system exists in V —and presumably has some additional interesting finer-grained states, such that and Now, given a system contained in V divided into N subsystems contained in V $_{i}$, for i = 1 , . . . , N , suppose we define a complete set of projection operators ˆ P α V$_{i}$ corresponding to each of the subsystems, as well as a set ˆ P α in defined on the in state for the system. Inserting Eq. (22) for these operators between the scattering operators for each subsystem in Eq. (19) gives where for α = ( α$_{0}$, · · · , α$_{N}$ ). This equation now has the essential form of Eq. (8) desired for chain operators at the end of Sec. II and Eq. (23) is analogous to the summation of Eq. (6) in the non-relativistic case. The intent here is that the projection operators ˆ P α$_{i}$ V$_{i}$ represent the propositions that either subsystem i does not exist (represented by ˆ P 0 $_{V$_{i}$}$) or that there is some “interesting” outcome α$_{i}$ as a result of the interactions generated by ˆ G$_{V}$$_{i}$ (represented by ˆ P α$_{i}$ V$_{i}$ for α$_{i}$ > 0). We will thus always assume in the following that the assertion made by each ˆ P α$_{i}$ V$_{i}$ , for α$_{i}$ > 0), depends only on outgoing particles from V i that have actually interacted within V $_{i}$. That is, for all i and any α$_{i}$ > 0, 17 ˆ P α$_{i}$ V$_{i}$ | χ, ¯ χ 〉 0 = ˆ P α$_{i}$ V$_{i}$ | χ $^{′}$, ¯ χ $^{′}$〉 $_{0}$, (25) ˆ C α = ˆ P α$_{N}$ V$_{N}$ ˆ G$_{V}$$_{N}$ ˆ P α$_{N}$ − 1 V$_{N}$ − 1 · · · ˆ P α 1 V 1 ˆ G$_{V}$$_{1}$ ˆ P α 0 in , (24) ˆ G$_{V}$ = ( ∑ α$_{N}$ ˆ P α$_{N}$ V$_{N}$ ) ˆ G$_{V}$$_{N}$   ∑ α$_{N}$ − 1 ˆ P α$_{N}$ − 1 V$_{N}$ − 1   · · · ( ∑ α 1 ˆ P α 1 V 1 ) ˆ G$_{V}$$_{1}$ ( ∑ α 0 ˆ P α 0 in ) = ∑ α ˆ C α , (23) ˆ P 0 V + ˆ ¯ P 0 V = 1 ˆ ¯ P 0 V = ∑ α> 0 ˆ P α V , where the projection operators for actually interesting system states are denoted by ˆ P α V for α > 0. By definition, the non-existence assertion ˆ P 0 V is never truly interesting (at least for the cases considered here), but it is only when this operator is included that the set of ˆ P α V is actually complete and sums up to the identity as in Eq. (22).