to add that Hans Georg Küssner (https://en.wikipedia.org/wiki/Hans_Georg_Küssner) published two books (in 1946 "Principia Physica" and 1976 "Grundlagen einer einheitlichen Theorien der physikalischen Teilchen und Felder") in which he tried to build a new foundation of physics and where Quaternions play an integral role.

Answer 6

Actually complex quaternions or "biquaternions", as Hamilton called them, are the same or isomorphic to Pauli matrices and are called Pauli Algebra or Pauli quaternions and are 8 dimensional. Vector analysis is a truncated form of quaternions. They are used throughout physics with many not realizing that they are a form of quaternions.