where ˆ V ( x ) ≡ g : ˆ ψ ( a $^{′}$) $^{‡}$( x ) ˆ ψ ( b $^{′}$) $^{‡}$( x ) ˆ ψ ( a $^{′}$) $^{′}$( x ) ˆ ψ ( b $^{′}$) $^{′}$( x ) : (18) with : · · · : representing normal ordering, that is, placing all ˆ ψ ‡ operators to the left of all ˆ ψ operators in any product. The interaction operator ˆ G = exp( − i ˆ V ) in this vertex operator will then properly include all loops. Since the ¯ n reverse particle types are only included in the formalism for this purpose, physical states, such as in and out scattering states, can be constructed using only the original particle types (that is, constructed using the original creation operators ˆ ψ ( n ) $^{‡}$). The need to include reverse propagation does complicate a bit the spacetime formalism for interaction presented here. However, the great advantage of the result is that Eq. (17) defining ˆ G does not involve time-ordering, as is found in the Dyson expansion of the usual scattering operator ˆ S [36–38]. This will be critical when considering the decomposition of the interactions within a system into a number of distinct subsystems, a topic to which we turn next. IV. SYSTEMS AND SUBSYSTEMS As reiterated by Zurek, “The Universe consists of systems” [34]. Quantum processes, measuring apparatuses and observers are all systems, and all subsystems of the system that is the universe as a whole. We make no a priori distinction between “quantum” systems and “classical” systems. We do, however, need to have a crisp definition of how to delineate what the systems of interest are in any given analysis of some or all of the universe. As discussed in Sec. II, it will be convenient here to define a system as being contained in a well-defined hypervolume V of spacetime, disjoint from the hypervolumes occupied by all other systems of interest. This hypervolume does not have to be continuous or connected, though it must have measure greater than zero. The effective identification of systems and subsystems with regions of spacetime is some- what more restrictive than the typical generic definition used in quantum mechanics, in which a system is taken to be any portion of the universe whose state can be represented by a vector in an appropriate Hilbert space. For example, identifying subsystems with hyper- volumes does not allow an overall set of degrees of freedom of a system to be divided between 13