Non-locality and Twistor functions

Question

Is there a nice intuitive way to visualize the concept of non-locality associated to twistor functions? And how is it related to the type of non-locality we encounter in Quantum Mechanics?

Answer 1

The machinery of sheaf cohomology has important consequences in the description of mass-less spinor fields in Twistor theory(Penrose Transform) and as an indicator for Quantum Contextuality.

Formally, given a bundle $(\mathscr{O},\pi,X)$ (for simplicity regard $\mathscr{O}$ as an abelian group with $\pi$ a local homeomorphism), the non-triviality of first cohomology group $H^1(X;\mathscr{O})$ gives an indication that one can't construct a globally defined cross-section by patching up local sections. In other words we have a family of data on $X$ which is locally consistent but globally inconsistent.

For visual representation, consider the example of impossible triangle which was originally due to Penrose (also check this related post):

The recipe to construct the cohomological group for above diagram is illustrated in the above linked article. In short, we have the diagram on the left which we break into three parts $Q_1,Q_2,Q_3$ (representing "local sections"). Without a priori knowledge about the size of each subpart $Q_i$, we define the ratio $d_{ij}=\frac{|\textbf{E}-\textbf{A}_{ij}|}{|\textbf{E}-\textbf{A}_{ji}|}$ from some fixed observer $\textbf{E}$. If $d_{ij}=q_i/q_j$ (i.e. a co-boundary), the group $H^1(Q;\mathbb{R}^+)$ is just $\{1\}$ (trivial). This would mean that if we place the three pieces $Q_i$'s at distance $q_i$'s from observer at $E$, then the whole arrangement will be indistinguishable from the impossible triangle. Non-triviality of $H^1(Q;\mathbb{R}^+)$ indicates that one can't glue the three pieces $Q_i|_{i=1,2,3}$ in any meaningful way to obtain the whole diagram.

We may generalize the above construction by considering a set of propositions which are not simultaneously satisfiable, e.g. say we have consistent propositions $\{\phi_i\}_{i=1}^{N-1}$ with associated probability $\{p_i\}_{i=1}^{N-1}$ and define the $N$th proposition to be true when at least one of the proposition is false, i.e. $\phi_N\implies \vee_{i=1}^{N-1}\neg\phi_i$. Assuming standard probability rules to hold, we obtain a Bell type inequality $\sum_{i=1}^Np(\phi_i)\leq N-1$. In fact any Bell inequality can be constructed in this way (see logical Bell inequalities). Experiments covering Bell or Kochen-Specker type scenarios can be given a bundle description as above where one could regard $X$ as a finite set of measurements and stalks $\mathscr{O}_x$ a finite non-empty set of possible outcomes for each $x\in X$. Existence of global sections will imply existence of local deterministic hidden variable model (see here)

In twistor theory, we see that classical mass-less spinor fields $\phi$ can be represented in terms of cocycles $f\in H^1(\mathbb{PT}^+;\mathscr{O}(\pm n-2))$; i.e. a notion of contextuality is already present at the level of twistor space . But, is this the same contextuality we see in Quantum Mechanics? Is there a natural "on-shell" way to define measurements in Twistor space which can be given a bundle description as above? This answer, as of now, is incomplete.