The theoretical framework establishes that rigorous perturbative calculations in interacting Quantum Field Theory require replacing naive free-vacuum expectations with Time-Ordered correlation functions computed relative to the full interacting vacuum, specifically defined via LSZ reduction limits of Green's functions. This formulation corrects topological errors arising from disconnected vacuum diagrams at higher orders by utilizing a unitary time-evolution operator ($U_{T}(t_2, t_1)$) that maps interaction-picture operators between arbitrary times while accounting for non-commuting Hamiltonian components and the unique spectral gap of the true vacuum. The resulting Green's functions serve as intrinsic observables whose analytic properties in momentum space dictate S-Matrix elements through residue extraction at physical poles, providing a systematic derivation where external propagators are amputated to yield scattering amplitudes independent of perturbative splitting definitions.
The theoretical framework establishes that rigorous perturbative calculations in interacting Quantum Field Theory require replacing naive free-vacuum expectations with Time-Ordered correlation functi…