I’ve just replaced the old version of my draft “spacetime is right-handed” paper (discussed here) with a new, hopefully improved version. If it is improved, thanks are due to a couple people who sent helpful comments on the older version, sometimes making clear that I wasn’t getting across at all the main idea. To further clarify what I’m claiming, here I’ll try and write out an informal explanation of what I see as the relevant fundamental issues about four-dimensional geometry, which appear even for $\mathbf R^4$, before one starts thinking about manifolds.

**Spinors, twistors and complex spacetime**

In complex spacetime $\mathbf C^4$ the story of spinors and twistors is quite simple and straightforward. Spinors are more fundamental than vectors: one can write the space $\mathbf C^4$ of vectors as the tensor product of two $\mathbf C^2$ spaces of spinors. Very special to four dimensions is that the (double cover of) the complex rotation group $Spin(4,\mathbf C)$ breaks up as the product

$$Spin(4,\mathbf C)=SL(2,\mathbf C)\times SL(2,\mathbf C)$$

where these two factors act on the spinor spaces.

While spinors are the irreducible objects for understanding complex four-dimensional rotations, twistors are the irreducible objects for understanding complex four-dimensional conformal transformations. Twistor space $T$ is a $\mathbf C^4$, with complex conformal transformations acting by the defining $SL(4,\mathbf C)$ action. A complex spacetime point is a $\mathbf C^2\subset T$ and conformally compactified complex spacetime is the Grassmannian of all such $\mathbf C^2\subset T=\mathbf C^4$. One of the spinor spaces at each point of complex spacetime is tautologically defined: it’s the point $\mathbf C^2$ itself (the other is of a different nature, with one definition the quotient space $T/\mathbf C^2$).

**Real forms**

While the twistor/spinor story for complex spacetime is quite simple, the story of real spacetime is much more complicated. When several different real spaces complexify to the same complex space, these are called “real forms” of the space. A real form can be characterized by a conjugation map $\sigma$ (an antilinear map on the complex space satisfying $\sigma^2=1$), with the real space the conjugation-invariant points. Using the obvious conjugation on $\mathbf C^4$, we get an easy to understand real form: the $\mathbf R^4$ with real coordinates, rotation group $SL(2,\mathbf R)\times SL(2,\mathbf R)$ and conformal group $SL(4,\mathbf R)$. Unfortunately, this real form seems to have nothing to do with physics, its invariant inner product is indefinite of signature $(2,2)$.

The real spacetime with Euclidean signature inner product has an unusual conjugation that is best understood using quaternions. If one picks an identification of the twistor space $T$ as $T=\mathbf C^4=\mathbf H^2$, then the conjugation is multiplication by the quaternion $\mathbf j$. The Euclidean conformal group is the group $SL(2,\mathbf H)$. The spinor spaces $\mathbf C^2$ are identified with two copies of the quaternions $\mathbf H$, with the rotation group now the group $Sp(1)\times Sp(1)$ of pairs of unit quaternions.

In this case the conjugation acts in a subtle manner. Since $\mathbf j^2$ is $-1$ rather than $1$, it’s not a conjugation on $T$, but is one on the projective space $PT=\mathbf CP^3$. It has no fixed points, so the twistor space has no real points. What is fixed are the quaternionic lines $\mathbf H\subset \mathbf H^2$, each of which corresponds to a point in the (conformally compacified, so $S^4=\mathbf HP^1$) real Euclidean signature spacetime. Using the decomposition as a tensor product of spinors, the action by $\mathbf j$ squares to $-1$ on each factor, but $1$ on the tensor product, where it gives a conjugation with fixed points the Euclidean spacetime.

The real spacetime with Minkowski signature is another real form of a subtle sort, with very different subtleties than in the Euclidean case. The conjugation $\sigma$ in this case doesn’t take the twistor space $T$ to itself, but takes $T$ to its dual space $T^*$. It takes spinors of one kind to spinors of the opposite kind (at the same time conjugating spinor coordinates to get anti-linearity). The Minkowski signature conformal group is the group $SU(2,2)$ and the rotation group is the Lorentz group $SL(2,\mathbf C)$ (acting diagonally on the two spinor spaces, with a conjugation on one side).

**Some philosophy**

The usual way in which the above real forms get used is that mathematicians ignore the Minkowski story and use the Euclidean signature real form to do four-dimensional Riemannian geometry, with the $Sp(1)\times Sp(1)$ decomposition at the Lie algebra level corresponding to the decomposition of two-forms into self-dual and anti-self-dual. Physicists on the other hand (especially Penrose and his school, but also those trying to do quantum gravity using Ashtekar variables) ignore the Euclidean story and use the Minkowski signature real form. In various places Penrose is quoted as explicitly skeptical of any relevance of the Euclidean story to physics. Working just with the Minkowski real form, one struggles with the fact that the Lorentz group is simple, but that one can get a very useful self/anti-self dual decomposition if one makes one’s variables complex.

The point of view I’m taking is that Wick rotation tells one that one should look simultaneously at both Euclidean and Minkowski real forms, understanding how to get back and forth between them. This is standard in usual geometry where one just looks at vectors, but looking at spinors and twistors shows that something much more subtle is going on. The argument of this new paper is that when one does this, one finds that the spacetime degrees of freedom can be expressed purely in terms of one kind of spinor (right-handed by convention), the one that twistor theory tautologically associates to each point in spacetime. The other (left-handed) half of the spinor geometry involves a purely internal symmetry from the point of view of Minkowski spacetime. This should correspond to the electroweak gauge theory, exactly how that works is still under investigation…

**Update**: Now posted on the arXiv here. Only reaction on social media I’ve seen so far is from Strinking42069, which seems to be a parody account trying to make fun of string theorists.