Electromagnetism is thoroughly misunderstood in modern physics although it has a highly quantitatively successful theory describing it via the Maxwell’s equations with symmetry. The universe has four rather than three macroscopic spatial dimensions and electromagnetism really has symmetry and is in reality a four-dimensional force — indeed, it is the ‘grand unification force’. The purpose of this note is to explicate how the Maxwell’s equations for electromagnetism arise naturally from the scalar wave equation on quaternions, in the process naturally producing Clifford algebras, spinors, and other parts of the machinery that is invoked during the study of Dirac operators in geometry.

Geometric machinery for Dirac operators have been built up in the following setting: for an even dimensional oriented riemannian manifold M, one defines the Clifford bundle C(M) whose fiber at each point is isomorphic to the Clifford algebra of the cotangent space. Recall that the Clifford algebra is isomorphic to the exterior algebra of a vector space but instead of the exterior product, the product operation on it satisfies . Then a Clifford module over is a vector bundle on which the Clifford module acts. A connection on compatible with the Clifford action is a Clifford connection, and then a Dirac operator is defined on sections on E as a composition of the connection and the Clifford action. On the other hand, there is much simpler level at which this machinery can be unpacked and give us a much simpler view of structure of electromagnetism. If we take the Dirac square root of the scalar wave equation on quaternions, we will naturally motivate this entire machinery naturally as follows: the Clifford algebra of a real four-dimensional Euclidean space consists of 2×2 matrices of quaternions, the spinors are pairs of quaternions, and the Dirac operator then becomes an operator on pairs of quaternions which squares to Laplacian on both factors. The square root of the scalar wave equation produces the classical Maxwell’s equations with “curl” replaced by the Dirac operators.

Recall that mathematically, waves are described by the scalar wave equation. A function that describes a wave with speed satisfies the equation

where is the standard Laplacian, which is just the sum of second space derivatives on and can be defined in terms of the metric on a riemannian manifold. This is a fairly generic equation describing waves. It is an interesting observation that this simple wave equation on quaternions automatically produce Maxwell’s equations, including the decoupling of electromagnetic fields, spinors, and Clifford algebra when we take the “Dirac square root”. Mathematically, this is a fairly simple exercise but it is a less prosaic observation physically, because it tells us that electromagnetism is a phenomenon that is encoded in the structure of the universe itself and not something that requires design by a deity, for example. With this in mind, the claim of S4 physics, that 4D electromagnetism is the grand unification force, gains potency.