Gauge-invariant construction of quantum cosmology

We present and analyze a gauge-invariant quantum theory of the Friedmann-Robertson-Walker universe with dust. We construct the reduced phase space spanned by gauge-invariant quantities by using the so-called relational formalism at the classical level. The reduced phase space thereby obtained can be quantized in the same manner as an ordinary mechanical system. We carry out the quantization and obtain the Schrödinger equation. This quantization procedure realizes a possible resolution to the problem of time and observables in canonical quantum gravity. We analyze the classical initial singularity of the theory by evolving a wave packet backward in time and evaluating the expectation value of the scale factor. It is shown that the initial singularity of the Universe is avoided by the quantum gravitational effects.

Figure 1

The action of the map ${\alpha }_C^t$ on phase space points and the gauge-invariance of $O_F^{\tau }(x)$.

Figure 2

The expectation value of the scale factor $⟨A⟩$ as a function of time $\tau$ for the four cases $(r,\lambda )=(0,0)$, (0, 1), (1, 0), and (1, 1), where $r$ represents the dimensionless parameter associated to the total energy of the radiation while $\lambda$ is the dimensionless cosmological constant. We show only the cases when the matter involves the radiation because the plots in the models with or without the radiation almost completely overlap. The solid line represents the cases with the cosmological constant, while the dashed line is for the cases without the cosmological constant. Figs. (a), (b), and (c) correspond to the three kinds of the operator ordering discussed in Sec. 4.

Figure 3

The absolute value of the wave function is plotted as a function of the time $\tau$ and the scale factor $A$ in the case when the matter consists of the dust, the radiation, and the cosmological constant. Figs. (a), (b), and (c) correspond to the three kinds of the operator ordering discussed in Sec. 4.