Micro-Macro Duality and Emergence of Macroscopic Levels

The mutual relation between quantum Micro and classical Macro is clarified by a unified formulation of instruments describing measurement processes and the associated amplification processes, from which some perspective towards a description of emergence processes of spacetime structure is suggested. 1 Sectors as Quantum-Classical Boundary To fix the setting of my discussion, let me start from a brief account of the notions of sectors, inter-sectorial structures, order parameters to parametrize sectors and so on. In terms of these we can formulate in a clear-cut manner the most important aspects of the mutual relations between the microscopic quantum world and the macroscopic classical levels, which is to be interpreted as a mathematical formulation [1] of the physically essential idea of “quantum-classical correspondence”: the “boundary” and the gap between the former with non-commutative algebras of quantum physical variables and the latter with commutative algebras can be described by means of the notion of a (superselection) sector structure consisting of a family of sectors (or pure phases). To define it, we need to classify representations and states of a C*-algebra A of quantum observables according to the quasiequivalence π1 ≈ π2 [2] defined by the unitary equivalence of representations π1, π2 up to multiplicity, which is equivalent to the isomorphism of representing von Neumann algebras π1(A) ′′ ≃ π2(A) . A sector, or, a pure phase in the physical contexts, is then defined by a quasi-equivalence class of factor representations and states corresponding to a von Neumann algebra with a trivial centre which is a minimal unit among quasi-equivalence classes. Representations belonging to different sectors πa, πb are mutually disjoint with no non-zero intertwiners, namely, any bounded operator T from the representation space Hπa of πa to that Hπb of πb vanishes, T = 0, if it satisfies Talk at the International Symposium, QBIC 2007.