Roles of Asymptotic Condition and S-Matrix as Micro-Macro Duality in QFT

Abstract: Various versions of "independence" are actively inverstigated in quantum probability. In the context of relativistic QFT, we show here that the physical origin of "independence" can be sought in the asymptotic condition through which asymptotic fields and states exhibiting the independence emerge from the non-independent interacting Heisenberg fields in a kind of "central limit". From the algebraic viewpoint, this condition is equivalent to the on-shell condition to pick up free one-particle modes, which also reduces to Einstein's famous formula $E=mc^{2}$. A scenario to reconstruct interacting Heisenberg fields as Micro-objects from these "independent" =free Macro-objects intertwined by an S-matrix as a measurable quantity is formulated according to the Micro-Macro Duality associated with a new notion of a \textit{cocycle of K-T operators}.