On the physical Hilbert space of loop quantum cosmology

In this paper we present a model of Riemannian loop quantum cosmology with a self-adjoint quantum scalar constraint. The physical Hilbert space is constructed using refined algebraic quantization. When matter is included in the form of a cosmological constant, the model is exactly solvable and we show explicitly that the physical Hilbert space is separable, consisting of a single physical state. We extend the model to the Lorentzian sector and discuss important implications for standard loop quantum cosmology.

Figure 1

Physical solution (50). The dots indicate the values of the solution for where $(\nu /4{\nu }_0)+[3/(2\Lambda {\nu }_0^3)]\in \mathbb{Z}$. The actual solution would be zero elsewhere; however the solid lines are interpolated values designed to bring out the behavior of the solution more clearly.

Figure 2

Difference equation solutions for both the simplified constraint (a) and standard loop quantum cosmology (b). The solution of (a) is the physical solution. The solution in (b) is the one picked out by the dynamical initial condition. The solid lines are the interpolated values. The actual physical solution of (a) would be zero except for the one sector indicated by the dots. For (b) the dynamical initial condition specifies that the wave function in the other sectors is the one that smoothly interpolates between the dotted solution.

Figure 3

Bessel $J$ solution of (58) for two different sectors. The solution of (a) is for the sector where the Bessel function order is an integer. This solution converges both for large negative and positive $\nu$. The solution of (b) is for a different sector and diverges for large $\nu$. The solid lines are interpolated values.

Figure 4

Normalizable solution to the difference equation of LQC (55) without the $\mathrm{s}\mathrm{g}\mathrm{n}(\nu )$ on the right-hand side. Here matter is only in the form of a cosmological constant. The solution is valued only in a single sector which approximately is given by the same sector of the simplified constraint where $-\nu /(4{\nu }_0)+9/(2{\gamma }^3{l}_p^2\Lambda {\nu }_0^3)\in \mathbb{Z}$.

Figure 5

Wave function of standard LQC (dots) compared with the corresponding Wheeler-DeWitt solution (solid line) for a positive cosmological constant. The deviation occurs when the value of the connection $A$ is of the order of $\pi /{\nu }_0$ which in this case occurs for large volumes.