Received 14 April 2016; accepted 25 June 2016; published 28 June 2016
Article  shows that the quantum theory allows building up the relativistically invariant state of matter, for which pressure p and energy density are related as follows
Interest in the investigation of such states is related to search for the physical interpretation of the cosmological constant in the Einstein’s equations  -  and dark energy nature  -  .
If a material medium exists, whose energy-momentum density tensor is expressible in the following form
then the cosmological constant in the Einstein’s equations  can be related to the energy density of such medium
Here G is the gravitation constant, c is the velocity of light. In case of the empty flat space, when the Minkovsky’s tensor with diagonal is the metric tensor, energy density and pressure obviously satisfy the relationship (1).
If, then negative pressure can be interpreted   as the cause of the antigravity. The opposite possibility, where   , is not also excluded. In such a case, the gravitational attraction between massive objects starts increasing at the distances exceeding some threshold one.
The cosmological constant problem is particularly relevant in the context of the experimental confirmation of the accelerated expansion of the Universe  . However, this phenomenon can be explained not only with the help of the General Relativity, but also by the different alternative versions of the Gravity Theory (see  and the articles cited there), where dark energy concept isn’t introduced. In paper  it is shown that the analysis of the interference of gravitational waves will allow making a choice for that or another Gravity Theory.
Experimental discovery of gravitational waves  allows hoping for increasing of our understanding of the nature of gravitation. Data obtained now confirm justice of the General Relativity  . But it is only the first experiments and it is not enough for final conclusions.
In any case, up to date, the material medium model-building problem, for which relationship (1) is true and density
is close, at least in the order-of-magnitude, to the value of
resulted from the interpretation of astrophysical data  that has not been solved. Solving this problem seems important for the understanding of potential physical phenomena leading to the formation of such material medium.
2. Energy-Momentum Tensor of Random Electromagnetic Field
Let us consider the model where the electromagnetic field tensor
is governed by the Maxwell’s equations
with random source on the right side. Here is the vector potential of electromagnetic field, and e is the electric charge.
We consider that for flow density the conservation law is true
and for potential the Lorentz’ gage condition is true
Suppose, devices available enable measuring only some time interval and/or volume mean values, which will be indicated with brackets. Suppose that
but at the same time for decomposition components on some set of functions,
the following relationship is true:
where is the tensor, whose exact form will be defined later on.
The formal solution of Equations (6) for meeting the initial condition
taking into account Decomposition (10), can be written down in the following form
where. According to Equations (5), (9), and (13)
Let us determine energy-momentum density tensors of electromagnetic field and its interaction with charged flow
These tensors can be presented using quadratic combinations, and therefore their average values may be non-zero. Using Formula (11), for the average value of the sum of tensors and, we obtain
Suppose, tensor describes transitions between acceptable states of spinor particles and antiparticles with mass m and charges e and −e in the binary mixture of their gases. Then it can be defined using the following relationships
Here indices e and h pertain to particles and antiparticles, respectively; is the Plank’s constant divided by;, and are four-dimensional momenta of particles and antiparticles, for which the following dispersion relationship is true
indices r and take on the values of ±1 and correspond to the two possible spin states of particles and antiparticles; and are chemical potentials of particles and antiparticles, which we will consider as constant;
is the Boltzmann’s constant, T is the temperature;
the Fermi-Dirac distribution function;
is the hydrodynamic velocity  that meets the relationship
Vectors correspond to flows of transitions between the states of particles and antiparticles
where is the Dirac gamma-matrices, and are spinors that satisfy Equations
and normalization conditions
It is easy to show that the relations of orthogonality are true
With account of these definitions, Expressions (25), and the formula for the transition from the summation over composite index to the integration over momentum p
we obtain from Relationships (16.1) and (16.2)
where is the fine-structure constant approximately equal to 1/137, if e is the electron charge.
3. Energy Density and Pressure of Interacting Mixture
Assume, the inequations are true
i.e. particle and antiparticle gases are degenerating ones. Let us consider the range of temperature and chemical potential values that satisfy the following conditions:
Here and are Fermi momenta defined in the following relationships:
In the frame of reference where, we obtain the expressions for energy density and pressure
related to the components of the energy-momentum tensor by simple relationships
The following formulas are true under Conditions (28) in the same frame of reference for concentrations of particles and antiparticles and, their energy densities and pressures
The condition of electroneutrality of the particle and antiparticle gases mixture has the following form
whence it follows that
In such a case, we obtain for the total energy density and pressure
The energy density and pressure of electrically neutral particle and antiparticle gases mixture will be relativistically invariant in two cases  where
and where condition (1) is met. In the first case, we obtain from Equations (35.1) and (35.2)
Furthermore, according to Formulas (32.1), concentrations of particles and antiparticles are zero.
In the second case, we obtain the condition of implementing the electrically neutral and relativistically invariant state for the mixture under consideration from Relationships (1), (35.1) and (35.2)
Here the Fermi momentum and temperature, at which conditions (1) is met, are indicated using and. It is easy to show that momentum satisfying Equation (37) satisfies also Inequations (28)
The following expressions are true for the concentrations of particles and antiparticles, energy density and pressure of the mixture
We receive the following from Formulas (3), (4), and (38.2) for the cosmological constant and dark energy density
Here the following is specified
−a dimensionless value representing a gravitational analogue of the fine structure constant in terms of the build-up method, if m is the mass of electron, then;
is the analogue of the Compton wave-length
for an object with the energy equal to.
If e and m are the charge and mass of electron, and dark energy density
The second of Formulas (39.2) for density has the strongly non-isotropic form. As if the dark energy is distributed in a parallelepiped with the very small height as compared with the typical dimensions of edges of bases
Note that the contribution of the random electromagnetic field, including the contribution of interaction of this field with particles and antiparticles, to the total energy density in absolute magnitude is thrice as much as the contributions of particles and antiparticles equal to each other
Just due to the random electromagnetic field generated by transitions between particles and antiparticles, the state that satisfies Condition (1) is possible in the system studied.
In the approximation considered, energy density and pressure are resulting mainly from particle- particle and antiparticle-antiparticle transitions. The energy density and pressure related to particle-antiparticle transitions are much less than and
Furthermore, the energy dominance  is disrupted for and, and their components.
The model of the material medium, for which the electrically neutral relativistically invariant state with non-zero energy density, pressure, concentrations of particles and antiparticles is possible, has been considered. At the same time particles and antiparticles are in the thermal equilibrium, but far from the chemical equilibrium state governed by the equality of their chemical potentials. In the considered case, the expression is true
This non-equilibrium electrically neutral state exists due to the random electromagnetic field generated by spontaneous transitions between particles and antiparticles being in different quantum states. The average vector potential and intensity of this field are zero. But the average components of the energy-momentum density tensor in the random process of transitions between the states of particles and antiparticles are non-zero.
The energy density of the above vacuum-like state can be expressed in terms of its temperature. The state electroneutrality requirement leads to the equality of Fermi momentum of particles and antiparticles, and the relativistic invariance requirement to the equation relating the Fermi momentum with the temperature.
If the energy density of the system considered is identified with the dark energy density, then temperature turns to be of the order of 10−5 K and Fermi velocity of particles and antiparticles to be approximately 1 cm/s (m is the mass of electron). Furthermore, the dark energy density and cosmological constant are negative.