The motivation for this comment came from a popular article " Quantum mischief rewrites the laws of cause and effect" (see this), which tells both about the theoretical work of Lucien Hardy (this) and related experimental work, in particular about the following experimental finding (this).
Photon beam goes through a spin splitter to form a superposition of photons going along two paths. At the first path they go through A and then through B having some effect on the photons. At second splitter the order of A and B is changed. After that the beams are superposed and it is found that the photons in a causally indefinite state in the sense that the effects of both AB and BA are superposed. In classical physics this is impossible.
The finding is claimed to demonstrate causal indefiniteness: one does not know whether A causes B or B causes A. Classically - that is in the framework provided by fixed causal order dictated by Minkowski space - this seems to be the case.
Is this interpretation correct? Is one really forced to give up causality in the standard form? The rules of standard quantum theory are consistent with the finding but should one change the views about the notion of space-time?
- Lightcone of M4 characterizes the causal structure of Minkowski space in special relativity and is the basic notion of QFTs. In curved space-time of GRT, the light-cone however depends on the metric of space-time. Causal structure is dynamical. The intuitive view is that in quantum gravity causality becomes somehow fuzzy since there is no unique space-time anymore. What this non-uniqueness means is not clear. For instance, could it correspond to what happens in the path integral over space-times?
- The problem is that one cannot compare the causal structure for different space-times because the light-cones characterizing them are in different space-times. If the space-times had common coordinates, the comparison would become possible but one cannot assume this.
Hardy proposes what he calls quantum Equivalence Principle (see this). It should be possible at least locally to compare small deformations of space-time metric by finding coordinates in which the light-cone defining the causal structure co-incide.
- The TGD based solution is much simpler. Space-times are identified as 4-surfaces in H=M4× CP2 and subset of preferred coordinate for H - Minkowski coordinates when the M4 projection of space-time surface is 4-D - provides universal space-time coordinates and one can compare space-time surfaces and their induced causal structure. In quantum sector the second quantized free spinor fields in H can be restricted to space-time surface and define fermionic propagators and causal structure.
The important point is that the configurations AB and BA appearing in quantum switch corresponds to a space-time surface represents a branching of 3-surfaces representing photon propagation to two pieces at beam splitter and recombination back to single 3-surface making it possible for the photon wave functions interfere. Causal indefiniteness in the proposed sense does not mean that the direction of the causal arrow as an arrow of time is changed and in TGD framework it is not natural to speak about causal indefiniteness.
See the chapter Topological Quantum Computation in TGD Universe or the article Still about the notion of causal indefiniteness in TGD framework.