# Kummer's quartic surface and the Dirac operator (source)

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincaré (from Mathematics, Art and Science of the Pseudosphere by Kenneth Brecher). I was surprised to learn that this fancy surface has something to do with the quantum theory of relativistic electrons.

In a 1928 paper A Symmetrical Treatment of the Wave Equation, Arthur Eddington, disappointed that Dirac’s equation did not appear in tensor form, tried to reformulate it in a tensor language and introduced so called E-numbers. Eddington's E-number is a linear combination (real or complex) of the imaginary unit i and fifteen basic elements $E_{\mu\nu}$, $\mu<\nu$, $\mu,\nu=0,1,2,3,4,5$, with the following properties (no summation is assumed over repeated indices) $$E_{\mu\nu}=-E_{\nu\mu},\;\;E_{\mu\nu}*E_{\mu\nu}=-1,$$ $$E_{\mu\nu}*E_{\mu\sigma}=-E_{\mu\sigma}*E_{\mu\nu}=E_{\nu\sigma},\;\; E_{\mu\nu}*E_{\sigma\tau}=E_{\sigma\tau}*E_{\mu\nu}=\pm i E_{\lambda\rho},$$ where $\mu,\nu,\sigma,\tau,\lambda,\rho$ are all different from each other and in the last relation the positive or negative sign is taken according as $(\mu,\nu,\sigma,\tau,\lambda,\rho)$ is an even or odd permutation of $(0,1,2,3,4,5)$. Soon (in 1932) it was shown by Oscar Zariski that the (projective) geometry behind this algebraic system was that of Kummer's quartic surface.

In fact Eddington discovered Majorana spinors and the modern account of this connection can be found in the papers Some remarks on the algebra of Eddington's E-numbers by Nikos Salingaros and The Kummer Configuration and the Geometry of Majorana Spinors by Gary W. Gibbons.

It is known (originally due to Majorana) that the quantum mechanics of photons can also be based on a Dirac-like equation (see also Photon Wave Function by Iwo Bialynicki-Birula. Is it possible to extend Eddington's considerations in this case too? And what projective geometry structure will be behind it? I would be grateful for references about the modern mathematical meaning(s) of Eddington's construction.

It is hard to imagine that Eddington's numbers could be anything but the imaginary part of the Clifford algebra $C$ of Minkowski space. Recall that the Clifford algebra for an n-dimensional real vector space endowed with a quadratic form of any signature has the same dimension as the full Grassmann algebra over that vector space, so is of dimension $2^n$. If the inner product is nondegenerate then it induces a nondegenerate inner product on the Clifford algebra and so, for $n=4$ the imaginary part of the Clifford algebra, by which I mean the orthogonal complement of the unit $1$ in $C$. has dimension $15$.