equivalent to it, namely the reinterpretation of the constraint as the Klein-Gordon-like equation $$\label{eq:constr-KG}
  [\partial^2_\phi\Psi](v,\phi) = -[\Theta\Psi](v,\phi) ,$$ defining the evolution of a free system along $\phi$, that is the mapping between the the spaces of the “initial data” – restrictions of $\Psi$ to the surfaces of constant $\phi$ $$\label{eq:evo-map}
  \re\ni\phi \mapsto \Psi(\cdot,\phi) \in \Hil_{\gr} .$$

The quite simple form of the operator $\Theta$ allows to easily define the physical Hilbert space $\Hil_{\phy}$ via its spectral decomposition. This step, as well as the notion of evolution will be described in more detail in Sec. \[sec:frame-phys\].

Before going to it let us note, that the structure of $\Theta$ and provides the natural division of the domain of $v$ onto the subsets (the *lattices*) $$\lat_{\varepsilon} = \{ \varepsilon+4n;\ n\in\integ\} ,\quad \varepsilon \in [0,4[$$ preserved by the action of $\Theta$. This division is naturally transferred to the splitting of $\Hil_{\phy}$ onto the superselection sectors. In consequence it is enough to fix particular value of $\varepsilon$ and work just with the restriction of the domain of $\Theta$ to functions supported on $\lat_{\varepsilon}$ only. For the clarity we will consider just the sector $\varepsilon=0$, however the presented treatment and its results generalize easily (at the 