Phase space regions operators and ISp(2) maps

Region operators (generalized observable) (RO), defined as operator valued probability measures over classical phase space ensuing from quantization of phase space characteristic functions are investigated. The problem of constructing of RO is treated by means of positive trace increasing maps (pti maps). Exemplary domains of RO include: sets of points, line segments and rotated line segments, circles and disks, straight and rotated lines, canonical polygons, squeezed line segments, and lines related to Radon transform in phase space. Phase plane RO constructing pti maps, in their operator sum representation, are generated by elements of the quantum symplectic in-homogeneous group ISp(2). The group ISp(2) and its double cover metaplectic group Mp(2) are the kinematical groups for RO construction on the plane.