A serious limitation of the Schrödinger equation is that it is not compatible with relativity. The Dirac equation solved that problem; it unites the concepts of quantum mechanics and special relativity, describing the quantum properties of particles like electrons, protons, neutrinos and quarks. The analysis of this equation elegantly explained some of the more elusive particle properties like spin, and provided a solid foundation for the so-called Pauli Exclusion Principle needed to explain atomic structure and the periodic table. Last but not least, the equation predicted the existence of antimatter: the fact that for any particle ‘species’ there exists an associated species with exactly the opposite properties (such as charge), but the same mass.

In spite of its tremendous successes, the Schrödinger equation had a serious drawback: it was not compatible with special relativity. This may be inferred from the fact that in the equation the space and time variables x and × do not appear on equal footing: it contains a first derivative with respect to time, but a second derivative with respect to the spatial coordinates. Dirac solved this problem with the equation carrying his name.

The Dirac equation has quite an involved mathematical structure, which is somewhat hidden by the compact notation, so let us take some time to comment on the notation used. There is an index μ which can take the values 0, 1, 2 or 3, indicating time and the three space components, indeed appearing on equal footing. The four Aμ fields, called ‘electromagnetic potentials’, describe the electromagnetic field in which (for example) the electron moves, and me is the electron mass. The electron field is here described by a four-component function ψ. The so-called ‘gamma matrices’ γμ are four numerical matrices (4×4 arrays of given numbers) which have to be multiplied in a standard mathematical way with the components of ψ. (Actually, we have suppressed an extra component index on ψ to prevent the notation from becoming even more involved).

Analysis of the equation revealed the meaning of the four components of the Dirac field. It includes the description of the somewhat mysterious property called spin, best described as some intrinsic rotational degree of freedom. We could say that the electron is the quantum equivalent of a tiny spinning top – and it can be left- or right-handed.