This appendix provides a discussion of the consistency condition (10.39) and of the construction of consistent families of histories. First, we should mention that other conditions have been proposed and used in the literature; in the initial article on histories [17], a weaker condition involving only the cancellation of the real part of (10.39) was introduced. For simplicity, here we limit ourselves to the stronger condition (10.39), which is a sufficient but not necessary condition to the weaker form; it turns out that, as noted in [519], it seems more useful in this context to introduce selectivity than generality in the definition of consistent histories.

At first sight, a natural question that comes to mind is whether or not it is easy, or even possible at all, to fulfil exactly the large number of conditions contained in (10.39). Actually, it has been proposed by Gell-Mann and Hartle to give a fundamental role to families that satisfy consistency conditions in only an approximate way [510], but here we leave aside this possibility and consider only exact consistency conditions. Let us assume for instance that the system under study is a particle propagating in free space; the various projectors may then define ranges of positions for the particle, playing a role similar to diaphragms or spatial filters in optics that confine an optical beam in the transverse direction. Then the consistency condition will appear as similar to a non-interference condition for the Huyghens wavelets that are radiated by the inner surface of each diaphragm.