as a basis for our analysis. The treatment is in fact an extension of the one applied to the FRW universe. Therefore, for shortness, here we will focus just on these aspects of it, which differ from the description presented in Sec. \[sec:frame-FRW\]. For the detailed description the reader is referred to [@awe].

### The classical model

Classically the Bianchi I flat spacetime admits the same foliation by homogeneous surfaces as the FRW one. The general form of the metric is $$g = -N^2\rd t^2 + a^2_1(t)\rd {x_1}^2 + a^2_2(t)\rd {x_2}^2 + a^2_3(t)\rd {x_3}^2$$ where $x_i$ are chosen (comoving) coordinates defining the fiducial Cartesian metric ${{\,{}^o\! q}}:= \rd {x_1}^2 + \rd {x_2}^2 + \rd {x_3}^2$ with orthonormal (co)triad (${{\,{}^o\! \omega}}^i_a$) ${{\,{}^o\! e}}^a_i$ . To construct the Hamiltonian formalism one has to introduce again a fiducial cell, which here is described by three fiducial (i.e. with respect to the metric ${{\,{}^o\! q}}$) lengths $L_i$ and fiducial volume $V_o:=L_1L_2L_3$. Here it is again possible to fix a gauge in which the Ashtekar connections and triads are represented by three pairs of canonically conjugated coefficients $c^i, p_i$

\[eq:Ashtekar-B1\]$$\begin{aligned}
  A^i_a &= c^i \, L_i^{-1} {{\,{}^o\! \omega}}^i_a  , &  
  E^a_i &= p_i \, L_i V_o^{-1} 