The twistor correspondence, at least in Penrose’s original formulation, is a natural isomorphism between sheaf cohomology classes on a real hypersurface of complex projective 3-space (i.e. twistor space) and massless Yang–Mills fields on Minkowski space. Mathematically speaking, the twistor space is the Grassmannian of lines in complexified Minkowski space, and the massless Yang–Mills fields correspond to the Grassmannian of planes in the same space. The correspondence space is therefore the Grassmannian of lines in planes in complexified Minkowski space, and it somehow encodes both the quantum mechanical structure of the Yang–Mills fields, and the geometrical structure of the underlying spacetime. This is directly analogous to the definition of the multiway causal graph, whose causal edges between branchlike-separated updating events encode the quantum mechanical structure of the multiway evolution graph, and whose causal edges between spacelike-separated updating events encode the relativistic structure of a pure (spacetime) causal graph.

]]>The relationship to higher-order mathematics, specifically homotopy theory and higher category theory, and their geometrical incarnation in the form of derived geometry, is somewhat more speculative. A possible connection exists via the “snake states” of multiway evolution, as described in our answer to the question about string theory above. The essential idea here would be to use the so-called “cobordism hypothesis”—a theorem which implies that functors on monoidal (∞,

Therefore, a black hole in the multiway causal graph may be characterized by the presence of two distinct horizons: a standard event horizon corresponding to regular causal disconnection, and an entanglement event horizon corresponding to multiway disconnection, which always lies strictly on the exterior of the causal event horizon. As such, from the point of view of an external observer in the multiway causal graph watching an infalling object to a black hole, the object will appear to “freeze” (due to quantum Zeno effects that are the multiway analog of time dilation) at the entanglement horizon, and will never get close to the true causal event horizon. Since Hawking radiation (which occurs as a consequence of non-convergent branch pairs in the multiway evolution graph) is emitted from the entanglement horizon and not the causal event horizon, the particles that get radiated from the black hole may be perfectly correlated with the information contained within the infalling object, without any apparent or actual violation of special relativity (since no information ever crossed a spacetime event horizon), thus resolving the black hole information paradox.

This resolution is formally quite similar to the standard resolution to the black hole information paradox implied by the holographic principle and the AdS/CFT duality.

Recall that the multiway causal graph encodes both the structure of the (purely quantum mechanical) multiway evolution graph, as well as the structures of the (purely relativistic) causal graphs corresponding to each branch of multiway evolution. Therefore, one can imagine “walling off” a certain bundle of causal edges in the multiway causal graph corresponding to some particular branch of multiway evolution, such that all of the causal edges inside the boundary of the wall correspond to edges in a purely relativistic causal graph (i.e. they designate causal relations between events in spacetime), whilst all of the causal edges intersecting the boundary of the wall correspond to edges in a purely quantum mechanical multiway graph (i.e. they designate causal relations between events in branchtime). As such, one immediately obtains a duality between the bulk gravitational theory on the interior of the wall, and the boundary quantum mechanical theory on the surface of the wall, just as in AdS/CFT. ]]>

More precisely, a formal statement of the ER=EPR conjecture is that the Bekenstein–Hawking entropy of a pair of entangled black holes is equivalent to their entanglement entropy. If Hawking radiation effects occur as a result of branch pairs that fail to reconverge as a consequence of disconnections in the multiway causal graph, the ER=EPR conjecture is really just a rather elementary statement about the geometry of branchtime (in other words, it states that the natural distance metric in branchtime is the entanglement entropy of pairs of microstates, which one can prove directly from the properties of the Fubini–Study metric tensor).

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