# A space more fundamental than Euclidean space

Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which is "more complicated". This raises several questions:

1. In what sense do pure spinors form a geometry? What sort of geometry is this?
2. Is there a precise sense in which this geometry is simpler than Euclidean geometry?
3. Did Cartan ever make such a direct statement, or is Budinich inferring that Cartan held such views from the overall philosophy of Cartan's work? What is the current status of this statement, e.g. vague philosophy, more or less precise body of conjecture, well-established theory?

Budinich also suggests that there is a connection of this more fundamental geometry to the geometry of minimal surfaces, in particular, to the property that they are generated by null vectors by means of the Weierstrass–Enneper parameterization. Is there some sort of geometry underlying the Weierstrass–Enneper parameterization that is more elementary or more fundamental than Euclidean geometry?

Background: In a filmed conversation (at $\color{blue}{\text{14:00}}$ mark) involving Paolo Budinich, Abdus Salam, Dennis Sciama, and Ed Witten following Witten's 1986 Dirac Medal lecture (the caption at the start of the film, dating it to 1990, is certainly in error), Budinich makes the following remark:

By which space-time is different than we thought. And by which I mean one could go back perhaps to Cartan who said the same thing in ’37, that Euclidean geometry is a very complicated geometry, while the geometry of zero vector is the more elementary one. Now there is a lot of zero vectors in these minimal surfaces—zero vectors. This was known 150 years ago, that zero vectors generate, by Weierstrass, minimal surfaces, so maybe that is also—that the Euclidean geometry is not the most elementary one.

It seems likely that Budinich meant to say "null vector" here rather than "zero vector".

The abstract of a contemporaneous paper by Budinich,

P. Budinich, Null vectors, spinors, and strings, Commun. Math. Phys. 107 455–465 (1986),

contains the following:

It is shown how, in the frame of the Cartan conception of spinors, the old theorems on minimal surfaces, as generated from null curves formulated by Enneper–Weierstrass (1864–1866) for 3-dimensional ordinary space, and by Eisenhart (1911) for 4-dimensional space time, may be reformulated in terms of complex 2- and 4- component projective spinors respectively.

A decade later, in the preprint Geometrical aspects of quantum mechanics in compactified momentum space, he writes of Cartan,

He stressed the simplicity and elegance of spinor geometry and, because of this, he formulated the hypothesis of its fundamental character, insofar all the properties of euclidean geometry may be naturally derived from it, by considering the euclidean vectors as squares of simple spinors* (their components as bilinear polynomia of spinor components) [1].

*The historical fact that euclidean geometry was discovered long ago, and was thereafter universally considered as the most elementary form of geometry, might be due to the fact that its elements like planes, lines, points, are well accessible to our common, everyday intuition, based on our optical sensorial perceptions, while spinors were discovered much later, in the frame of advanced mathematics, and they are less accessible to our intuition, in fact a simple spinor may be conceived as a totally null plane, that is a plane whose (null) vectors are all orthogonal to each other, however, once the abstract mathematical reasoning is adopted, one is easily convinced of their geometrical simplicity and elegance which induced Cartan to formulate his conjecture on their fundamental role in elementary geometry.

These quotations come from page 2 of the preprint. Reference [1] is E. Cartan, Leçon sur la Théorie des Spinors (Hermann, Paris, 1937), which was translated into English as The Theory of Spinors. The term "simple spinor" is used interchangeably with "pure spinor". Similar statements can be found in many of Budinich's papers, spanning several decades.

Remark: Michael Atiyah is quoted by G. Farmelo in his biography of Paul Dirac as having written

No one fully understands spinors. Their algebra is formally understood but their geometrical significance is mysterious. In some sense they describe the 'square-root' of geometry and, just as understanding the concept of the square root of $-1$ took centuries, the same might be true of spinors.

This seems to suggest that the idea that there is a simpler spinor geometry underlying Euclidean geometry is more of an uncompleted hope than a fully-developed theory.

• Would recommend trying MO, this doesn't look like it would be too out of place there. – Morgan Rogers Aug 6 '16 at 17:10