br After quantization the Laplace Beltrami operator where re

After quantization, the Laplace-Beltrami operator where , replaces the quadratic form of the momenta (11). In this case, the “kinetic energy” operator has the form
The first term in (14) is a “radial” part of the kinetic energy of the Universe similar to the radial part of the kinetic energy of an hcv virus in quantum mechanics [13]:
The measure of integration in the configuration space of the model is
Initial state of minimal excitation of the Universe If we accept the semi-classical approximation for the inflation stage of the evolution of the Universe [9] in which the role of a cosmic time parameter is played by the inflaton scalar field ϕ, next we will have to determine the initial state of the Universe corresponding to the initial value ϕ0 of the field. In Ref. [9] the part of the kinetic energy operator corresponding to the scale factor a is determined as follows: and a new variable , is introduced in place of the scale factor. Then, a Gaussian wave packet centered in is taken for the initial state of the Universe. Excitations of all other physical degrees of freedom (here, the parameters of anisotropy β±), excluding the scale factor a, are not taken into account. Such a choice of the initial state is natural for quantum mechanics but it is rather arbitrary. In the present work, we shall adhere to the traditional interpretation of the scale factor a as radial variable in the geometry sector of the configuration space with the corresponding operator ordering, and determine the initial state of the Universe by the principle of its minimum excitation. Such an initial state of the Universe may be called its basic, or vacuum state. In the case of a closed Universe, the dynamical structure of GR has a measure of excitation of its physical degrees of freedom similar to the positive definite energy [10] in an asymptotically flat space-time geometry. Such a measure is a minimal eigenvalue of a 3D Dirac operator defined on a spatial slice of the Universe [12]. In the case of a homogeneous anisotropic Universe considered in the present work, the square of the minimal eigenvalue is equal to the positive part of the super-Hamiltonian H. After quantization, this quantity becomes a positive definite operator
In analogy with the ground state of an electron in quantum mechanics determined by the minimum condition of its mean energy, let us define the initial state of minimal excitation of the Universe by the condition of the minimum mean value of the operator (18) on a space of wave functions . This initial state will indeed correspond to the minimal excitation of physical degrees of freedom β±, if an additional constraint condition (the classical GR super-Hamiltonian equals zero, i.e. H = 0) is taken into account. In the quantum theory of the initial state of the Universe proposed here, the mean value of the super-Hamiltonian operator equals zero. Therefore, we come to the initial state of the Universe as determined by the conditional extremal principle with the functional where λ is an indefinite Lagrange multiplier. Let us further simplify the model by excluding the anisotropy physical degrees of freedom β±. Then, according to (8), the initial state we are looking for will be a state of the minimal volume of the Universe. Let us estimate the minimal volume taking for the initial state a simple exponential function with the only variation parameter , where a0 is the required minimal radius. In order to determine the only parameter μ in this case, it is sufficient to solve the weak constraint equation
The function (20) is selected so that all integrals that arise in Eq. (21) may be calculated precisely. For instance, a mean value of the volume of the Universe in this state equals and Eq. (21) is reduced to the power equation for μ A solution of Eq. (23) may be found approximately, assuming that a0 > >. In that case the first term in (23) may be neglected and we obtain