content_image/328294.jpg content_image/328293.jpg content_image/328292.jpg content_image/328289.jpg content_image/328291.jpg content_image/328290.jpg
interaction process (as, e.g., captured in the interaction operator ˆ G defined in Sec. III) one can consider incoming and outgoing particles to be on-shell sufficiently far outside the interaction area [25]. 

Multiple particle states can be straightforwardly introduced as members of a Fock space over the Hilbert space of position states | x ; λ 〉 . First, in order to allow for multiparticle states with different types of particles, extend the position state of each individual particle with a particle type index n , such that 

<img src='content_image/328294.jpg'>

Then, construct a basis for the Fock space of multiparticle states as symmetrized products of N single particle states: 

<img src='content_image/328293.jpg'>

where the sum is over all permutations P of 1 , 2 , . . . , N . (When including Fermions, one needs to, of course, antisymmetrize rather than symmetrize the products [26].) 

It is then convenient to introduce a creation field operator ˆ ψ $^{†}$( x, n ; λ ) such that 

<img src='content_image/328292.jpg'>

with the corresponding annihilation field ˆ ψ ( x, n ; λ ) having the commutation relation 

Further, define 

so that 

<img src='content_image/328289.jpg'>

<img src='content_image/328291.jpg'>

<img src='content_image/328290.jpg'>

Identifying the field operators ˆ ψ ( x, n ; λ$_{0}$ ) and ˆ ψ ( x, n ) defined here with ˆ ψ ( n ) 0 ( x ) and ˆ ψ ( n $^{)}$( x ) defined in Sec. III then completes the grounding of the formalism used in the main text. 

[1] R. B. Griffiths, J. Stat. Phys. 36 , 219 (1984). 

