k \bar{\tilde{\Psi}}(k)\tilde{\Phi}(k) .$$

Since here we are dealing with the constrained system, there is no natural notion of time and evolution. It can be provided via the unitary mapping . An alternative way to define an evolution is the construction of the family of *partial observables* [@par-obs], parametrized by one of the dynamical variables and of the elements related via unitary transformation. Here we construct family $\ln|\hat{v}|_{\phi}$ [@aps-det] interpreted as $\ln|v|$ at given “time” $\phi$. The systematic way of constructing such observables is presented in [@GaSch]. For the models considered in the article the expectation values and dispersions of $\ln|\hat{v}|_{\phi}$ for the physical state $\Psi$ equal the analogous quantities of the kinematical observable $\ln|\hat{v}|$ acting on the initial data $\psi_{\phi}:=\Psi(\cdot,\phi)\in\Hil_{\kin}$ $$\label{eq:lnv}
  [\ln(v)_{\phi_o}\Psi](v) = e^{i\sqrt{\Theta^2}(\phi-\phi_o)} \ln(v)\Psi(v,\phi_o) .$$ For completeness we introduce one more observable, corresponding to the constant of motion $\ln(\omega)$ – an operator $\ln(\hat{\omega})$ acting as follows $$\label{eq:lnw}
  [\ln(\hat{\omega}/\sqrt{G})\tilde{\Psi}](k) = \ln(\omega(k)/\sqrt{G})\tilde{\Psi}(k) .$$ This operator will be useful later in the paper as (it will be shown that) its dispersion bounds the growth of the spread in $\ln|\hat{v}|_{\phi}$.

Wheeler-DeWitt analog {#sec:frame-WDW}
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The systems studied in this article can be also quantized via methods of the geometrodynamics. Indeed, the geometric component 