Generalized cluster decomposition principle illustrated in waveguide quantum electrodynamics

We show that the form of the cluster decomposition principle, commonly used in quantum field theory, needs to be significantly generalized. As an illustration, we consider the general structure of two-photon S matrix for a waveguide coupled to a local quantum system that supports multiple ground states. The presence of the multiple ground states results in a noncommutative aspect of the system with respect to the exchange of the orders of photons. Consequently, the two-photon S matrix significantly differs from the standard form in quantum field theory.