level (ignoring the constraints).* Here, the as the basic objects instead of $A^i_a$, $E^a_i$ we select the holonomies of $A^i_a$ along the straight lines and fluxes of $E^a_i$ along the unit square $2$-surfaces, which form a closed algebra. The direct implementation of the procedure used in LQG [@lqg2] leads to the gravitational Hilbert space $\Hil_{\gr} = L^2(\bar{\re},\rd\mu_{\Bohr})$, where $\bar{\re}$ is the Bohr compactification of the real line. The basic operators are respectively the holonomies $\hat{h}^{(\lambda)}$ along the edge of the length $\lambda$ and triad $\hat{p}$ (corresponding to the flux across the unit square). The basis of $\Hil_{\gr}$ is built of the eigenstates of $\hat{p}$ and parametrized by $v$ such that $\hat{p}\ket{v} = (2\pi\gamma\lPl^2\sqrt{\Delta})^{2/3} \sgn(v)
|v|^{2/3}\ket{v}$, where the parameter $\Delta$ is the so called *area gap* specified in the next point. The inner product on $\Hil_{\gr}$ is given by $$\label{eq:gr-ip}
  \sip{\psi}{\chi} = \sum_{v\in\re} \bar{\psi}(v)\chi(v).$$

The matter degrees of freedom are quantized via standard methods, thus attaining the Schroedinger-like representation. In consequence the full kinematical Hilbert space takes the form $$\Hil_{\kin} = \Hil_{\gr} \otimes \Hil_{\phi} , \quad \Hil_{\phi} := L^2(\re,\rd\phi).$$ The basic operators on $\Hil_{\phi}$ are the field value $\hat{\phi}$ and its momentum $\hat{p}_{\phi} = -i\hbar\partial_{\phi}$.

$\bullet$ *Promoting the constraint to quantum 