Quantum noncomutativity in quantum cosmology

In the present work, we study the noncommutative version of a quantum cosmology model. The model has a Friedmann-Robertson-Walker geometry, the matter content is a radiative perfect fluid and the spatial sections have positive constant curvatures. We work in the Schutz's variational formalism. We quantize the model and obtain the appropriate Wheeler-DeWitt equation. In this model the states are bounded. Therefore, we compute the discrete energy spectrum and the corresponding eigenfunctions. The energies depend on a noncommutative parameter ($\theta$). The solutions to the Wheeler-DeWitt equation are function of the scale factor ($a$) and a time variable ($\tau$), associated to the fluid. They also depend on an integer ($n$) and $\theta$. The most general solution ($\Psi(a,\tau)$) to the Wheeler-DeWitt equation is a sum, in the integer $n$, of the solutions mentioned above. We observe that, there is no $\Psi(a,\tau)$ satisfying the appropriate boundary conditions. Therefore, we conclude that it is not possible to obtain a wavefunction satisfying the appropriate boundary conditions for the present model with the considered noncommutativity.