In the last posting I explained how a fundamental problem shows up if you try to Wick rotate a Weyl spinor. Wick rotation is supposed to be analytic continuation in a four dimensional complex spacetime which, in terms of spinors, is $S_L\otimes S_R$ (or equivalently, linear maps from $S_R^*$ to $S_L$). This is acted on by the complex spin group $$Spin(4,\mathbf C)=SL(2,\mathbf C)_L\times SL(2,\mathbf C)_R$$ where the two factors act on the two kinds of spinors. The subgroup preserving the four real dimensional Minkowski spacetime is $SL(2,\mathbf C)$, embedded by
$$g\in SL(2,\mathbf C)\rightarrow (\overline g, g)\in SL(2,\mathbf C)_L\times SL(2,\mathbf C)_R$$
The subgroup preserving the four real dimensional Euclidean spacetime is
$$Spin(4)=SU(2)_L\times SU(2)_R$$
so pairs of $SL(2,\mathbf C)$ elements that are in $SU(2)$.
In Minkowski spacetime you only need one kind of spinor (and its complex conjugate), but analytic continuation to Euclidean spacetime requires two independent kinds. You have to introduce Dirac spinors (pairs of Weyl spinors with both chiralities) to do this kind of Wick rotation. My “right-handed spacetime” proposal is essentially that you should find a version of Wick rotation that just uses $S_R$, not analytically continuing spinors in complex spacetime in the usual way.
A way to do this is to use twistor theory, where a point in spacetime can be identified with the spin space $S_R$ at the point. Twistor space is a space $T=\mathbf C^4$, with a complex spacetime point a $\mathbf C^2\subset T$. Projectivizing (modding out by the action of non-zero complex numbers), projective twistor space is $PT=\mathbf CP^3$, and spacetime points are $\mathbf CP^1$s inside $PT$. In twistor theory one can exploit the complex structure of the complex three dimensional space $PT$ rather than that of the complex four dimensions parametrizing the space of $\mathbf CP^1$s.
To understand how Minkowski spacetime appears in $PT$, it may be a good idea to start with something simpler that has much of the same structure, thinking about $\mathbf CP^1$ instead of $\mathbf CP^3$. $\mathbf CP^1$ is the space of complex lines through the origin in $\mathbf C^2$, and it is acted on transitively by $SL(2,\mathbf C)$. It is a version of the Riemann sphere, with $SL(2,\mathbf C)$ the group of conformal transformations acting on the sphere. The unitary subgroup $SU(2)$ is a real form of $SL(2,\mathbf C)$ and also acts transitively on this sphere. One can construct finite dimensional representations of $SU(2)$ or $SL(2,\mathbf C)$ on spaces of holomorphic sections of holomorphic line bundles on $\mathbf CP^1$ (Borel-Weil), or on cohomology spaces $H^1$ (Borel-Weil-Bott).
If one instead looks at the other real form, $SL(2,\mathbf R)=SU(1,1)$, one finds that it acts on $\mathbf CP^1$ with three orbits: the upper hemisphere $\mathbf CP^1_+$, the lower hemisphere $\mathbf CP^1_-$, and the equator that is their boundary. One can construct infinite dimensional discrete series unitary representations of $SU(1,1)$ on holomorphic sections of holomorphic line bundles over $\mathbf CP^1_+$ or $\mathbf CP^1_-$ (or on cohomology spaces $H^1$). These representations can be characterized by their behavior at the boundary $S^1$, where one can make various choices of function space, with hyperfunctions a very natural one.
The Cayley transform relates the action of $SL(2,\mathbf R)$ on $\mathbf C=\mathbf R^2$ (with orbits the upper/lower half planes and the real number line) to the action of $SU(1,1)$ on $\mathbf CP^1$ as above. The action of $SL(2,\mathbf R)$ on the upper half plane is a central object in mathematics, especially in number theory (which comes into play through the subgroup $SL(2,\mathbf Z)$).
One way to understand how Minkowski spacetime appears in twistor theory is to generalize the above story from $\mathbf CP^1$ to $\mathbf CP^3$. One now has $SL(4,\mathbf C)=Spin(4,2,\mathbf C)$ and its real form $SU(4)$ acting transitively. The analog of $SU(1,1)$ is another real form, $SU(2,2)=Spin(4,2)$, which again acts with three orbits: $PT^+, PT^-$ and their common boundary, which we’ll call $N$.
$SU(2,2)=Spin(4,2)$ is the conformal group of Minkowski spacetime. In twistor theory, points in Minkowski spacetime are the $\mathbf CP^1$s that lie inside $N$. $N$ can be identified physically with the space of light-rays in Minkowski spacetime, and topologically with $S^3\times S^2$. The Penrose transform identifies solutions of massless wave equations in Minkowski spacetime with representations of $SU(2,2)$ on infinite dimensional spaces of holomorphic sections of holomorphic line bundles over $PT^+$ or $PT^-$ (or cohomology spaces $H^1$). As in the $\mathbf CP^1$ case, it is natural to characterize the function spaces involved as spaces of hyperfunctions on $N$.
What about Wick rotation in this context? Something happens in the twistor case that did not happen in the $\mathbf CP^1$ case. One can choose an identification of $\mathbf C^4$ with $\mathbf H^2$ and get a different real form of $Spin(4,2,\mathbf C)$, $SL(2,\mathbf H)=Spin(5,1)$, which acts transitively on $PT$. This is the Euclidean signature conformal group. Using the identification $\mathbf C^4=\mathbf H^2$, acting on a point in $\mathbf CP^3$ by quaternion multiplication, gives a fibering of $\mathbf CP^3$ over $S^4=\mathbf HP^1$ with fiber $\mathbf CP^1=S^2$. $S^4 $ is conformally compactified Euclidean space-time, with its points identified with the fibers in $PT$.
From the point of view that Minkowski spacetime is all $\mathbf CP^1$s inside $N$, Wick rotation starts with a choice of which Euclidean spacetime we want to Wick rotate to. This will be characterized by the $S^3\subset S^4$ where they intersect. $N$ will be fibered over this $S^3$ by $S^2$s: it no longer is just topologically $S^3\times S^2$ but there is an explicit geometrical identification.
I’m running out of time today, and want to get to a blog posting explaining how this is all supposed to relate to physics. I may or may not add a bit more to this later, or just go on tomorrow to the next topic.
