br For the integral to be

For the integral (8) to be finite we set the regularization parameter ξ equal to –1 at this stage. Then the Gauss integral (8) can be calculated without effort. The following simple example illustrates the regularization procedure proposed here (for ):
The only irritant in our regularization procedure is an occurrence of the constant multiplier , which arises in Eq. (8) after orexin manufacturer over a(τ). But this multiplier does not depend on the dynamical variables, so it can be omitted. Notice that the regularization parameter ξ in the integral (3) can be inserted into the lapse function , so that we derive two independent variables and, correspondingly, two parameters which should be identified at the final stage. In the former case (i) the Gauss integral (8) equals where the action in the exponent is calculated on the classical trajectory ((t), (t)), t ∈ [0, C] with the corresponding boundary condition:
As a result, the action is equal to
In the latter case (ii), and
Let us now restore the ``normal\'\' value of the regularization parameter , and return to the real time . As the result, the Euclidean action in the exponent of formula (10) becomes an imaginary phase function, which defines a real phase which we consider as a quantum action corresponding to the birth of the universe. In the former case (ψ is smooth in the South Pole) the quantum action is
At the last step in our definition of the wave function, we fix the time of birth Tusing the additional condition of the extreme value of the quantum action: from which
The solution of Eq. (17) can be interpreted as the time of the universe\'s birth in the intimated state from “nothing”. The corresponding stationary value of the quantum action is
It is easy to check that the stationary wave function is one of the solutions of the WDW Eq. (5) with . In the latter case (ϕ is smooth in the South Pole) the corresponding quantum action is
The condition for it to be stationary with respect to the variation of the time of birth T implies . Therefore, the stationary wave function would be taken as
It is also a solution of the WDW Eq. (5). The time of this state\'s birth is not defined.
Conclusion This procedure can be interpreted as a complex amplitude of the universe\'s birth from “nothing” with the time parameter not defined yet. Considering the phase of the complex amplitude as a quantum equivalent of the classical action, at the last step in our definition of the wave function we proposed to fix the time of birth Tusing the additional condition of the parameter extremum of the quantum action.
Acknowledgment
Introduction The modern cosmological paradigm includes (as an inevitable part) the existence of an inflation stage with the exponential expansion of the universe [1–4]. A quantum epoch takes its own place substantially at the beginning of the universe in just the same way. Inflation theories do not explicitly specify the initial state of the universe and its size before the inflation. For instance, the inflation may be supposed to begin exactly following the quantum epoch with the Planck initial size [5]. On the other hand, in different quantum theories of the Beginning [6, 7] the classical inflation stage is considered as a natural continuation of the history of the universe. According to the Vilenkin tunneling theory [6], the de Sitter stage of the exponential expansion of the universe is sewn together with the de Sitter instanton at a definite radius determined by a vacuum energy density in the framework of the Grand Unified Theory. In order to accommodate the tunneling theory with inflation theories where an effective scalar field (inflaton) is present, the vacuum energy density should be identified with an initial value of the inflaton potential energy. In Ref. [8], the classical stage of the inflation in the quasi-classical approximation of the Hartle–Hawking no-boundary wave function of the universe was obtained as well.