Note that in the spacetime state framework we may

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Note that in the spacetime state framework, we may replace talk of states of particles relative to elements of a foliation with direct talk of the states of spacetime regions, such as R j ,S j and T j . 18
sense in which non-unitary dynamics has consequences going beyond those of unitary dynamics. To this end we shall now allow ourselves to talk directly of the states of the pertinent spacetime regions shown in Figure 1; considering first the unitary case, before seeing how things differ under non-unitary dynamics. Thus suppose that in Figure 1 the transformations performed at X 1 and X 2 were unitary—say, | 0 ) −→ | 1 ) | 1 ) −→ − | 0 ) ; (5) a spin-flip rather than a measurement. The states of the various relevant regions are easily calculated: hatwide ρ R 1 = hatwide ρ S 1 = | α | 2 | 0 ) ( 0 | + | β | 2 | 1 ) ( 1 | hatwide ρ R 2 = hatwide ρ S 2 = | β | 2 | 0 ) ( 0 | + | α | 2 | 1 ) ( 1 | hatwide ρ T 1 = | β | 2 | 0 ) ( 0 | + | α | 2 | 1 ) ( 1 | hatwide ρ T 2 = | α | 2 | 0 ) ( 0 | + | β | 2 | 1 ) ( 1 | and hatwide ρ Λ i = | ψ i ) ( ψ i | where | ψ 1 ) = α | 0 ) | 1 ) + β | 1 ) | 0 ) | ψ 2 ) = α | 1 ) | 1 ) − β | 0 ) | 0 ) | ψ 3 ) = α | 0 ) | 0 ) + β | 1 ) | 1 ) | ψ 4 ) = α | 1 ) | 0 ) − β | 0 ) | 1 ) . We observe that, as expected, the state on both Λ 2 and Λ 3 is changed from Λ 1 , since either X 1 or X 2 lies in the past light cone of each region. The states of the individual particles, however, are unambiguously fixed by the dynamics, and neither particle’s individual state is affected by a transformation outside its light cone — X 1 lies outside the past light cone of S 2 , for instance, so hatwide ρ S 2 = hatwide ρ R 2 even though S 2 is part of Λ 2 and the state of Λ 2 is changed by X 1 . Things are otherwise in the event of collapse, where we make a measurement rather than a spin flip at X i . The (hypothetical) stochastic dynamics produce perfectly reasonable (non-deterministic) predictions for all the joint states— assigning a non-zero probability to, say, hatwide ρ Λ i = | φ i ) ( φ i | where | φ 1 ) = α | 0 ) | 1 ) + β | 1 ) | 0 ) | φ 2 ) = | φ 3 ) = | φ 4 ) = | 0 ) | 1 ) . But there is no unique prescription for the one-particle states. Take S 2 , for instance—if it is regarded as part of Λ 1 then it has state | β | 2 | 0 ) ( 0 | + | α | 2 | 1 ) ( 1 | ; (6) 19
if it is regarded as part of Λ 2 then it has state | 1 ) ( 1 | . We might interpret this (at the risk of confusion with a terminology adopted elsewhere in philosophy of QM) as a contextuality of local states—the state of a region depends on which larger region it is regarded as a part of. We might also interpret it as denying the existence of states for many regions of spacetime—regions only have states at all if they are assigned the same state by all foliations. We might call this nihilism about local states, a name which will be particularly apt if, as we are inclined to suspect (but have not proved), in realistic models virtually no regions smaller than entire foliations are assigned states. To conclude this section: there is an “hierarchy of counterintuitiveness” here. We have 1. Full locality (classical physics). All spacetime regions have states; the state of a region supervenes on the states of its subregions.

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