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

After the mathematical notion of “Micro-Macro Duality” for understanding mutual relations between microsopic quantum systems (Micro) and their macroscopic manifestations (Macro) is explained on the basis 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. 1 Quantum-Classical Correspondence and MicroMacro Duality The essence of “quantum-classical correspondence” (q-c correspondence, for short) is usually understood in such an intuitive way that macroscopic classical objects arise from a microscopic quantum system as condensates of infinite quanta in the latter. Aiming at a satisfactory understanding of mutual relations among different hierarchical levels in the physical world, we try here to provide this heuristic idea with a mathematically sound formulation on the basis of what we call “Micro-Macro duality” [1]; this is the mathematical notion of duality (or, categorical adjunction in more general situations), which allows us to connect microscopic and macroscopic levels in the physical nature in bi-directional ways from Micro to Macro and vice versa. To be precise, the contrasts of [Micro vs. Macro] (according to length scales) and of [Quantum vs. Classical] (due to the essential differences in their structures) are to certain extent independent of each other, as