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We consider some generalization of the theory of quantum states, which isbased on the analysis of long standing problems and unsatisfactory situationwith the possible interpretations of quantum mechanics. We demonstrate that theconsideration of quantum states as sheaves can provide, in principle, more deepunderstanding of some phenomena. The key ingredients of the proposedconstruction are the families of sections of sheaves with values in thecategory of the functional realizations of infinite-dimensional Hilbert spaceswith special (multiscale) filtration. Three different symmetries, kinematical(on space-time), hidden/dynamical (on sections of sheaves), unified (onfiltration of the full scale of spaces) are generic objects generating the fullzoo of quantum phenomena (e.g., faster than light propagation).

Quantum points/patterns, Part 1. From geometrical points to quantum points in a sheaf framework