Geometric structure of a classifying space in the sector theory

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At the starting point, the aim of this project on the mathematical structures of infinite quantum systems was to construct a mathematical framework for classifying, describing and interpreting the quantum states according to the symmetries and their breakings or to the thermodynamic properties in terms of geometric structures of classifying spaces of sectors ; this goal has been achieved by my research in these three years. The attained results lead to a new perspective concerning not only quantum states but the algebra and the group action on it to characterize a specific physical system as a whole. The classification scheme of states is divided into two levels, the first one consisting of sectors specified by order parameters in the centre algebra and the second one concerning the internal structure of a sector. Generalizing the Doplicher-Haag-Roberts sector theory for unbroken symmetries, I proposed in [1-4] a unified scheme for generalized sectors applicable also to broken symmetries and thermal situations. To jump into the inside of a sector, we need to replace the centre by a maximal abelian subalgebra and to introduce a Kac-Takesaki operator controlling the group duality, by which all the necessary ingredients are implemented [6] to construct a measurement' process, such as the notion of instrument. Once this is done, it becomes possible to recover the infinite-dimensional non-commutative algebra of microscopic quantum system from the mathematical structure of measured data through the Takesaki duality for crossed products. It is evident here that the dynamics with an outer action is crucial for a local subalgebra of quantum fields of type III to be restored, which forces us to face mathematically the problem of determining a dynamics in an operational way (I.Ojima & M.Takeori, in preparation).