I’ve been very interested in the search for a theory of quantum gravity. General Relativity and Quantum theory, the twin crowning achievements of twentieth century physics, are not compatible, and the hunt has been on for a successor theory which would underlie or reconcile the two.
Approaches include trying to extend or modify the quantum field theory programs which were so successful for building models of particle interactions and forces, but which failed to accommodate gravity (superstring theory falls broadly into this category). Alternatively, some researchers have explored approaches which feature some conceptual rethinking of the issues involved. I’ve been intrigued by recent research programs which posit that the space-time geometry of GR emerges from a more fundamental background theory, such as a dimensionless quantum causal framework of some kind.
In thinking about the conceptual, rather than technical issues involved, it is worth reflecting on the fact that there may be a basic conflict between ordinary quantum mechanics and relativity, which predates the issues of reconciling quantum field theories and Einstein’s theory of gravity.
M.P. Seevinck reviews the issue of compatibility in his white paper (pdf), “Can quantum theory and special relativity peacefully coexist?” The source of the potential conflict is the nonlocal aspect of quantum phenomena, as described by Bell’s inequalities and demonstrated in EPR-style experiments (see also this comprehensive SEP article).
Seevinck argues that non-locality is simply not consistent with the local causal structure inherent in SR. Now, there is a weaker sense in which one might argue the theories are compatible: while the nonlocal correlations which arise in entangled systems are themselves well demonstrated empirically, we have been unable to utilize these phenomena to create an experimental conflict with SR such as superluminal signaling. Theoreticians also have characterized no-signaling as an essential part of quantum theory, developing no-signaling theorems.
Seevinck is critical of no-signaling theorems, saying they are either circular or else serve as consistency proofs (QM and no-signaling can, rather than must, be compatible). In the case of some theorems derived from QFT it seems clear why they might be circular: quantum field theory obscures the issue at hand because it is formulated against a backdrop of SR – so compatibility is enhanced by construction.
But the compatibility between QM and SR is not all about no-signaling. It can be argued that the spirit of SR is a geometric causal structure of space-time, and there can be no story of nonlocal correlations arising causally in this structure.
Seevinck briefly reviews several general approaches to resolving the conflict through interpretation or modification of the theories, without endorsing one. Part of his discussion references the ideas of Nicolas Gisin. In several papers, Gisin has also argued the case for incompatibility. He has been critical of the traditional discussion of Bell’s results which describe it as presenting a choice to reject either locality or “realism”. He finds no sensible "irrealism" option which gives a reason to reject the conclusion of nonlocality (see brief Gisin papers here and here).
Gisin’s view is that we must accept that nonlocal correlations can’t originate in space-time, and he ponders the alternative, which is that they must emerge from “outside” space-time. What this means needs to be fleshed out, but it seems compatible with the idea that space-time geometry is an emergent rather than fundamental aspect of nature.