How to Observe Quantum Fieldsand Recover Them From Observational Data? — Takesaki Duality as a Micro-Macro Duality

On the basis of the mathematical notion of “micro-macro duality” for understanding mutual relations between microsopic quantum systems (micro) and their macroscopic manifestations (macro) in terms of the notion of sectors and order parameters, a general mathematical scheme is proposed for detecting the state-structure inside of a sector through measurement processes of a maximal abelian subalgebra of the algebra of observables. For this purpose, the Kac-Takesaki operators controlling group duality play essential roles, which naturally leads to the composite system of the observed system and the measuring system formulated by a crossed product. This construction of composite systems will be shown to make it possible for the micro to be reconstructed from its observational data as macro in the light of the Takesaki duality for crossed products.