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The physical nature of the fundamental scalar field generation and hence the origination of the Universe is a matter of the discussions for many years. We propose to use the statistical approach to the description of the steady states of the quasi stationary systems with the elements of the quantum field theory methods as a basis to explain the appearance of the cosmological scalar field. Particularly, we apply two fundamental principles,
*i.e.*, the H-theorem and least-energy principle to show principal possibility of the scalar field origination. Along with the basic statement that in the presence of the fundamental scalar field, the energy of the vacuum ground state is lower than the ground state energy of the vacuum with no scalar field (primary vacuum), and with regard to the nonlinear interaction of fluctuating physical fields with the scalar field, these principles are employed to reveal probable phase transitions that may be associated with origin and further evolution of the Universe. Thus, we propose the possible physical justification of the spontaneous cosmological scalar field generation.

Modern notions (rather hypotheses) on the reason for the formation of the Universe imply an instability of some (hypothetical) scalar fields associated with the quantum nature of the matter [

In the case of spontaneous generation of the scalar field in vacuum, the ground-state energy of the “new” vacuum (i.e. the initial vacuum plus the scalar field) for the fields of other nature should be lower than the ground state energy of the “initial” vacuum. Moreover, the self-consistent interaction of the scalar field with fluctuations of any other field provides energy conservation for the new state of the system. Calculations of the partition function for this system reveal a probability of a phase transition from the state with zero scalar field to the state with finite spontaneously generated scalar field.

According to Gibbs [

ρ ( q , p ) d Γ = e x p { F − H ( q , p ) Θ } d Γ (1)

where H ( q , p ) = E is the Hamiltonian on the hypersurface of the constant energy E, d Γ = ∏ i d q i d p i is an element of the phase space, Θ = k T , T is the temperature, and F is the free energy that can be found from the normalization

condition ∫ e x p ( F − H ( q , p ) Θ ) d Γ = 1 . The phase space is known [

ρ ( E ) d E = C e x p { F − E Θ + Σ ( E ) } d E . (2)

The normalization condition yields ∫ c e x p ( F − E Θ + Σ ( E ) ) d E = 1 . In order to select the states with dominant contributions in the partition function, we employ the condition for the temperature given by d Σ d E = 1 Θ .

We assume that the relation between the changes of the value of the phase space from the energy E is known. In terms of this definition and within the context of fundamental principles of statistical mechanics [

reproduces the entropy of the system bearing in mind that the temperature describes dependence of entropy only on energy but not on the other thermodynamic functions. It also follows that integration over energy in the continual sense yields an expression for the partition function. It is obvious that the extreme contribution in the partition function is associated with the states for which F = E − θ S and that for any deviations from the latter condition the contribution in the partition function is negligibly small similarly to the contribution of quantum corrections to the classical trajectories [

The Universe is non-equilibrium from origination, so in order to describe its evolution we introduce an additional intrinsic parameter “time”. We assume that both the statistical distribution and the evolution of the Universe can be described in terms of the distribution function that depends only on energy. An example of how this idea is applied to describe the properties of the statistical distribution is given in [

Now let us apply the above speculations to the description of the Universe. First, we suggest that the vacuum ground state possesses energy. We also assume that fluctuations of all fields existing in vacuum can occur and thus we can write the equation of state for the vacuum. The thermodynamic relations yield the

pressure given by P = − Θ d S d V where V is the volume. For pure vacuum, we have P = − Θ d S d E v d E v d V = − d E v d V = − ρ v under the assumption that energy with

density ρ v is additive, i.e., E v = ρ v V and constant entropy. Obtained equation reproduces the known equation of state for the vacuum. In order to describe its evolution we introduce an additional intrinsic parameter, “time”, and write

S ˙ = d S d E E ˙ = 1 Θ E ˙ . The latter equation implies that time changes of the entropy

are related to the time changes of energy. Inasmuch as S ˙ > 0 , relaxation to the equilibrium state occurs for E ˙ > 0 , i.e., energy growth is accompanied by the increase of entropy. Thermodynamics regards heat as energy distributed between the degrees of freedom that are not macroscopic observable. Hence we suggest that under the change of the vacuum state the heat d Q = Θ d S varies as

d Q d t = d E d t which in turn implies that heating can occur only under the relaxation towards equilibrium state.

The above consideration and relations are well known. Now we employ them to propose one more possible explanation of the origin of fundamental scalar field. We begin with the assumption that the phase transition from the “initial” vacuum to the new vacuum state is accompanied by the appearance of new scalar field. This means that the presence of the scalar field makes the “new” vacuum different from the “primary” vacuum for any field that may exist. The scalar field decreases the energy of the “new” vacuum with respect to the energy of the “primary” vacuum. Hence, the ground-state energy of the “new” vacuum is given by

E = E v − μ 0 2 2 φ 2 (3)

Here the second term is the scalar field energy; the coefficient μ 0 2 describes the coupling of the new field and the “primary” vacuum, i.e., the self-consistent interaction of the new field with the fluctuations of all other fields that can exist in the “primary” vacuum. Notice, that the coupling coefficient is now positive so there is no need to use the explanations accepted in the standard approach. The contribution of the above interaction to the partition function (2) is given by

Z ~ ∫ D φ ∫ D ξ e x p 1 Θ { − E v + 1 2 μ 2 φ 2 − 1 2 ξ φ 2 − ξ 2 σ 2 } (4)

where the coupling coefficient is presented in terms of its average value plus the fluctuation caused by the nonlinear coupling of the scalar field with a fluctuation field of other nature. We also assume that the mean-square value of the fluctuation is equal to μ 0 2 = μ 2 + ξ . σ 2 is dispersion of the couples coefficient fluctuations. Integration over fluctuation fields yields

Z ~ ∫ D φ e x p 1 Θ { − E v + 1 2 μ 2 φ 2 − σ 2 φ 4 4 } (5)

This means that we have a system with the effective energy (averaged over the fluctuations of the other field coupled with the scalar field) given by

E = E v − 1 2 μ 2 φ 2 + σ 2 φ 4 4 (6)

where the last two-term is the well-known expression for the energy of the fundamental scalar field V ( φ ) = − 1 2 μ 2 φ 2 + σ 2 φ 4 4 . The total effective energy of the “new” vacuum with the fundamental scalar field is given by

E = E v − μ 4 4 σ 2 + σ 2 4 ( φ 2 − μ 2 σ 2 ) 2 (7)

In the case of no scalar field φ = 0 , E = E v while for φ 2 = μ 2 σ 2 the expression for the effective ground state energy of the “new” vacuum reduces to E = E v − μ 4 4 σ 2 . As follows from last relation, the energy of the “new” vacuum is lower than the energy of the primary vacuum and can vanish for E v = μ 4 4 σ 2 .

This relation can be applied to estimate the maximum dispersion of the field fluctuations. If σ 2 tends to infinity, then the energy of the new state tends to the initial energy of the ground state. Thus, we come to the standard form of the energy of the fundamental scalar field, but with different behavior of the energy of vacuum at the presence of the scalar field. The coefficient of non-linearity in the potential energy is determined by the coupling of the fundamental scalar field with the fluctuations of the field of different nature. This means that there could be a new scenario of the Universe formation. In this scenario, the energy of pure vacuum does not contribute to the energy-momentum tensor and thus we cannot introduce dynamic presentation (and geometry) for such state. Only, if the fundamental scalar field appears and the matter is originated we can tell about the geometry. In this sense, the potential of scalar field determines the vacuum state of the Universe.

Standard cosmological models involve a scenario of the Universe nucleation and expansion based on a scalar field which is of fundamental importance for the unified theories of weak, strong, and electromagnetic interactions with spontaneous symmetry breaking [

As mentioned above, only the fundamental scalar field and its symmetry breaking generate the matter, and equilibrium distribution of this matter determines the geometry [

Thus, the assumption of the vacuum ground state energy decrease for all physical fields due to the presence of the fundamental scalar field makes it possible to reveal the probability of a phase transition caused by the spontaneous generation of the field, i.e. the phase transition from vacuum with zero scalar field to the “new” vacuum with the spontaneously generated field. Combining this assumption with the idea that the Universe interacts with the fluctuations of various physical fields in vacuum we can get a consistent picture of the Universe origination and evolution. The decrease of the initial ground state energy does not contradict the H-theorem, because the distribution functions describing the evolution governed by the Fokker-Planck equation are known to satisfy it [

This work was supported by the Target Program of Fundamental Research of the Department of Physics and Astronomy of the National Academy of Sciences of Ukraine (N0120U100857).

The authors declare no conflicts of interest regarding the publication of this paper.

Lev, B.I. and Zagorodny, A.G. (2020) Noise-Induced Origin of the Fundamental Scalar Field. Journal of Modern Physics, 11, 502-508. https://doi.org/10.4236/jmp.2020.114032