describe the structure of the physical Hilbert space and relevant observables for LQC theory (in Sec. \[sec:frame-phys\]), as well as its WDW analog (Sec. \[sec:frame-WDW\]).

Isotropic flat FRW universe {#sec:frame-FRW}
---------------------------

### Classical theory {#sec:frame-FRW-class}

On the classical level spacetimes of this class admit a (parametrized by a time $t$) foliation by homogeneous surfaces ${\mathcal{M}}=\Sigma\times\re$, where $\Sigma$ is topologically $\re^3$. Their metric tensor is $$\label{eq:metric}
  g = -N^2\rd t^2 + a^2(t) {{\,{}^o\! q}},$$ where $N$ is a lapse function, $a$ is a scale factor (or equivalently a size of certain selected region ${\mathcal{V}}$, see the discussion below ) and ${{\,{}^o\! q}}$ is a fiducial Cartesian metric. To describe the system further we apply the canonical formalism, expressing the geometry in terms of Ashtekar variables: connections and triads, selecting the gauge fixing in which they can be expressed in terms of the real connection and triad coefficients $c,p$

\[eq:Ashtekar\]$$\begin{aligned}
  A^i_a &= c \, V_o^{-\frac{1}{3}} {{\,{}^o\! \omega}}^i_a  , &  
  E^a_i &= p \, V_o^{-\frac{2}{3}} \sqrt{{{\,{}^o\! q}}} {{\,{}^o\! e}}^a_i , 
\tag{\ref{eq:Ashtekar}}\end{aligned}$$

where ${{\,{}^o\! e}}^a_i$/${{\,{}^o\! \omega}}^i_a$ is an orthonormal triad/cotriad corresponding to the fiducial metric ${{\,{}^o\! q}}$ and $V_o$ is the fiducial volume of ${\mathcal{V}}$. The variables $c,p$ are canonically conjugated with $\{c,p\}=8\pi G\gamma/3$ 