Discretization parameter and operator ordering in loop quantum cosmology with the cosmological constant

In loop quantum cosmology, the Hamiltonian reduces to a finite difference operator and quantum dynamics are controlled by the difference equation. In this framework, Bojowald [M. Bojowald, Phys. Rev. Lett. 86, 5227 (2001).] showed that the initial singularity is absent in the twofold sense: (i) the spectrum of the inverse scale factor operator is bounded from above; (ii) the wave function of the Universe can be uniquely extended beyond the point which was the initial singularity in classical theory. In this paper, we study the initial singularity in this sense and the large-volume limit against the ambiguities in the discretization and the operator ordering within a homogeneous, isotropic and spatially flat model with the cosmological constant. We find that the absence of the singularity strongly depends on the choice of the operator ordering and the requirement for the absence singles out a very small class of orderings. Moreover we find a general ordering rule required for the absence of the singularity. We also find that the large-volume limit naturally recovers a smooth wave function in the discretization where each step corresponds to a fixed volume increment but not in the one where each step corresponds to a fixed area increment. If loop quantum cosmology is to be a phenomenological realization of full loop quantum gravity, these results are important to fix the theoretical ambiguities.

Figure 1

The wave function $\Psi (\mu )$ for (a) $0\le \mu /{\mu }_0\le 700$ and (b) $2100\le \mu /{\mu }_0\lesssim 2450$, and (c) the logarithmic scale of its absolute value are plotted as a function of $\mu$ with $\tilde{\Lambda }{\mu }_0^3=0.005$, where we choose the initial values as $\Psi (-4{\mu }_0)=-1$ and $\Psi (0)=0$.

Figure 2

The wave function $\Phi (v)$ for (a) $0\le v\le 400$ and (b) $9400\le v\le 10000$ are plotted as a function of $v=4n$ with $\sqrt{3}\tilde{\Lambda }=0.1$, where we choose the initial values as $\Phi (-4)=-1$ and $\Phi (0)=0$.