_{1}

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It is shown that the non-equilibrium electrically neutral and relativistically invariant vacuum-like state with the negative energy density and positive pressure may exist at the non-zero temperature in the system of spinor particles, antiparticles, and random electromagnetic field generated by particle-particle, particle-antiparticle, and antiparticle-antiparticle transitions. At the temperature of the order of 10
^{-5} K, the energy density of its state corresponds to the dark energy density in absolute magnitude. The cosmological constant for such material medium turns out to be negative.

Article [

Interest in the investigation of such states is related to search for the physical interpretation of the cosmological constant in the Einstein’s equations [

If a material medium exists, whose energy-momentum density tensor is expressible in the following form

then the cosmological constant ^{ }

Here G is the gravitation constant, c is the velocity of light. In case of the empty flat space, when the Minkovsky’s tensor

If

The cosmological constant problem is particularly relevant in the context of the experimental confirmation of the accelerated expansion of the Universe [

Experimental discovery of gravitational waves [

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 [

Let us consider the model where the electromagnetic field tensor

is governed by the Maxwell’s equations

with random source

We consider that for flow density

and for potential

Suppose, devices available enable measuring only some time interval and/or volume mean values, which will be indicated with brackets

but at the same time for decomposition components

the following relationship is true:

where

The formal solution of Equations (6) for

taking into account Decomposition (10), can be written down in the following form

where

Let us determine energy-momentum density tensors of electromagnetic field

These tensors can be presented using quadratic combinations

Suppose, tensor

Here indices e and h pertain to particles and antiparticles, respectively;

indices r and

the Fermi-Dirac distribution function;

Vectors

where

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

we obtain from Relationships (16.1) and (16.2)

where

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

In the frame of reference where

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

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 [

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

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

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

Furthermore, the energy dominance [

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

If the energy density of the system considered is identified with the dark energy density, then temperature ^{−5} K and Fermi velocity of particles and antiparticles

Ilya A. Obukhov, (2016) Cosmological Constant and Energy Density of Random Electromagnetic Field. Journal of High Energy Physics, Gravitation and Cosmology,02,312-319. doi: 10.4236/jhepgc.2016.23028