In May, Losada and Laura wrote a paper^{a } pointing out the equivalence between two conditions on a set of “elementary histories” (i.e. fine-grained histories^{b }). Let the elementary histories be defined by projective decompositions of the identity at time steps (), so that

(1)

where are the class operators. Then Losada and Laura showed that the following two conditions are equivalent

- The set is consistent
^{c }for any state: . - The Heisenberg-picture projectors at all times commute: .

However, this is not as general as one would like because assuming the set of histories is elementary is very restrictive. (It excludes branch-dependent sets, sets with inhomogeneous histories, and many more types of sets that we would like to work with.) Luckily, their proof can be extended a bit.

Let’s forget that we have any projectors and just consider a consistent set .… [continue reading]