We present a phase-space noncommutative extension of quantum cosmology and study the Kantowski-Sachs cosmological model requiring that the two scale factors of the Kantowski-Sachs metric, the coordinates of the system, and their conjugate canonical momenta do not commute. Through the Arnowitt-Deser-Misner formalism, we obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system. The Seiberg-Witten map is used to transform the noncommutative equation into a commutative one, i.e. into an equation with commutative variables, which depend on the noncommutative parameters $\theta$ and $\eta$. Numerical solutions are found both for the classical and the quantum formulations of the system. These solutions are used to characterize the dynamics and the state of the universe. From the classical solutions we obtain the behavior of quantities such as the volume expansion, the shear, and the characteristic volume. However, the analysis of these quantities does not lead to any restriction on the value of the noncommutative parameters $\theta$ and $\eta$. On the other hand, for the quantum system, one can obtain, via the numerical solution of the WDW equation, the wave function of the universe for commutative as well as for the noncommutative models. Interestingly, we find that the existence of suitable solutions of the WDW equation imposes bounds on the values of the noncommutative parameters. Moreover, the noncommutativity in the momenta leads to damping of the wave function, implying that this noncommutativity can be of relevance for the selection of possible initial states of the early universe.

• Received 22 December 2007

DOI:https://doi.org/10.1103/PhysRevD.78.023516