The spin reduced matrix defined by taking partial trace with the momentum variable has been shown not to transform covariantly under Lorentz transformations. In this paper, I suggest another representation space of the Poincare group, the elements of which are in a sense apparent wave functions. In this new Hilbert space, the naive spin observable
21τ from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.
Quantum theory can be formulated as a theory of operations, more specific, of complex represented operations from real Lie groups. Hilbert space eigenvectors of acting Lie operations are used as states or particles. The simplest simple Lie groups have three dimensions. These groups together with their contractions and subgroups contain - in the simplest form - all physically important operations which come as translations for causal time, for space and for spacetime, as rotations, Lorentz transformations and as Euclidean and Poincare transformations with scattering and particle states and also - via the Heisenberg groups - as the operational structure of nonrelativistic quantum mechanics. The classification of all those groups and their contractions is given together with their Hilbert spaces, constituted by energy-momentum functions. The groups representation matrix elements can be written as residues of energy-momentum poles - simple poles for abelian translations, e.g. in Feynman propagators, and dipoles for simple Lie group operations, e.g. in the Schroedinger wave functions for the nonrelativistic hydrogen atom.