A space more fundamental than Euclidean space

Question

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.

Answer 1

I can't say with any authority what Cartan had in mind, and I can't claim to have any real insight into the geometry of spinors, but here are some simple things one can say from the point of view of representation theory and Klein geometry.

The idea of Klein geometry is that what distinguishes different "geometries" is their groups of symmetries. So, for example, Euclidean geometry in $n$ dimensions is associated with the orthogonal group $O(n)$ or special orthogonal group $SO(n)$ and its usual vector representation on $\mathbb{R}^n$. From this point of view "spinor geometry," whatever that means, is associated with the spin group $Spin(n)$ and its spinor representations.

Let's see how this works when $n = 3$. Here "Euclidean geometry" means the action of the special orthogonal group $SO(3)$ on $\mathbb{R}^3$. The corresponding spin group $Spin(3)$ is the universal cover of $SO(3)$ which, as is well known, is the special unitary group $SU(2)$. Here the corresponding spinor representation is the usual representation $V$ of $SU(2)$ on $\mathbb{C}^2$.

One can obtain the vector representation of $SO(3)$ from the spinor representation $V$ of $SU(2)$ as follows. Starting from $V$ we can construct the tensor product $V \otimes V^{\ast}$, which can be canonically identified with the algebra $M_2(\mathbb{C})$ of complex $2 \times 2$ matrices, on which $SU(2)$ acts by conjugation. From here we can consider the trace $\text{tr} : V \otimes V^{\ast} \to \mathbb{C}$ and take its kernel, giving us the action of $SU(2)$ on traceless complex $2 \times 2$ matrices; this is the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. This is a $3$-dimensional complex representation, and we can cut it down to a $3$-dimensional real representation by identifying a suitable map $c : \mathfrak{sl}_2(\mathbb{C}) \to \mathfrak{sl}_2(\mathbb{C})$ which plays the role of complex conjugation and is also preserved by $SU(2)$. This map can be constructed using a suitable isomorphism $j : V \to V^{\ast}$ but the details seem a bit annoying to work out and in any case aren't too important to this story. The usual Euclidean inner product is built out of the Killing form.

The point of saying all this is that

the spinor representation can be used to construct the vector representation but not conversely.

This is the sense in which the spinor representation is "more fundamental" than the vector representation. From the point of view of the spinor representation, the vector representation no longer appears "fundamental"; instead its elements are built out of pairs of spinors. (I think this is one way to make precise that "square root of geometry" claim but I'm not sure.)

The physically relevant version of this story concerns the indefinite spin group $Spin(3, 1)$, which turns out to be $SL_2(\mathbb{C})$. It has two spinor ("half-spinor"?) representations, namely the usual representation $V$ of $SL_2(\mathbb{C})$ on $\mathbb{C}^2$ and its dual, which I believe are referred to as $\mathbf{2}$ and $\mathbf{2}^{\ast}$ in physics. It is again the case that the vector representation of $SO(3, 1)$ on $\mathbb{R}^{3, 1}$ can be constructed from the spinor representations of $Spin(3, 1)$ but I am even less clear on the specific details; I believe the vector representation shows up inside $\mathbf{2} \otimes \mathbf{2}^{\ast}$ and the Lorentzian inner product can again be built out of the Killing form, so it is again the case, as above, that vectors can be thought of as built out of pairs of spinors, and it is again the case that from this point of view spinors are "more fundamental" than vectors.

Answer 2

This is an interesting question, and I have a few thoughts. I unfortunately do not have an answer to question 3 however, but maybe a guess from what I know historically.

For question 1 & 2, I will take the view that Euclidean space is a fiber-bundle. It is also a Cartan Geometry, however I am not an expert here and I digress. Nevertheless, Euclidean space can be made by taking the $N$-dimensional Euclidean group and quotienting out the group $SO(p,q)$, such that $p+q=N$. Then we can talk about equivalence up to rotations. We can also translate objects because the space is flat and talk about equivalence up to translation and rotation.

For spinors, I know that in 4D they can be described by the group $Spin(p,q)$ which happens to be isomorphic to $SL(2,\mathbb{C})$ (More generally however, $Spin(p,q)$. Now, to my knowledge, $Spin(p,q)$ is almost $SO(p,q)$ but with antipodal points identified (double cover). Topologically, the fundamental group is different. Again, I am not an expert here and welcome clarifying comments.

I do not know what happens in $N$-dimensions, but in $4D$ you can think of spinors as a fiber bundle over a manifold ($\mathbb{R}^4$ for example & consistency) with structure group $SL(2,\mathbb{C})$. In this case, for a Lorentzian manifold, you get a splitting from the manifold to two spinor spaces (these are vector spaces). This dimensional reduction simplifies things calculation wise, and comes with some nifty and incredibly useful theorems. I would say at least in this limited context that this is what makes spinor geometry simpler than Euclidean geometry. The dimension of the spinor space being less than that of the corresponding Euclidean/Lorentzian space is precise for question 2; there is more, but I hope that helps!

For question 3, Cartan thought in terms of bundles a lot, it is one amazing thing that let him do so much. I am unsure if Cartan ever made a direct statement, but more "elementary" or more "fundamental makes sense in terms of the group structures occurring. This is just my opinion however. Maybe they are more elementary or fundamental because $\mathbb{C}$ is algebraically closed, and in that sense easier to work with (factoring of polynomials, etc.)

Again, I'd welcome comments expanding on this. To be specific, I would call it a philosophical statement probably made by Cartan (I think I saw it in his spinor book, but I am not sure... or maybe it was Weyl or Penrose? ("The Classical Groups" & "Spinors and Spacetime volume 1), since "more elementary" or more "fundamental" can be interpreted differently in different contexts. Hope this helps!

Answer 3

Consider a scalar as having only magnitude (unit).

Consider a vector as having a magnitude and direction (unit up or unit down).

Now consider a spinor as having only direction (up and down).

So naturally, if a spinor only has direction, when you "rotate" anything it will change orientations, flip from up to down, the only rotation available to it.

Hence her use of "zero" not "null" vector as a zero vector has zero magnitude but still has direction while a null vector would have neither direction or magnitude (as I understand it).

Thus a spinor is "elementary" given that it embodies only direction, think of the "spin" of the electron or photon, while Euclidean space embodies both direction and magnitude.