Quantum cosmological intertwining: Factor ordering and boundary conditions from hidden symmetries

We explore the implications of hidden symmetries present in a particular quantum cosmological setting, extending the results reported in \cite{10,11}. In more detail, our case study is constituted by a spatially closed Friedmann-Lema\^{\i}tre-Robertson-Walker universe, in the presence of a conformally coupled scalar field. The $su(1,1)$ hidden symmetries of this model, together with the Hamiltonian constraint, lead to the gauge invariance of its corresponding Bargmann indices. We subsequently show that some factor-ordering choices can be related to the allowed spectrum of Bargmann indices and hence, to the hidden symmetries. Moreover, the presence of those hidden symmetries also implies a set of appropriate boundary conditions to choose from. In summary, our results suggest that factor ordering and boundary conditions can be intertwined when a quantum cosmological model has hidden symmetries.