real-world applications of the algebra of the quaternions:

electromechanics, quantum mechanics, 3D animation

Quaternions have found a permanent place in engineering and computer description of moving objects. In physics, their use has been controversial. Although they appear natural to the description of 4-dimensional space and entities therein, they have not been widely used in the 20th century. However, in quantum field theory, quaternions have always been present in the guise of “spinors”.

In Maxwell’s 1891 A Treatise on Electricity and Magnetism, quaternions are mentioned in several places, and quaternion references are cited, although the author says that he has “endeavoured to avoid any process demanding of the reader a knowledge of the Calculus of Quaternions” (see vol II, art. 619).

Defining the conjugate quaternion gradient operator by

where *c* indicates the speed of light,
Deavours writes Maxwell’s equations as

This amounts to a notational compression by a factor of four over the conventional notation, at least. But, do quaternions do more than provide a notational convenience (or create obfuscation)?

The non-commutativity of quaternions has a deep consequence in quantum mechanics. The famous Heisenberg uncertainty principle can be expressed in a relation

where *P* and *Q* are linear operators
that represent observations of momentum and position, respectively, and
*h* is Planck’s constant.

It is easy to show, however, that if these are conventional linear operators over a field (the real or complex numbers), then they have to be from an infinite dimensional space (!). That they are from an infinite-dimensional space is usually taken as one of the starting points of the theory.

Physics isn’t restricted to conventional linear algebra, however. If we relax the requirement that the operators are from a linear algebra over a field (which by definition, has a commutative multiplication operator) to be that the operators come from a linear algebra over the quaternions, the operator space no longer has to be infinite-dimensional. For many applications, such a relaxation is not appropriate, but for some, such as the solution of the equation for the electron, it is.

Objects related to quaternions arise from the solution of the Dirac equation for the electron. The non-commutativity is essential there.

The quaternions are closely related to the various “spin matrices” or “spinors” of quantum mechanics. Exactly how, I haven’t quite understood—partly, it seems, because the physics notion is a little hazy and refers to several different kinds of object.

The most widespread use of quaternions to date is in computer animation; there, they are used to represent transformations of orientations of graphical objects. They provide an elegant solution to problems that plagued early animated programs, gimbal lock, instability, and convenient interpolaton.

Let’s start by looking at the problems encountered with more naive representation of orientation transformations.

The gimbal lock problem has been known for a long time, particularly from gyroscopic avionics controllers. Here, a gyro would be mounted in two nested “gimbals”, each consisting of a ring with two pivots on which the next ring, and finally the gyro, is mounted. The pivots of the next ring are set at 90° from the ring on which it is mounted.

Now, it is possible to align the rings so they are all in one plane. In this configuration, the gyro cannot rotate easily about the axis perpendicular to the rings. This situation is called gimbal lock. In practice, the effect of gimbal lock is that the gyro swings wildly when even a small change is made to its orientation.

An analogous effect occurs in 3D animation, when 3×3 matrices are naively used to model changes of attitude of the viewer. Sometimes movement from one position to another will seem very unnatural, and sometimes get stuck, or swing wildly.

Mathematically, what has happened is, in the vicinity of the configuration where gimbal lock occurs, the angles yield transformation matrices with unboundedly large elements.

One solution to the gimbal problem is to use instead three gimbals, and arrange that their pivots never line up in a plane. This has been done, but it raises several other complications. With the conventional gyro described above, each attitude of the gyro housing corresponds uniquely to angles between the gimbals; if the number of gimbals is increased, this uniqueness is lost.

The modern solution for the 3D animation problem, considered to be by far the simplest and most robust, is to model the attitude of the viewer with quaternions.

Interpolating between quaternions produces smooth changes in attitude. Also, it is easy to compose quaternions to produce a composite rotation. Matrix and Quaternion FAQ

The loss of algebraic structure in each step of the Cayley-Dickson
construction is not an isolated mathematical oddity. It is deeply
related to the topological properties of manifolds (topological spaces
that locally look like
**R**^{n}).
It turns out that, after
**R**^{4}, the
topology of manifolds is greatly simplified.
(**R**^{3} is bad, but
**R**^{4} is absolutely
the worst.)

Finkelstein, Jauch, Schiminovich, and Speiser
Foundations of Quaternion Quantum Mechanics,
J. Math. Phys, **3** (1962) 207-220

Finkelstein, Jauch, Schiminovich, and Speiser
Some Physical Consequences of General Q-Covariance,
Helvetica Physica Acta, **35** (1962) 328-329

Finkelstein, Jauch, Schiminovich, and Speiser
Principle of General Q-Covariance,
J. Math. Phys. **4** (1963) 788-796.

William Rowan Hamiltion, On Quaternions

John Baez, The Octonions

Jack B. Kuipers, Quaternions and rotation sequences: a primer with applications to orbits aerospace and virtual reality, Princeton University Press, 1998

Deavours, C. A.,
The Quaternion Calculus,
Amer. Math. Monthly **80** (1973) 995-1008.

Guide to History of Calculus, keyed to Thomas’ Calculus