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The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth Brecher: http://archive.bridgesmathart.org/2013/bridges2013-469.pdf  ). 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 http://adsabs.harvard.edu//abs/1928RSPSA.121..524E 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: http://www.jstor.org/discover/10.2307/2370888?uid=3738936&uid=2129&uid=2&uid=70&uid=4&sid=21103286171827

In fact Eddington discovered Majorana spinors and the modern account of this connection can be found in papers Some remarks on the algebra of Eddington's E-numbers by Nikos Salingaros http://link.springer.com/article/10.1007%2FBF00738296 and The Kummer Configuration and the Geometry of Majorana Spinors by Gary W. Gibbons http://link.springer.com/chapter/10.1007/978-94-011-1719-7_5

It is known (originally due to Majorana) that quantum mechanics of photons can also be based on a dirac-like equation: http://link.springer.com/article/10.1007%2FBF02812391 (see also Photon Wave Function by Iwo Bialynicki-Birula http://www.sciencedirect.com/science/article/pii/S0079663808703160 ). Is it possible to extend Eddingtons considerations in this case too? And what projective geometry structure will be behind it? I would be grateful for references about modern mathematical meaning(s) of Eddington's construction.

(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.

                 (source)

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth Brecher: http://archive.bridgesmathart.org/2013/bridges2013-469.pdf ). 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 http://adsabs.harvard.edu//abs/1928RSPSA.121..524E 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: http://www.jstor.org/discover/10.2307/2370888?uid=3738936&uid=2129&uid=2&uid=70&uid=4&sid=21103286171827

In fact Eddington discovered Majorana spinors and the modern account of this connection can be found in papers Some remarks on the algebra of Eddington's E-numbers by Nikos Salingaros http://link.springer.com/article/10.1007%2FBF00738296 and The Kummer Configuration and the Geometry of Majorana Spinors by Gary W. Gibbons http://link.springer.com/chapter/10.1007/978-94-011-1719-7_5

It is known (originally due to Majorana) that quantum mechanics of photons can also be based on a dirac-like equation: http://link.springer.com/article/10.1007%2FBF02812391 (see also Photon Wave Function by Iwo Bialynicki-Birula http://www.sciencedirect.com/science/article/pii/S0079663808703160 ). Is it possible to extend Eddingtons considerations in this case too? And what projective geometry structure will be behind it? I would be grateful for references about modern mathematical meaning(s) of Eddington's construction.

(source)

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth Brecher: http://archive.bridgesmathart.org/2013/bridges2013-469.pdf ). 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 http://adsabs.harvard.edu//abs/1928RSPSA.121..524E 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: http://www.jstor.org/discover/10.2307/2370888?uid=3738936&uid=2129&uid=2&uid=70&uid=4&sid=21103286171827

In fact Eddington discovered Majorana spinors and the modern account of this connection can be found in papers Some remarks on the algebra of Eddington's E-numbers by Nikos Salingaros http://link.springer.com/article/10.1007%2FBF00738296 and The Kummer Configuration and the Geometry of Majorana Spinors by Gary W. Gibbons http://link.springer.com/chapter/10.1007/978-94-011-1719-7_5

It is known (originally due to Majorana) that quantum mechanics of photons can also be based on a dirac-like equation: http://link.springer.com/article/10.1007%2FBF02812391 (see also Photon Wave Function by Iwo Bialynicki-Birula http://www.sciencedirect.com/science/article/pii/S0079663808703160 ). Is it possible to extend Eddingtons considerations in this case too? And what projective geometry structure will be behind it? I would be grateful for references about modern mathematical meaning(s) of Eddington's construction.

Kummer surface plaster model http://www.inp.nsk.su/%7Esilagadz/Kummer_Surface.jpg

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth Brecher: http://archive.bridgesmathart.org/2013/bridges2013-469.pdf ). I was surprised to learn that this fancy surface has something to do with the quantum theory of relativistic electrons. In 1928 paper A Symmetrical Treatment of the Wave Equation http://adsabs.harvard.edu//abs/1928RSPSA.121..524E 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 summution 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: http://www.jstor.org/discover/10.2307/2370888?uid=3738936&uid=2129&uid=2&uid=70&uid=4&sid=21103286171827

In fact Eddington discovered Majorana spinors and the modern account of this connection can be found in papers Some remarks on the algebra of Eddington's E-numbers by Nikos Salingaros http://link.springer.com/article/10.1007%2FBF00738296 and The Kummer Configuration and the Geometry of Majorana Spinors by Gary W. Gibbons http://link.springer.com/chapter/10.1007/978-94-011-1719-7_5

It is known (originally due to Majorana) that quantum mechanics of photons can also be based on a dirac-like equation: http://link.springer.com/article/10.1007%2FBF02812391 (see also Photon Wave Function by Iwo Bialynicki-Birula http://www.sciencedirect.com/science/article/pii/S0079663808703160 ). Is it possible to extend Eddingtons considerations in this case too? And what projective geometry structure will be behind it? I would be grateful for references about modern mathematical meaning(s) of Eddington's construction.

               (source)

The picture shows 1936 photograph by Man Ray of a Kummer surface plaster model in the collection of the Institute Henri Poincare (from Mathematics, Art and Science of the Pseudosphere by Kenneth Brecher: http://archive.bridgesmathart.org/2013/bridges2013-469.pdf ). 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 http://adsabs.harvard.edu//abs/1928RSPSA.121..524E 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: http://www.jstor.org/discover/10.2307/2370888?uid=3738936&uid=2129&uid=2&uid=70&uid=4&sid=21103286171827

In fact Eddington discovered Majorana spinors and the modern account of this connection can be found in papers Some remarks on the algebra of Eddington's E-numbers by Nikos Salingaros http://link.springer.com/article/10.1007%2FBF00738296 and The Kummer Configuration and the Geometry of Majorana Spinors by Gary W. Gibbons http://link.springer.com/chapter/10.1007/978-94-011-1719-7_5

It is known (originally due to Majorana) that quantum mechanics of photons can also be based on a dirac-like equation: http://link.springer.com/article/10.1007%2FBF02812391 (see also Photon Wave Function by Iwo Bialynicki-Birula http://www.sciencedirect.com/science/article/pii/S0079663808703160 ). Is it possible to extend Eddingtons considerations in this case too? And what projective geometry structure will be behind it? I would be grateful for references about modern mathematical meaning(s) of Eddington's construction.