# In both approaches the dynamics of

In both approaches the dynamics of the scale factor of the universe at the inflation stage was considered as a classical one. However, quantum effects, for instance, the ordinary matter creation, are important at the final stages. This also concerns the dynamics of the scale factor of the universe. In order to formulate quantum dynamics of the scale factor, a cosmic time should be defined in the quantum universe. In Ref. [9], a canonical time parameter related to the slow-rolling inflaton scalar field was introduced in the minisuperspace model of the universe. As a result, the Wheeler–DeWitt (WDW) equation for quantum geometry [10] took the form of the Schrödinger equation with that cosmic time. This equation implies the exponential growth of the average volume of the universe, provided the initial state of the scale factor is a Gauss wave packet. The width of the packet is an arbitrary parameter in that approach.
In fact, in recent years a heated argument based on the chaotic theory of inflation [11] has developed on the debated topic whether there was a Beginning of the universe or whether it did not exist at all [12, 13]. Taking into account the positivity of the parp apoptosis pathway of space in a closed universe [14] we have come to the conclusion that a ground state with minimal excitation of the energy of space does exist and it can be taken as the Beginning of the universe. At the same time, the ground state is a state of maximal (Planckian) vacuum energy density. It is similar to Planck energy density in the loop quantum gravity [15], which serves as a “quantum bridge” between large classical universes, one contracting and the other expanding. The minimal energy principle was used first for definition of the ground state of the universe in Ref. [16]. Notice that this state is not stationary and evolves with time. For instance, it admits quantum fluctuations in which a universe with high initial value φ0 of the inflaton scalar field may be nucleated. The space energy in such a proto-inflation quantum state remains minimal admitted by the Hamiltonian constraint in a closed universe.
In this paper we propose a toy model for determination of the proto-inflation quantum state of the universe and its subsequent quantum dynamics on the condition that a cosmic time related to the inflaton scalar field is introduced [9]. Together with the initial ground state of space, excited states are introduced as well, in terms of which the exponential expansion obtained in Ref. [9] is represented as quanta of space birth. It is pertinent to note that these quanta are not the same as those of a spatial volume obtained in the loop quantum gravity [17]. Here, we call quanta the excitations of the space energy.

Minisuperspace quantum inflationary model of the universe
Let us consider a homogeneous Friedman–Robertson–Walker (FRW) model of the universe with the metric
and a scalar field ϕ with the four degree potential
described by the action

Here N is the lapse function [18], a is the scale factor of the universe, (G is the Newton gravitational constant).
By varying the action I with respect to the lapse function N, we obtain the Hamiltonian constraint equation
where and are canonical conjugate momenta to a and φ, respectively.
In conventional quantum theory Eq. (3) is replaced by the WDW equation for a quantum state
with

A problem of ordering of non-commuting multipliers (in the first round brackets) will arise in the definition of the operator . It will be solved below.
We are interested in the slow-roll regime with the slowly varying scalar field φ. So, we shall consider the momentum
to be small, and, following Ref. [9], replace the kinetic energy of the scalar field in the constraint (3) by

Then, a formula
gives an appropriate canonical time parameter [9], where ϕ0 is an initial value of ϕ.
According to Ref. [9], let us choose the following prescription for the operator ordering:
where the new variable