# Kummer's quartic surface and the Dirac operator

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.

## 2 Answers

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 $15$.

But this may also be a dimensional coincidence. (I doubt it.)

This question is interesting. I have done a little work in this direction, looking at brachistochrones over the Dirac matrices, like a geodesic idea. Here's a link to a paper that might be relevant: "Time Dependent Biqubits", Peter Morrison (Wayback Machine link) (current link).

I agree with Prof. Eddington in that there are definitely links between tensors or complex matrices and the Dirac equation. My work looked at whether it was possible to formulate the Dirac equation in terms of fundamental non-commutative matrix calculus instead of the standard approach that uses differential operators to define the algebra. It is, but by the same caveat, I would say it is a a little more involved; however, the simplicity of representation is attractive.

Thank you for posting this idea. It's good to know I'm not alone in looking at the properties of these types of curious surfaces etc. Incidentally, in looking at geodesics on a 4-dimensional submanifold of a 16-dimensional space, the electron rest mass came out neatly as a constant of motion, consistent with what we observe.