  We propose an improved variant of the third-quantization scheme, for the
spatially homogeneous and isotropic cosmological models in Einstein gravity
coupled with a neutral massless scalar field. Our strategy is to specify a
semi-Riemannian structure on the mini-superspace and to consider the quantum
Klein-Gordon field on the mini-superspace. Then, the Hilbert space of this
quantum system becomes inseparable, which causes the creation of infinite
number of universes. To overcome this issue, we introduce a vector bundle
structure on the Hilbert space and the connection of the vector bundle. Then,
we can define a consistent unitary time evolution of the quantum universe in
terms of the connection field on the vector bundle. By doing this, we are able
to treat the quantum dynamics of a single-universe state. We also find an
appropriate observable set constituting the CCR-algebra, and obtain the
Schr\"odinger equation for the wave function of the single-universe state. We
show that the present quantum theory correctly reproduces the classical
solution to the Einstein equation.
