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are being used here in the sense of particle paths in spacetime, not in the sense of time—that is, the position states | x 〉 are not being separated into particle and antiparticle states. 

Such an interaction can be modeled using a vertex operator constructed from the appro- priate number of annihilation and creation operators. For example, consider the case of an interaction with two incoming particles, one of type a and one of type b , and two outgoing particles of the same types. The vertex operator for this interaction is 

<img src='content_image/358399.jpg'>

where the coefficient g represents the relative probability amplitude of the interaction. 

In the following, it will be convenient to use the special adjoint ˆ ψ ‡ defined by 

With this notation, the expression for ˆ V becomes 

<img src='content_image/358398.jpg'>

<img src='content_image/358397.jpg'>

To account for the possibility of any number of interactions, we just need to sum up powers of ˆ V to obtain the interaction operator 

<img src='content_image/358396.jpg'>

where the 1 /m ! factor accounts for all possible permutations of the m identical factors of ˆ V . The − i factors are introduced so that ˆ G is unitary relative to the special adjoint (that is, ˆ G ‡ ˆ G = ˆ G ˆ G ‡ = 1), so long as ˆ V is self-adjoint relative to it (that is, ˆ V ‡ = ˆ V ). 

The self-adjointness of ˆ V implies that an interaction must have the same number of incoming and outgoing particles, of the same types, at least when only one possible type of interaction is involved (as is the case with the example of Eq. (16)). The formalism can be easily extended to allow for multiple types of interactions by adding additional terms to the definition of ˆ V . In this case, only the overall operator ˆ V needs to be self-adjoint, not the individual interaction terms. 

As mentioned above, normal particle states are obtained in the + ∞ time limit, while antiparticle states are obtained in the −∞ time limit. Moving to a momentum representation then results in (multiparticle) on-shell scattering in and out states. These states can be used with the interaction operator ˆ G to compute multipoint interaction amplitudes. Expanding 

