Quantum Vacuum : The Structure of Empty Space-Time and Quintessence with Gauge Symmetry Group SU ( 2 ) ⊗ U ( 1 )

We consider the formation of structured and massless particles with spin 1, by using the Yang-Mills like stochastic equations system for the group symmetry SU(2) ⊗ U(1) without taking into account the nonlinear term characterizing selfaction. We prove that, in the first phase of relaxation, as a result of multi-scale random fluctuations of quantum fields, massless particles with spin 1, further referred as hions, are generated in the form of statistically stable quantized structures, which are localized on 2D topological manifolds. We also study the wave state and the geometrical structure of the hion when as a free particle and, accordingly, while it interacts with a random environment becoming a quasi-particle with a finite lifetime. In the second phase of relaxation, the vector boson makes spontaneous transitions to other massless and mass states. The problem of entanglement of two hions with opposite projections of the spins +1 and −1 and the formation of a scalar zero-spin boson are also thoroughly studied. We analyze the properties of the scalar field and show that it corresponds to the Bose-Einstein (BE) condensate. The scalar boson decay problems, as well as a number of features characterizing the stability of BE condensate, are also discussed. Then, we report on the structure of empty space-time in the context of new properties of the quantum vacuum, implying on the existence of a natural quantum computer with complicated logic, which manifests in the form of dark energy. The possibilities of space-time engineering are also discussed.


I. Introduction
From a mathematical and philosophical point of view, the vacuum can be comparable with the region of absolutely empty space or, which is the same, with the region of the space where there are no massive particles and fields. From the physical point of view, the real vacuum is empty only "on average". Moreover, as it is known, due to the principle of quantum-mechanical uncertainty, there is not any way screening a certain area of space to exclude the appearance of virtual particles and fields in it. The Lamb Shift [1], Casimir effect [2], Unruh effect [3], anomalous magnetic moment of electron [4], Van der Waals forces [5], Delbrück scattering [6], Hawking radiation [7], the cosmological constant problem [8,9], vacuum polarization at weak electromagnetic fields [10,11] -here is an incomplete list of phenomena, part of which has been experimentally discovered. All of them are conditioned by the physical vacuum or, more accurately, by a quantum vacuum (QV). The issues of cosmic acceleration [9] and dark energy-quintessence [12] are often discussed in the framework of QV theories, which necessarily include scalar fields. In [13], the mass and electric dipole moment of the graviton are evaluated, which is identified with a particle of dark matter that radically change our understanding of dark matter and possibly of dark energy. The properties of QV can be studied within the framework of quantum field theory (QFT), ie quantum * 1 Institute for Informatics and Automation Problems NAS of RA, 2 Institute of Chemical Physics NAS of RA ; Electronic address: g˙ashot@sci.am electrodynamics and quantum chromodynamics (QCD). Note that QFT could accurately describe QV if it were possible to exactly summarize the infinite series of perturbation theories, that is typical of field theories. However, it is well known that perturbation theory for QFT is destroyed at low energies, which does not allow describing, for example, nonzero values of the vacuum expectation, called condensates in QCD or the BCS superconductivity theory. In particular, as shown in [14], the radiative corrections of the massless Yang-Mills theory lead to instability of the vacuum state, which in fact completely corresponds to the asymptotic freedom of gauge theories and is due to infrared features.
Moreover, in the Standard Model (SM) precisely the non-zero vacuum expectation value of the Higgs field [15,16], arising from spontaneous symmetry breaking, is the principled mechanism allowing to acquire masses of other fields of theory.
To overcome these difficulties and conduct a consistent and comprehensive study of the quantum vacuum, we propose to carry out a study within the framework of a system of complex stochastic differential equations (SDE) of the Langevin type. Note that such a system of equations makes it possible to describe the massless quantum fields with multiscale fluctuations in Hilbert space, where subspaces with single-particle states of zero mass and spin 1 exist.
The purpose of this article is to give an exhaustive answer to a number of important questions in the theory of quantum vacuum that remain insufficiently studied: a. Can random fluctuations of massless quantum fields lead to the formation of stable massless particles with spin 1 (vector boson-hion)?
b. Is it possible to form vector bosons, what are the space-time features of this particle and how does its quantum state change when multiscale character of random fluctuations of massless fields is taken into account? c. What quantum states of the scalar boson are possible at entangled of hion, respectively, with projections of spins 1 and -1? How do these quantum states evolve taking into account the random influence of the environment and what is the degree of stability of these states in the asymptotic of space-time? Whereby is the symmetry of the quantum state broken which leads to spontaneous transitions of the boson to other states? d. What is the structure of the quantum vacuumempty space-time? Finally, what is the structure of the space-time of scalar massless VF on large scales?

II. Quantum motion of a photon in empty space
The questions of correspondence between the Maxwell equation and the equation of quantum mechanics was of interest to many researchers at the dawn of the development of quantum theory [17][18][19].
As shown (see, for example, [20]), the quantum motion of a photon in a vacuum can be considered within the framework of a wave function representation, writing it in vector form similarly to the Weyl equation for a neutrino: (1) where c 0 denotes the speed of light in an empty 4D Minkowski space-time, Ψ + (r, t) and Ψ − (r, t) denote the photons wave functions of both helicities, corresponding to left-handed and right-handed polarizations. In addition, in (1) the set of matrices S = (S x , S y , S x ) describes infinitesimal rotations of particles with spin projections ±1, respectively: Note that the absence of electric and magnetic charges in the equation (1) ensures the following conditions: If we represent the wave function in the form: then from the Eqs. (1) and (3) it is easy to find Maxwell's equations in an ordinary vacuum or in empty space: where ǫ 0 and µ 0 describe the dielectric and magnetic constants of the vacuum, respectively. It is important to note that the dielectric and magnetic constants provide the following equations: Recall that the only difference between the equations (1) and (5) is that the Maxwell equations system does not explicitly take into account the spin of the photon, which is fundamentally for further constructions.
Since the refractive indices ǫ 0 and µ 0 are constants that do not depend on external fields, and characterize the state of unperturbed or ordinary vacuum, a reasonable idea arises, namely, to consider a vacuum or, more accurately, QV, as some energy environment with unusual properties and structure.
In (6) ψ ± r, t; f(t, ε) ∈ L 2 (R 4 ⊗ R {f} ) denotes the wave function, when R {f} the functional space, f(t, ε) is a random vector probabilistic process characterizing QV fluctuations, which will be described in detail below and c denotes the field propagation velocity, which differs from the speed of light in a vacuum c 0 . In addition, we need similar to conditions (3) the following equations: It is also assumed that at infinitesimal time intervals δt << τ 0 , where τ 0 = min{τ } is the relaxation time of the minimum duration, the complex SDE (6) transits to the Weyl type equation (1). In other words, the equation (1) in this case plays the role of the principle of local correspondence, and therefore further the equation (6) will conventionally be called the Langevin-Weyl equation.
Theorem. If QVF obeys Langevin-Weyl SDE (6) -(7), then in the first phase of relaxation (τ 0 , ǫ 0 ) in the statistical equilibrium limit, a massless Bose particle with spin 1 can be formed, as a 2D topological structure in 3D space.
It is obvious that in the case of a localized quantum state, the 4D-interval of the propagated signal will be zero, and the points of the Minkowski space (events) should be related by the relation, like the light cone: The vector field satisfying the equations (6), depending on the value of the spin projections, can be described by the wave function: Substituting (9) into (6) and taking into account (7), we can obtain the following independent system of SDE: In addition, taking into account (7), we can find the following relations between the fields: Bearing in mind that particles with projections of spins +1 and −1 are symmetric, below we will study only the wave function of a particle with spin projection +1.
Taking into account (10) and (11), we can be obtained: where = △ − c −2 ∂ 2 t denotes the D'Alembert operator, △ is the Laplace operator, c ,σ = ∂c/∂σ and σ = (x, y, z, t). To determine the explicit form of equations (12), we need to calculate the derivatives c ,σ . Using the equations (8), we get: To study of the problem, we need to reduce the system of equations (12) to the canonical form, when the field components are separated, and each of them is described by an autonomous equation.
Taking into account the fact that in the considered problem all the fields are symmetric, the following additional conditions can be imposed on the field components: It is easy to verify that these conditions are symmetric with respect to the components of the field and are given on the hyper-surface of four-dimensional events. Further, using the conditions (14), the system of equations (12) easily reduces to the canonical form: For further investigations, it is convenient to represent the wave function component in the form: where σ = x, y, z, and ζ σ (t) denotes the random function and f σ (t, ε) denotes the corresponding projection of the vector f(t, ε).
Hereinafter, based on the symmetry of the considered problem, we can assume that; ζ x = ζ y = ζ z = ζ and Substituting (16) into (15) and taking into account (13), we get the following system of differential equations: In the equations (17) the following notation is made: whereζ(t) = dζ(t)/dt.
It is easy to verify that the coefficients in the equations (17), are random functions of time. It will be natural if we average these equations on the scale of the relaxation time τ 0 .
Averaging (17), we obtain the following system of stationary differential equations of the second order: where ω and ̺(ω) are regular parameters of the problem, which are defined as follows: In the (19) the bracket ... denotes the averaging operation by the relaxation time τ 0 . Now the main question is that: is it possible the emergence of statistical equilibrium in the system under consideration, which can lead to the stable distribution of the parameter ̺(ω)?
Note that the latter circumstance, for obvious reasons, eliminates the nontrivial question connected with the unitary transformation of the state vector, since the quantum system in this problem is not isolated. Obviously, in this case it is necessary to require the conservation of the norm of the average value of the state vector: where Recall that now the key question for the representation will be the proof of existence of an average value of the state vector φ ± (r) in the limit t → ∞.
Using the first relation in (19), we can determine the following non-linear SDE of the Langevin type: In (21) the term f (t, ε 0 ) denotes the random force satisfying of the white noise correlation relations: where ε 0 is a constant denoting the fluctuations power.
Using SDE (21) and relations (22), one can obtain the equation for the probability distribution [21,22]): The equation (23) is solved exactly and has the following form [23]: whereζ = ζ/ε 1/3 0 and Φ(ζ) = (ζ 3 + 3ω 2ζ )/3. As regards the coefficient J (ω), it is determined from the condition of normalization of the distribution P(ζ;ω) to unity: is the dimensionless frequency and the function J (ω) has the meaning of the probability density of states.
Finally, using the von Neumann mean ergodic theorem [24] and also the Birkhoff pointwise ergodic theorem [25], we can calculate the function ̺(ω): Note that the function ̺(ω) has the dimension of frequency. Following the standard procedures (see in detail [28]), we can construct a measure of the functional space R σ {f } and, accordingly, to calculate the functional integral entering into the expression (20): where P 1 (ζ σ , t) is a function satisfying the following second-order partial differential equation: Now it is important to show that the integral (26) converges. As proven (see Appendix A), in the limit of statistical equilibrium; lim t→∞ I 1 (t) ≤ M = const, or, which is the same thing, the integral (26) converges. The latter means that the function P 1 (ζ σ , t), we can give the sense of the probability density and normalize it on unity.
Thus, we have proved that on the scale of the relaxation time τ 0 , the system goes to a statistical equilibrium state and describing by the stationary wave function (20). Obviously, in this case the parameter ̺(ω) is a regular function of the frequency.
A. The wave function of a massless particle with spin 1 Since the equations in the system (18) are independent, we can investigate them separately. For definiteness, we consider the first equation of the system (18).
Representing the wave function in the form: from the first equation of the system (17), we can get the following two equations: where the parameter: has the dimension of the inverse distance.
As is easy to show, these equations are transformed into each other by simple substitutions φ (r). The latter means that these solutions are equal in absolute value and differ only in sign. In other words, the symmetry properties mentioned above make it possible to obtain two independent equations of the following form: Now we will analyze the possibility of obtaining a solution in the form of a localized state for the term φ +(r) (r). For this we consider the following equation of the plane: where µ is a some parameter. The changing range of this parameter will be defined below.
Taking into account (32) the first equation in (31), can be written in the form: It is convenient to carry out further investigation of the problem in spherical coordinates. Rewriting the equation (33) in the spherical coordinate system (x, y, z) → (r, θ, ϕ), we obtain: Representing the wave function in the form: we can conditionally separate the variables in the equation (46) and write it in the form: and, respectively; where Λ ′ = dΛ/dr and ν is a constant, which can represented in the form ν = l(l + 1), in addition, l = 0, 1, 2... Note that the conditional separation of variables means to impose an additional condition on the function µ(θ, ϕ) = const. Writing equation (32) in spherical coordinates, we obtain the following trigonometric equation: Analysis of the equation (38) shows that the range of the ]. The solution of the equation (37) is well known, these are spherical Laplace functions Y l,m (θ, ϕ), where m = 0, ±1, ..., ±l.
As for the equation (36), we will solve it for a fixed value µ, which is equivalent to the plane cut of the three-dimensional solution. In particular, we will seek a solution Λ(r) tending to finite value for r → 0 and, respectively, to zero at r → ∞.
For a given parameter µ 0 > 0, we can write the equation (36) in the form: where ρ = r/a p and a p = 2/(|λ|µ 0 ) denotes the characteristic spatial dimension of a hypothetical massless Bose particle with spin projection +1 and β = (ωa p /c). It is important to note that from the symmetry and noncoincidence of the components φ , it follows that µ 0 = 2. This fact will be taken into account in further calculations.
As well-known the solution of the equation (39) describes the radial wave function of the hydrogen-like system, which is written in the form [26]: where b = (2/na p ) and L 2l+1 n−l−1 (br) is the generalized Laguerre polynomials.
Note that the solution (40) takes place if the following condition is satisfied: where n r is the radial quantum number, n is the principal quantum number and l denotes the quantum number of the angular momentum limited by the principal quantum number l ≤ n − 1.
In other words, the condition of quantization is the integer value of the member β −1 or, when the following equations are exactly satisfied: is a dimensionless function, the brackets [...] and {...} denote integer and fractional parts of the function, respectively.
As follows from the calculations (see FIG. 1), in the frequency rangeω ∈ {0.05, 0.34} there are 8 points, that are highlighted in red, satisfy the quantization conditions (42). The latter means that in specified frequency range there are only eight quantum states, however the number of these states is growing atω → 0. Table 1. The average-statistical dimensionless frequency of the system in different quantum states (see (41)). lution φ . Taking into account the fact that µ 0 = 2, the equation (38) can be written in the form: In particular, as follows from the equation (43), all solutions (40) are localized on the manifold S r The imaginary part of the wave function φ is calculated similarly and has the same form, but in this case the solution must satisfy the following trigonometric equation: Obviously, the equation (44) } shows that the separation of variables in corresponding equations is possible taking into account the following algebraic equations: Analysis of the equations (45) shows that the projections of the wave function φ + (r) are localized on the following manifolds; {φ The theorem is proved.
Thus, we have proved the possibility of formation of massless Bose particles with spin 1 as a result of random fluctuations of QVF. As can be seen the obtained solutions (40) combine the properties of quantum mechanics , respectively. and the theory of relativity and, respectively, maximally reflect to the ideas of string theory. It is interesting to note that the ground state of the vector boson characterized the highest frequency. In the future we will call the particle of a vector field hion.

IV. Hion distribution in different quantum states
Let us consider the solution of the equation (39) in the ground state.
Taking into account (35) and (40), for the solution; φ x(1,0,0) (r) = Λ 10 (r)Y 0,0 (θ, ϕ) localized on the manifold S r x , the following expression can be obtained: where the indices (1, 0, 0) of the wave function denote the quantum numbers (n, l, m), accordingly, the constant C is defined below from the normalization condition of the wave function, in addition, a p is the characteristic spatial dimension of the vector boson in the ground state, which can be calculated taking into account the equations (25) and (30): Recall that in (47) the frequency ω 1 = ε 1/3 0ω 1 , where dimensionless frequency of the ground state is equalω 1 = 0.34 (see table and FIG. 1). Within the framework of the developed representation, it is impossible to determine the constant a p , since c and ε 0 remain free parameters of the theory. Apparently, these parameters will have to be refined experimentally and introduced into the theory as fundamental constants.
As for the wave function φ +(i) x(1,0,0) (r), it is also described by the expressions (46), but with the only difference that in this case the wave function is localized on the manifold S i x . In a similar way one can obtain solutions for the wave functions φ + y(1,0,0) and φ + z(1,0,0) localized on the corresponding manifold. Now we can write down the normalization condition for the full wave function: where φ+ T = φ+ x ,φ + y ,φ + z is the transposed vector. Taking into account that the projections of the total wave function φ + (r) are localized on different manifolds that do not intersect, the integral (48) can be represented in the form of the following sum: Below, for simplicity, the indices denoting the quantum numbers is omitted. It is obvious that all terms in the expression (49) are equal and, therefore, each of them is equal to 1/3. As an example, we calculate the first term of the integral, considering the case of the ground state. Taking into account that the wave function can be represented in the form φ + x = φ , and that its components φ , are localized in a region, which is the union of two manifolds (−Y, Z) ∪ (Y, −Z) ⊂ (Y, Z), we can write the following equality: where ̺ = r(0, y, z) = y 2 + z 2 denotes radius-vector r on the plane (−Y, Z), in addition, in calculating the in- tegral, we assume that the wave function in the direction x perpendicular to the plane (−Y, Z) is the Dirac delta function.
Taking into account (46), we can calculate the integral in the expression (50): Considering (50) and (51), we can determine the normalization constant of the wave function (46), which is equal to C = 4/ √ 3. Note that in a similar way one can obtain the hion wave function with the spin projection -1 (see Appendix B). Now we can calculate the probability distribution of the hion's x-projection in the ground state. Using (46), we obtain the following expression for the probability distribution on the surface element dS: where C ′ = C/a 3/2 p and dS = ρdρdϑ. Recall that the angle ϑ coincides with the angle θ on the fixed plane.
Integrating the expression (52) by the angle ϑ ∈ [0, π/2], we obtain the probability distribution of the ground state depending on radius: Finally, calculating the expression (53), we find that for the value ρ 0 = 1/2 and, respectively, for r(0, x, y) = ρ = a p /2, the probability distribution has a maximum (see FIG. 3). Now we consider the first three excited quantum states, which are characterized by the principal quantum number n = 2. Using the solution (40), we can write the explicit form of these wave functions: where Taking into account expressions (54), we can construct a radial probability distribution for the first four excited states of the hion: Recall that at deriving of expressions for the probability distributions W 2,1,0 (ρ), W 2,1,+1 (ρ) and W 2,1,−1 (ρ), averaging over the angle ϑ is performed. Note that the probability distributions (see FIG. 4), and also the energies of considered three states coincide. In particular, the quantum state described by the wave function φ +(r) x(2,0,0) has the energy E 2,0,0 = −0.2 ε 1/3 0 , whereas three different quantum states φ +(r) x(2,1,0) , φ +(r) x(2,1,+1) and φ +(r) x(2,1,−1) are characterized by the same energy E 2,1,0 = E 2,1,±1 = −0.14 ε can be a few. In this connection, the natural question arises: namely, how the state of the hion changes if we take into account random fluctuations of QVF of the next order, ie, consider the change of the particle on the next evolutionary scale {ε 1 , τ 1 }.
Let us consider the evolution of hion with the spin projection +1 taking into account the influence of the random environment in the framework of SDE of the type: and also the equation: where η + (s) = (η + x , η + y , η + z ) denotes the generator of random forces, and ds 2 = c 2 dt 2 − dx 2 − dy 2 − dz 2 is the 4D-interval in which these random influences are carried. The equation (55) can be represented in matrix form: and the equation (56), respectively, in the form: whereψ + σ = ∂ψ + σ /∂s. For further constructions, the system of equations (57)-(58) must be reduced to the canonical form: wheres = s/a p and η(s) =s −1 η x =s −1 η y =s −1 η z , in addition, the following notations are made: For simplicity, we assume that a random generator η(s) satisfies the white noise correlation conditions: where ε 1 ≡ ǫ = ǫ r + iǫ i and ǫ r = ǫ i = ǫ 0 , in addition, it is assumed that the bracket ... means averaging over the relaxation time τ 1 . The joint probability distribution of QVF can be represented in the form (see for example [27]): where the set of functions {ψ In (62) the function δ(ψ + σ (s; r, t) −φ + σ ) denotes the Dirac delta function in the three-dimensional Hilbert space, in addition, by default we will assume that the wave function is dimensionless, ie, it is multiplied by a constant value a 3/2 p (see (46)). Now using the system of SDE (59) and (60)-(61), for the conditional probability (62) the following second order partial differential equation can be obtained [28]: whereψ + σ denotes the complex conjugate of the wave functionψ + σ and ǫ + σ (r, t) which is a dimensionless quantity and denotes the effective fluctuations power. In the equation (63) the following nota-tions also are made: ψ . The general solution of the equation (63) is convenient to represent in the integral form: where the function f ({φ + }) denotes the initial condi-tion of the equation (63) at s = 0, before switching on the interaction with the random environment. Since before switching on the interaction, the hion (the vectorboson) is in a pure quantum state, ie, in the Hilbert space is determined by a fixed vector φ + , then we can put; f ({φ + }) = P 0 ({φ + }), where P 0 ({φ + }) has the sense of the distribution hion, which is defined as follows: where Substituting the expressions (65)-(66) into (64) and integrating over the variablesφ + within [ψ , we obtain the expression for the deformation of the initial quantum distribution P 0 ({φ + }), taking into account the evolution of hion in a random environment: where Note that the function F +(̟) σ characterizes the deformation of the initial distribution: Integrating (64) taking into account (67)-(68), we obtain the quantum distribution of hion with consideration of the random influence of an environment. It is easy to see that before the relaxation, the 4D-interval is zero, ie, s = 0 and, correspondingly, the deformation coefficient F = 1, as expected.
With the help of similar reasoning, we can calculate the deformation of components of the hion state vector: Thus, it is obvious that the deformation of the hion quantum state leads to a breaking of the symmetry the state, which entails spontaneous transitions, including from the ground state, to other, massless, and also massive states.

VI. Formation of singlet and triplet pairs of hions
In the second phase of evolution in the ensemble of hions, it is possible formation of pairs of hions, which can be in singlet or triplet states [29].
As is well known [30], there are four possible entangled states of the so-called Bell states, which can be represented as: where the radius vectors r + and r − determine positions of the first and second hions, respectively. Note that the first equation denotes possible two singlet states, and the second -two triplet states. In (70) also the following notations are made: where we recall that the wave functions | ↑ 1 and | ↑ 2 denote the pure states of hions with the spin projections +1 and -1, respectively. In (70) the dash " − " over a wave function denotes complex conjugation, [...] T denotes the transposed vector and the symbol ⊗, respectively, denotes the tensor product between the vectors. The explicit form of a direct tensor product between vectors with opposite spins have the following form: whereas a direct tensor product between vectors with parallel spins, respectively, can be represented as: where A and A ⇈ denote the third-rank matrixes, whilē A andĀ ⇈ = A are their complex conjugate matrices.
A. The zero-spin particles and the scalar field Now the main question is the question of the so-called quintessence-is it possible to form particle-like excitations in the form of some dynamic scalar field [12]?
Using the first equation of the system (70), we can construct the wave function of a zero-spin boson, repre-senting it in the form (see Appendix B): where the matrix elements B ∓ ij = A ij ∓Ā ij are calculated explicitly and have the form: The matrix elements with the plus sign are zero B + ij = 0, since it is possible easy to show that the components of the corresponding wave functions are localized on disjoint manifolds. The latter means that the wave state φ + (r + , r − ) does not exist. In other words, in the case of the Minkowski space-time there is only one singlet state for zero-spin boson.
The quantum distribution of the scalar boson in the singlet state before the onset of the relaxation process, ie, for s = 0, can be represented as: where is a diagonal third rank matrix, the elements of which have the following form: Taking into account the fact that the spins of two vector states φ + and φ − are directed oppositely, and also considering features of spatial localization these quasiparticles, we obtain the following expressions for the matrix elements: Now let us consider how the density of the quantum distribution of a scalar boson changes taking into account the random influence of the environment.
To study this problem, we will use the following system of complex stochastic matrix equations: and also the equations: where r ± denotes the radius-vector of corresponding hion, in addition, the complex generators η ± (s ± ) = (η ± x , η ± y , η ± z ) describing random fluctuations of charges and currents, which continuously arise in 4D-interval ds 2 ± = c 2 dt 2 − dx 2 ± − dy 2 ± − dz 2 ± . For further studies, the system of equations (79), it is useful to write in the matrix form: and, respectively, whereψ ς σ = ∂ψ ς σ /∂s ς and ς = ±. As in the case of one hion (see (59)-(60)), the system of SDE (81) can be reduced to the canonical form: wheres + = s + /a p ands − = s − /a p , in addition, the following notations are made: Below in the equations (82)-(83), we will assumed that the following relations are satisfied: which is quite natural. As in the case of a single hion, we will assume that the random generator η(s) satisfies the correlation properties of white noise (see Eqs. (61)).
Using the system of SDE (82), for the conditional probability the following second order partial differential equation can be obtained (see [28]): whereψ ς σ denotes the complex conjugate of the function ψ ς σ and ǫ ς σ (r ς , t) = ǫ 0 [b ς σ (r ς , t) + d ς σ (r ς , t)] 2 . For further analytical calculations, it is convenient to represent the general solution of the equation (85) in the integral form: where, as in the case of a single hion, we assume that is the initial distribution of the scalar boson before the relaxation begins. It is obvious that integration over the space-time, ie, by the spectrum, in accordance with the ergodic hypothesis, is equivalent to integration over the full 12D space. Substituting Recall that C σσ denotes the mean value and is defined as follows: in addition, performing similar calculations, as in the case of a single hion (see (64)-(68)), we can get: Note that, using analogous arguments, we can construct the wave function of entangled two hions with consideration of its relaxation in a random environment.
Thus, we have shown that, as a result of the multiscale evolution of QVF, a scalar field is formed, as a sort of Bose-Einstein condensate of massless scalar bosons. However, such a condensate differs significantly from a conventional substance, since it consists of massless particles that have a large Compton wavelength that can not "thicken" unlimitedly and to form large-scale structures like as stars, planets, etc. In other words, the described substance meets all the characteristics of the quintessence requirements and, accordingly, it can be asserted that the quintessence hypothesis is theoretically proved.

B. Triplet state of two hions and the vector field
The wave function of the triplet state, formed by entangled two hions, can be represented as: where, as shown by simple calculations, the following equalities hold: From these equalities in particular it follows that there is only one triplet state, which described by the matrix of the third rank φ ⇈ − (r + , r − ). The relaxation of the triplet state can be taken into account using a similar constructing, as in the case of the singlet state. In this case, however, the following substitutions must be made in the expressions (83): that ensures the transformation of the singlet state into a triplet state. Note that these replacements in the expression (86) changes the power of fluctuations ε ς σ .

VII. Conclusion
Although no single fundamental scalar field has been experimentally observed so far, such fields play a key role in the constructions of modern theoretical physics.
There are a few important hypothetical scalar fields, for example the Higgs field for the Standard Model, the dark energy-quintessence for a theory of the quantum vacuum, etc. The presence of each of them is necessary for the complete classification of the theory of fundamental fields, including new physical theories, such as, for example, String Theory. Recall that despite the great progress in the representations of modern particle theory within the framework of the SM, it does not give a clear explanation of a number of fundamental questions of the modern physics, such as "What is dark matter ?" or "What happened to the antimatter after the big bang?" and so on. Note that as modern astrophysical observations show, not less than 74 percent of the energy of the universe is associated with a substance called dark energy, which has no mass and whose properties are not sufficiently studied and understood. It was obvious to assume that this substance should be associated with a quantum vacuum or simply to be QV itself.
As it is known, in the modern understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space". Recall that in the vacuum state, electromagnetic waves and particles continuously appear and disappear, so that on the average their value is zero. It would be reasonable to think that these fluctuating or flickering fields are born as a result of spontaneous decays of quasi-stable states of some, very inert to any external interactions massless field.
The main purpose of this work was the theoretical justification for the possibility of forming a scalar field consisting of uncharged massless zero-spin particles in the region of negative energies. As the basic equation describing the multiscale evolution of vector fields, a complex SDE (1) is used, which in infinitesimal 4D-intervals is in accordance with the Weyl-type equation for neutrinos.
Assuming that on a small scale 4D space-time is pseudo-Euclidean, ie, of the Minkowski type (8), we have proved, that on the first scale of evolution of vector quantum fields (τ 0 , ε 0 ), the quantum 2D-structures or randomly oscillating 2D strings (brane) arise. These branes eventually go over to statistical equilibrium, and corresponding quantum states in Hilbert space are described by autonomous second-order equations (18), which combine relativism and quantumness. This 2D topological formation with all its properties is a stable massless vector boson with spin 1, which we call hion. It is shown that the state of the hion is quantized, where the ground state has the highest frequency.
It is shown that on the second scale of evolution (τ 1 , ε) two hions with spin projections +1 and -1 can form a zero-spin boson.
It is shown that hions with spin projections +1 and -1, in general, are the excited topological objects, described by wave functions (see (35) and (40)) and localized on corresponding two-dimensional manifolds (see FIG. 2). Note hion is described by the vector state consisting of three components characterizes the electrical and magnetic properties of the massless particle. The problem of deformation of the hion wave vector under the influence of a random environment is investigated. It is shown that such an influence leads to a violation of the symmetry of the state vector of the hion, which makes possible spontaneous transitions to other massless and mass states, including transitions from the ground state.
It is shown that on the second scale of evolution (τ 1 , ε 1 ) two hions with spin projections +1 and -1 can form a zero-spin boson. The ensemble of zero-spin bosons forms a Bose-Einstein condensate, which is a scalar field with all the necessary properties. In other words, the work is a theoretical proof of the quintessence-dark energy hypothesis. Note that a small part of the energy of a quantum vacuum is concentrated in vector fields, which consist of ensembles of hions and bosons with spin 2, formed by entangling of two hions with spin projections +1 and -1, respectively.
A very important question connected to the value of the parameter a p (see (47)), which characterizes the spatial size of the hion, remains open within the framework of the developed representation. Apparently, we can get a clear answer to this question by conducting a series of experiments. In particular, if the value of the constant a p will be significantly different, from the Planck length l P , then it will be necessary to introduce a new fundamental constant characterizing the hion size.
Thus, the conducted studies allow us to speak about the structure and properties of QVF and, accordingly, about the structure and properties of the "empty" spacetime. Preliminary studies show that the properties of space-time can be changed on experimentally measurable values at relatively weak external fields, which is very important for future technologies. At that it is obvious that the external fields will be especially effective for influencing the vector vacuum fields, which will allow us to seriously talk for the first time about the possibility of performing space-time engineering.
Finally, it is important to note that, in the light of the above proofs, the quantum vacuum-quintessence is nothing more than a natural quantum computer with complex logic, different from that currently being implemented in practice.

VIII. Appendix
A.
Let us consider the limit of statistical equilibrium lim t→∞ P 1 (ζ, t) =P 1 (ζ). In this case, the partial differential equation (27) is transformed into the ordinary differential equation of the form: Proposition. If the functionP 1 (ζ) the solution of the equation (91), then the integral: where M = const > 0.
Let the function P + (ζ) satisfies the equation: then it will look like: where C + is the arbitrary constant.
If we choose C + > 0 then the function P + (ζ) will be positive on all axis ζ ∈ (−∞, +∞). Substituting (93) into (91), we get: where As follows from the analysis of the coefficient K(ζ), on the axis ζ ∈ (−∞, +∞), it has types of uncertainties 0 0 or ∞ ∞ . Applying the L'Hôpital's rule, for the coefficient near the critical points ζ i , where uncertainties appear, we obtain the expression: In asymptotic domains, ie, when |ζ| ≫ 1 in the equation (97), we can neglect the term 3ζ and to obtain the following solution: whereas the asymptotic solution of the equation (91), respectively, is: where L >> 1.
For the first and third integrals, we can write down the following obvious estimates (see (24)): then it is obviously converges M * < ∞, since the function P 1 (ζ) is bounded, and the integration is performed on a finite interval. Thus taking into account the estimations (102)-(104) we can approve that the integral (92) converges, ie, the estimation where M = const < ∞, is right. The proposition is proved.