Here for that purpose we select $\lambda_3$, finally labeling the basis elements as $\ket{\lambda_1,\lambda_2,v}$.

The space $\Hil_{\phi}$ and the set of basic operators corresponding to the matter are the same ones as in the FRW case. The full kinematical Hilbert space is also a product $\Hil_{\kin}=\Hil_{\gr}\otimes\Hil_{\phi}$.

The quantum Hamiltonian constraint is constructed out of the classical one by, first reexpressing it in terms of the holonomies and fluxes, and next promoting these components to operators. In particular the field strength $F^k_{ab}$ is again represented via holonomies along closed rectangular loops of the physical area equal to $\Delta$. Unlike in isotropic case however fixing the loop area does not allow to uniquely fix the fiducial lengths of its edges. This apparent “ambiguity” gave rise to several distinct prescriptions present currently in the literature (including the one of [@chiou-b1]). On the other hand the relation of the LQC degrees of freedom with the full LQG ones constructed in [@awe] allowed to fix the relation uniquely. The constraint resulting from this operation (defined on the dense domain in $\Hil_{\kin}\otimes\Hil_{\phi}$) is of the Klein-Gordon form $$\label{eq:constr-b1}
  \hat{C} =\hat{{\openone}} \otimes \partial_{\phi}^2 + \hat{{\openone}}\otimes\Theta_{\operatorname*{B1}} .$$ Similarly to FRW model we can restrict our interest to just 