In May, Losada and Laura wrote a papera pointing out the equivalence between two conditions on a set of “elementary histories” (i.e. fine-grained historiesb ). 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 consistentc 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]