Wave Packet in Quantum Cosmology and Definition of Semiclassical Time

We consider a quantum cosmology with a massless background scalar field \pb and adopt a wave packet as the wave function. This wave packet is a superposition of the WKB form wave functions, each of which has a definite momentum of the scalar field \pb. In this model it is shown that to trace the formalism of the WKB time is seriously difficult without introducing a complex value for a time. We define a semiclassical real time variable \tp from the phase of the wave packet and calculate it explicitly. We find that, when a quantum matter field \pq is coupled to the system, an approximate Schr\"odinger equation for \pq holds with respect to \tp in a region where the size $a$ of the universe is large and |\pb| is small.Comment: 20 pages, latex, no figure

Third quantization of f(R)f(R)-type gravity

We examine the third quantization of $f(R)$-type gravity, based on its effective Lagrangian in the case of a flat Friedmann-Lemaitre-Robertson-Walker metric. Starting from the effective Lagrangian, we execute a suitable change of variable and the second quantization, and we obtain the Wheeler-DeWitt equation. The third quantization of this theory is considered. And the uncertainty relation of the universe is investigated in the example of $f(R)$-type gravity, where $f(R)=R^2$. It is shown, when the time is late namely the scale factor of the universe is large, the spacetime does not contradict to become classical, and, when the time is early namely the scale factor of the universe is small, the quantum effects are dominating.Comment: 9 pages, Arbitrary constants in (4.19) are changed to arbitrary functions of $\varphi$. Conclusions are not changed. References are added. Typos are correcte