Gravity has often been suggested as playing a role in quantum theory, principally as a mechanism that induces quantum state vector collapse. However, at an ontological level, the Invariant Set Postulate does not require superposed states and hence does not require a collapse mechanism, gravitational or otherwise.
On the other hand, the order-of-magnitude estimates provided by Penrose, that gravitational processes can be locally significant when a quantum sub-system and a measuring apparatus interact, seem persuasive. Here, we would interpret these estimates as supporting the notion that gravity plays a key role in defining the state space geometry of the invariant set, in particular in defining the regions of relative stability (small local Lyapunov exponents) and relative instability (large local Lyapunov exponents). Black-hole thermodynamics may additionally provide the mechanism which leads to the dimensional reduction of the invariant set compared with that of the embedding state space.
Indeed this leads to the following rather radical suggestion. If the geometry of invariant set I is to be considered primitive, then the geometric properties of the invariant set which lead to certain regions being relatively stable and other regions unstable should be considered a generalization of the notion introduced by Einstein that the phenomenon we call ‘gravity’ is merely a manifestation of some more primitive notion of geometry—here the geometry of a dynamically invariant subset of state space. As such, a challenge will be to try to unify the notions of pseudo-Riemannian geometry for space–time, and fractal geometry for state space. This is a very different perspective on ‘quantum gravity’.
From this we can make two gravitationally relevant predictions. Firstly, since gravitational processes are not needed to collapse the quantum state vector, experiments to detect gravitational decoherence may fail. By contrast with Objective Reduction, I could be seen as providing the preferred basis, with respect to which conventional non-gravitational decoherent processes operate. Secondly, if gravity should be seen as a manifestation of the heterogeneity in the geometry of the invariant set, then attempts to quantize gravity with the framework of standard quantum theory will also fail. As such, it is misguided to assume that ‘theories of everything’ can be formulated within conventional quantum theory.