for all ¯ χ and ¯ χ ′ and any χ and χ ′ that differ only in positions outside of V $_{i}$. (Note that this assumption now presumes a specific ordering of the subsystems, but that will actually be convenient in the following, as further discussed in Sec. VIII.) Given this assumption, each ˆ P α$_{i}$ V$_{i}$ ˆ G$_{V}$$_{i}$ ˆ P α$_{i}$ − 1 V$_{i}$ − 1 effectively defines the probability amplitude for the outcome α$_{i}$ for subsystem i given the outcome α$_{i}$$_{−}$$_{1}$ for subsystem i − 1. This cannot exactly be called a “transition” amplitude, since there is not necessarily any ordering in time. Nevertheless, subsystem outcomes of interest will generally be chosen such that certain outcomes of subsystem i − 1 preclude the interactions required to generate certain other outcomes of subsystem i , in which case ˆ P α$_{i}$ V$_{i}$ ˆ G$_{V}$$_{i}$ ˆ P α$_{i}$ − 1 V$_{i}$ − 1 will be identically zero. Indeed, in the following it is usually the case that outcome α$_{i}$$_{−}$$_{1}$ for subsystem i − 1 implies a specific outcome A$_{i}$ ( α$_{i}$$_{−}$$_{1}$ ) for subsystem i . That is The situation represented in Eq. (26) is essentially a dynamic process, in which the cross-correlation terms ˆ P α$_{i}$ V$_{i}$ ˆ G$_{V}$$_{i}$ ˆ P α$_{i}$ − 1 V$_{i}$ − 1 for α$_{i}$ = A$_{i}$ ( α$_{i}$$_{−}$$_{1}$ ) are driven to zero. (Such a process may actually only take the cross-correlations to approximately zero, not identically zero, but I will always assume they are zero in the following.) The dynamics in time are not explicit, of course, in the spacetime formalism used here. However, the hypervolume V i for any subsystem will generally have a finite temporal extent and it can be arranged for the projection operators ˆ P$_{i}$ to represent states at the upper bound of that extent. Thus, when looked at from a time-evolution viewpoint, various processes may be taking place within the hypervolume V i leading to zero cross-correlations for outgoing particle states. We will also assume in the following that either all subsystems of interest “exist”, or none do. That is, ˆ P α$_{N}$ V$_{N}$ ˆ G$_{V}$$_{N}$ ˆ P α$_{N}$ − 1 V$_{N}$ − 1 · · · ˆ P α 1 V 1 ˆ G$_{V}$$_{1}$ ˆ P α 0 in is identically zero if some, but not all, of the α$_{i}$ are zero. Which of the terms for other values of the α$_{i}$ are non-zero depends on the specific physical situation under consideration. Clearly, with a full theory of interactions, all such dependencies should be determinable from first principles. However, for the purposes of the following sections, it will be enough to simply assert the physical dependencies between subsystems required in each situation considered. 18 ˆ P α$_{i}$ V$_{i}$ ˆ G$_{V}$$_{i}$ ˆ P α$_{i}$ − 1 V$_{i}$ − 1 = δ$_{α}$$_{i}$$_{,A}$$_{i}$$_{(}$$_{α}$$_{i}$$_{−}$$_{1}$$_{)}$ ˆ P A$_{i}$ ( α$_{i}$ − 1 ) V$_{i}$ ˆ G$_{V}$$_{i}$ ˆ P α$_{i}$ − 1 V$_{i}$ − 1 (26)