On the Notion of Causal Indefiniteness in the TGD Framework

Abstract

The light-cone of M⁴ characterizes the causal structure of Minkowski space in special relativity and is the basic notion of QFTs. In  of GRT,  the light-cone however depends on the metric of space-time. 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. 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.  Lucien  Hardy proposes what he calls quantum Equivalence Principle. 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 x CP₂and 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. A closely related experimental finding is that it seems possible to construct  causally indefinite photon states. In quantum switch,  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. In the TGD framework, 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.