content_image/362008.jpg content_image/362007.jpg
factors in the limiting case of widely separated, non-interacting clusters of particles (the so- called “cluster decomposition principle” [37, 41]), which is not very useful for the analysis of interacting subsystems. But note also that the commutivity of the interaction operators ˆ V ( x ) necessary for Eq. (19) requires the inclusion of reverse particle types (as discussed in Sec. III). 

We can convert the product of operators in Eq. (19) into a product of matrix elements using the usual trick of inserting resolutions of the identity. For the extended Fock space, such resolutions have the form 

<img src='content_image/362008.jpg'>

where χ, ¯ χ represent complete configurations of particles ( χ ) and reverse particles (¯ χ ). That is, | χ, ¯ χ 〉 0 is a position state of the entire universe, with χ symbolically representing the positions of all particles of normal particle types and ¯ χ representing the positions of all particles of reverse particle types. (For particles of non-zero spin, appropriate spin indices should also be considered included.) The integration measure ∫ d χ d¯ χ is intended to not only include integration of particle positions over all spacetime, but also summation over all configurations with all possible numbers of particles of each type (and summation over spin indices, as appropriate). 

Inserting the resolution of the identity from Eq. (20) between the operators in Eq. (19) then gives: 

<img src='content_image/362007.jpg'>

Each 0 〈 χ$_{i}$, ¯ χ$_{i}$ | ˆ G$_{V}$$_{i}$ | χ$_{i}$$_{−}$$_{1}$, ¯ χ$_{i}$$_{−}$$_{1}$ 〉 0 factor is the scattering amplitude for the interaction of subsys- tem V i with the rest of the universe. The interfaces on each “side” are bidirectional : on the i − 1 side, the particles given by χ$_{i}$$_{−}$$_{1}$ are incoming into V $_{i}$, but the reverse particles in ¯ χ$_{i}$$_{−}$$_{1}$ are effectively outgoing from V $_{i}$. Similarly, on the i side, the particles in χ$_{i}$ are outgoing, but the reverse particles in ¯ χ$_{i}$ are effectively incoming . Note again that “incoming” and “out- going” are used in the sense of the propagation along a particle path, not time—incoming particles are thus not necessarily “initial” and outgoing particles are not necessarily “final”. 

It is important that the integrations in the resolution of the identity in Eq. (20), and hence also in Eq. (21), cover all of spacetime, not just the hypervolume of any one system 

