Quantum Vacuum and the Structure of Empty Space-Time

# Quantum Vacuum and the Structure of "Empty" Space-Time

## Abstract

We have considered the possibility of formation a massless particles with spin 1 in the region of negative energies, within the framework of the Weyl type equation for neutrinos. It is proved that, unlike quantum electrodynamics, the developed approach allows in the ground state the formation of such stable particles. The structure and properties of this vector-boson are studied in detail. The problem of entangling two vector bosons with projections of spins +1 and -1 and, accordingly, the formation of a zero-spin boson is studied within the framework of a complex stochastic equation of the Langevin type. The paper discusses the structure of the Bose particle of a scalar field and the space-time’s properties of an empty space (quantum vacuum).

## 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 Lamb (), Casimir effect Casimir (), Unruh effect Unru (), anomalous magnetic moment of electron Proh (), Van der Waals forces Van (), Delbrück scattering Del (), Hawking radiation Haw (), the cosmological constant problem Weinb (); Wein (), vacuum polarization at weak electromagnetic fields Ash1 (); Ash2 () - 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 quintessence (dark energy) and cosmic acceleration often are discussed in the framework of QV theories, which necessarily include scalar fields Wein (). The properties of a quantum vacuum can be studied within the framework of quantum field theory (QFT), ie quantum 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 the perturbation theory for QFT breaks down at low energies (for example, QCD or the BCS theory of superconductivity) field operators may have non-vanishing vacuum expectation values called condensates. Moreover, in the Standard Model precisely the non-zero vacuum expectation value of the Higgs field, arising from spontaneous symmetry breaking, is the principled mechanism allowing to acquire masses of other fields of theory.

To overcome these difficulties and to conduct a consistent and comprehensive study of the quantum vacuum, we developed a nonperturbative approach based on a system of complex stochastic equations of the Langevin type describing the motion of a massless particle with spin 1 in a random environment of similar particles.

The main objectives pursued by this study are as follows:

a. to study the possibility of formation a stable bound state of a massless particle with spin 1 (vector boson) in the region of negative energies,

b. to investigate the possibility of forming a massless particle with zero spin as a result of entangling the ground states of two vector bosons, respectively, with the projections of the spins 1 and -1,

c. to justify of the possibility the formation of a scalar field as a result of relaxation and Bose-Einstein (BEC) condensation of vector fields. To study the features the wave state of the boson of a scalar field, in particular, the possibility of its decay and spontaneous transitions to other vacuum and extravacuum states.

## 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 Opp (); Mol (); Wein1 ().

As shown BIALYNICKI (), the quantum motion of a photon in a vacuum can be considered within the framework of a wave function representation, writing it analogously to the Weyl equation for a neutrino in a vector form:

 ∂tΨ+(r,t)−c0(S⋅∇)Ψ+(r,t)=0, ∂tΨ−(r,t)+c0(S⋅∇)Ψ−(r,t)=0,∂t≡∂/∂t, (1)

where denotes the speed of light in an empty Minkowski space-time, and denote the photons wave functions of both helicities, corresponding to left-handed and right-handed polarizations. In addition, in (1) the set of matrices describes infinitesimal rotations of particles with spin , respectively:

 Sx=⎡⎢⎣00000−i0i0⎤⎥⎦,Sy=⎡⎢⎣00i000−i00⎤⎥⎦, Sz=⎡⎢⎣0−i0i00000⎤⎥⎦. (2)

Recall, that in (1) the absence of electrical and magnetic charges provides the following conditions:

 ∇⋅Ψ±(r,t)=0. (3)

If we represent the wave function in the form:

 Ψ±(r,t)=1√2{D(r,t)√ϵ0±iB(r,t)√μ0},c0=1√μ0ϵ0, (4)

then from the equations (1) and (3) it is easy to find Maxwell’s equations in an ordinary vacuum or in empty space:

 ∂tD−∇×H=0,∇⋅E=0, ∂tB+∇×E=0,∇⋅H=0, (5)

where and 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:

 D=ϵ0E,B=μ0H.

Recall that the only difference between the equations (1) and (5) is that the Maxwell equations system does not take into account the spin of the photon, which will be important for further constructions. Since the refractive indices and 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.

### ii.1 Vector fields and their fundamental particle

Let us make the following substitutions in the equations (1):

 c0↦c(x,y,z,τ),t↦τ(t), (6)

where is the velocity of propagation of the field in the structural particle, which differs from the speed of light in a vacuum, is the chronological parameter of a closed quantum system (internal time), which, naturally, must be quasi-periodic, and in the most general case can be represented as:

 τ(t)=T(t)sin(t/T(t)),−∞

where is a bounded function-quasiperiod.

Proposition. The equations system (1)-(3), taking into account the remarks (6) and (7) describes the quantum states of the massless particles with spin projections .

The particle of the vector field in depending on the value of the spin projection, can be described by one of the following sets of wave functions:

 ψ+(r,τ)=⎡⎢ ⎢⎣ψ+x(r,τ)ψ+y(r,τ)ψ+z(r,τ)⎤⎥ ⎥⎦,ψ−(r,τ)=⎡⎢ ⎢⎣ψ−x(r,τ)ψ−y(r,τ)ψ−z(r,τ)⎤⎥ ⎥⎦, (8)

which satisfy equations of the type (1)-(3) taking into account the above remarks.

Obviously, in the bound state, the 4D-interval of the propagated signal will be zero, and the points of the Minkowski space (events) are related to the relation, like the light cone:

 s2=c2τ2−r2=0,r2=x2+y2+z2. (9)

Thus, the key question arises: can such equations describe the spatial localizations of massless fields, characteristic of a stable structured formations called a particles?

Substituting (8) into (1) and taking into account (2), we can find the following two independent system of first order partial differential equations:

 ic−1∂τψ±x=±∂yψ±z∓∂zψ±y, ic−1∂τψ±y=±∂zψ±x∓∂xψ±z, ic−1∂τψ±z=±∂xψ±y∓∂yψ±x. (10)

In addition, the following relation can be found between the fields, taking into account the equation (3):

 ∂xψ±x+∂yψ±y+∂zψ±z=0. (11)

Since particles with spins and are symmetric in any sense, it is sufficient to study in detail only the wave function describing a particle with spin .

Using the equations (10) and (11) we can obtain the systems of second order partial differential equations for vacuum fields:

 □ψ+x=c−1c,y(∂xψ+y−∂yψ+x)−c−1c,z(∂zψ+x−∂xψ+z)−c,τc−3∂τψ+x, □ψ+y=c−1c,z(∂yψ+z−∂zψ+y)−c−1c,x(∂xψ+y−∂yψ+x)−c,τc−3∂τψ+y, □ψ+z=c−1c,x(∂zψ+x−∂xψ+z)−c−1c,y(∂yψ+z−∂zψ+y)−c,τc−3∂τψ+z, (12)

where denotes the D’Alembert operator, is the Laplace operator, and . To determine the explicit form of equations (12), we need to calculate the derivatives . Using the equation (9), we can find:

 c,τ=−c2r,c,x=cxr2,c,y=cyr2,c,z=czr2. (13)

To continue the 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 circumstance that in this problem all fields are symmetric, the following additional conditions can be imposed on the field components:

 (c,z−c,y)∂τψ+x=c,z∂τψ+y−c,y∂τψ+z, (c,x−c,z)∂τψ+y=c,x∂τψ+z−c,z∂τψ+x, (c,y−c,x)∂τψ+z=c,y∂τψ+x−c,x∂τψ+y. (14)

It is easy to verify that these conditions are symmetric with respect to the components of the field and are given on the hypersurface of four-dimensional events. Further, using the conditions (14), the system of equations (12), can be reduced to the following canonical form:

 {□+[i(c,z−c,y)+c,τc−1]c−2∂τ}ψ+x=0, {□+[i(c,x−c,z)+c,τc−1]c−2∂τ}ψ+y=0, {□+[i(c,y−c,x)+c,τc−1]c−2∂τ}ψ+z=0. (15)

Note that unperturbed QV fields must satisfy the conditions of autonomy, that imply the separation of spatial and temporal components of the wave function. Proceeding from this, it is convenient to represent the wave function in the form:

 ψ+σ(r,τ)=exp(Eστℏ)ϕ+σ(r),σ=x,y,z, (16)

where is the energy of the one mode of QV field. It is obvious that the symmetry of the problem implies the equality of the modes energies .

Finally, substituting (16) into (15), taking into account (13), we can get the following system of equations:

 {△+λ[−λ−1r+iz−yr2]}ϕ+x(r)=0, {△+λ[−λ−1r+ix−zr2]}ϕ+y(r)=0, {△+λ[−λ−1r+iy−xr2]}ϕ+z(r)=0, (17)

where , in addition, parameter has dimensionality of inverse distance.

## Iii The wave function of a massless particle with spin +1

Representing the wave function in the form:

 ϕ+x(r)=ϕ+(r)x(r)+iϕ+(i)x(% r), (18)

from the first equation of the system (17), we can get the following two equations:

 {△−λ(λ+1r)}ϕ+(r)x(r)−λz−yr2ϕ+(i)x(r)=0, {△−λ(λ+1r)}ϕ+(i)x(r)+λz−yr2ϕ+(r)x(r)=0. (19)

It can be verified that by using simple substitutions of unknowns and , these equations pass into each other. The latter means that the solutions in modulus are equal and differ only in sign. In other words, the symmetry properties mentioned above make it possible to obtain two independent equations of the form:

 {△−λ[λ+r−y+zr2]}ϕ+(r)x(r)=0, {△−λ[λ+r+y−zr2]}ϕ+(i)x(r)=0. (20)

Now we analyze the possibility of solving for the term in the form of a localized state. Let us consider the following equation:

 r−y+z=μr, (21)

where is a some parameter. The changing range of this parameter will be defined below.

Substituting (21) into the first equation of the system (20), we get:

 {△+[−λ2−λμr−1]}ϕ+(r)x(r)=0. (22)

To continue the research, it is useful to rewrite the equation (22) in the spherical coordinates system :

 (23)

Using for the wave function the representation:

 ϕ+(r)x(r)=Λ(r)Y(θ,φ), (24)

we can conditionally separate the variables in the equation (31) and write it in the form of the following two equations:

 (25)

and, respectively;

 1sinθ{1sinθ∂2∂φ2+∂∂θ(sinθ∂∂θ)}Y+νY=0, (26)

where and is a constant, which is convenient to represent in the form and Recall that the conditional separation of variables means to impose an additional condition on the function . Writing equation (21) in spherical coordinates, we obtain the following trigonometric equation:

 μ(θ,φ)=1−sinθsinφ+cosθ. (27)

As the analysis of the equation (27) shows, the range of variation of the parameter for real angles is .

The solution of the equation (26) is well known, these are spherical Laplace functions and .

As for the equation (25), we will solve it for a fixed value , which is equivalent to the plane cut of the three-dimensional solution. In particular, we will seek a solution tending to finite value for and, respectively, to zero at .

For a given parameter , we can write the equation (25) in the form:

 d2Λdϱ2+2ϱdΛdϱ+[−β2+2ϱ−l(l+1)ϱ2]Λ=0, (28)

where is a dimensionless distance, denotes the characteristic spatial dimension of a hypothetical massless Bose particle with spin 1 and . As can be seen, this equation describes the radial wave function of a hydrogen-like system Land (). Recall that it can have a solution describing the bound state if the following condition is satisfied:

 nr+l+1=n+l=1β,nr=0,1,2..., (29)

where is the radial quantum number, and , respectively, is the principal quantum number.
It is easy to see that there is only one integer for which and, accordingly, the equation (29) has a solution. In other words, the equality is a necessary quantization condition, which, with consideration (27), is equivalent to the trigonometric equation:

 1+sinθsinφ−cosθ=0. (30)

In particular, from this condition it follows that the solution of the equation (28) is localized on the plane (see FIG. 1).

Proceeding from the foregoing, we can write the solution of equation (28) on this plane:

 ϕrx(r,θ,φ)=Λnl(r)Yml(θ,φ),

which, subject to the condition (30), has a solution only for the ”ground state”:

 Λ10(r)=C√a3pe−r/ap,Y00(θ,φ)=12√1π, (31)

where denotes a constant, which will be defined below from the normalization condition of the wave function. It is easy to see that the additional condition does not lead to a contradiction between the equations (25) and (26), and the solution (31) is the only solution of the first equation in the system (19) in the region of negative energies.

The imaginary part of the wave function is calculated similar way, but in this case we obtain following underdetermined trigonometric equation:

 1−sinθsinφ+cosθ=0, (32)

which determines the plane on which the solution of the ground state is localized.

Similar solutions are obtained for the projections of the wave function and . In this case, however, the components of the wave function are localized on the planes and , accordingly. So, the orient of first pair of planes are determined by the equations:

 1±cosθsinφ∓cosθ=0, (33)

while, the orientations of the second pair of planes are given by the equations:

 1±sinθsinφ∓cosθsinφ=0. (34)

It should be noted that the coefficient of the radial wave function (see (31)) is chosen so that the total vector wave function of the ”ground state” is normalized to unity.

Now we can write down the normalization condition for the total wave function for the vector boson:

 J=∫ϕ+(¯ϕ+)TdV=1,dV=dxdydz, (35)

where is the transposed vector.
We can represent the integral (35) as a sum of three terms:

 J=∫ϕ+x¯ϕ+xdV+∫ϕ+y¯ϕ+ydV+∫ϕ+z¯ϕ+zdV. (36)

It is obvious that all terms in the expression (36) are equal and, therefore, each of them is equal to 1/3.

As an example, consider the first summand. Taking into account that the wave function can be represented in the form , and its components and , are localized in a region that is the union of two quaternary planes , we can write:

 ∫ϕ+x¯ϕ+xdV=∫(|ϕ+(r)x|2+|ϕ+(i)x|2)dV=2∫|ϕ+(r)x|2dV=2∫|ϕ+(i)x|2dV=ap2π∫(−Y,Z)|Λ10(ρ)|2dydz=1/3. (37)

where denotes radius-vector on the plane , in addition, in calculating the integral, we assume that the wave function in perpendicular to the plane direction is the Dirac delta function.

Taking into account (31), we can accurately calculate the integral in the expression (37):

 ∫(−Y,Z)|Λ10(ρ)|2dydz=π8apC2. (38)

Now using (37) and (38), we can determine the normalization constant of the wave function (31), which is equal to . Note that similarly we can find a solution of the boson wave function with the spin projection -1 (see Appendix A).

Finally, we turn to the characteristic scale of the brane, which, as follows from the quantization condition (see (28)-(29)), is defined by the equation:

 ap=1|λ|=cℏ|E|, (39)

where is the speed of propagation of a wave in the brane, which can significantly differ from the speed of light in a vacuum. Unfortunately, it is impossible to determine from equation (28). Nevertheless, if we assume that the spatial scale brane has the Planck length , ie, , and the energy particle is , then the propagation velocity of the wave in the brane will be equal to .

Using the solution of the ground state, on the corresponding planes, we can calculate the location of the maximum probability of one of the projections of the brane. The probability of finding the maximum amplitude of the projection of a brane wave into an element of the area can be determined as follows:

 dW(ρ)=C′2e−2ρρdρdϑ,C′=C/√a3p, (40)

where
Investigating the expression , it is easy to find that for the value and, respectively, for , the probability distribution has a maximum FIG 2. Proposition is proved.

## Iv Formation of a scalar field

Let us consider QV, taking into account the fluctuations of the vector fields. As the basic equations we will use the system of stochastic matrix equations of the Langevin type:

 ∂t˘ψ+(r+,t)−c(S⋅∇)˘ψ+(r+,t)=η+(s+), ∂t˘ψ−(r−,t)+c(S⋅∇)˘ψ−(r−,t)=η−(s−), (41)

and also the equations:

 ∇˘ψ+(r+,t)=0,∇˘ψ−(r−,t)=0, (42)

where denotes the radius-vector of corresponding vector particles, is a usual time. In equation (41) complex stochastic generators , describing random charges and currents arising in 4D-intervals .

For further research, it is useful to write these equations in matrix form:

 ⎡⎢⎣\vspace2mmict−z+y+\vspace2mmz+ict−x+\vspace2mm−y+x+ict⎤⎥⎦⋅⎡⎢ ⎢ ⎢ ⎢ ⎢⎣\vspace0.7mm˙˘ψ+x\vspace0.7mm˙˘ψ+y\vspace0.7mm˙˘ψ+z⎤⎥ ⎥ ⎥ ⎥ ⎥⎦=s+⎡⎢ ⎢⎣\vspace2mmη+x\vspace2mmη+y\vspace2mmη+z⎤⎥ ⎥⎦,

and, respectively,

 ⎡⎢⎣\vspace2mmictz−−y−\vspace2mm−z−ictx−\vspace2mmy−−x−ict⎤⎥⎦⋅⎡⎢ ⎢ ⎢ ⎢ ⎢⎣\vspace0.7mm˙˘ψ−x\vspace0.7mm˙˘ψ−y\vspace0.7mm˙˘ψ−z⎤⎥ ⎥ ⎥ ⎥ ⎥⎦=s−⎡⎢ ⎢⎣\vspace2mmη−x\vspace2mmη−y\vspace2mmη−z⎤⎥ ⎥⎦, (43)

In addition, from (3) we can find the following two relations between the stochastic components of the wave function:

 x+˙˘ψ+x+y+˙˘ψ+y+z+˙˘ψ+z=0, x−˙˘ψ−x+y−˙˘ψ−y+z−˙˘ψ−z=0. (44)

For the further analytical study of the problem, it is important to reduce the system of equations (43) to the canonical form:

 (45)

 bςx(rς,t)=(zς−yς)/ap, bςy(rς,t)=(xς−zς)/ap, bςz(rς,t)=(yς−xς)/ap, dςx(rς,t)=(c2t2−x2ς−xςyς−xςzς)/(apct), dςy(rς,t)=(c2t2−y2ς−xςyς−yςzς)/(apct), dςz(rς,t)=(c2t2−z2ς−xςzς−yςzς)/(apct). (46)

Note that when deriving the equations (46) - (45) we assumed that the following relations hold:

 ηςx=ηςy=ηςz=ης,d¯s+=d¯s−=d¯s,

which looks quite natural in the context of considered problem.

Now we will assume that the random generators satisfy the following correlation properties:

 ⟨η(¯s)⟩=0,⟨η(¯s)η(¯s′)⟩=2εδ(¯s−¯s′), (47)

where and is a complex constant denoting the fluctuation power; in addition, we assume that

We can represent the joint probability distribution of fields in the form (see AshG ()):

 (48)

where denotes a set of fluctuations of vacuum fields and . In the (48) , denotes the Dirac delta function generalized on a 6D-Hilbert space, in addition, by default we will assume that the wave function is dimensionless, that is, it is multiplied by a constant value (see (31)).

Using the system of stochastic equations (45), for the conditional probability the following second order partial differential equation can be obtained (see Ashg1 ()):

 {∂∂¯s−12∑ς,σεςσ(rς,t)∂2∂˘ψςσ∂¯˘ψςσ}P=0, (49)

where denotes the complex conjugate of the function and , which is a dimensionless quantity and denotes fluctuations power.

Recall that in the equation (49) the following notation is used:

 ∂2∂˘ψςσ∂¯˘ψςσ=∂2∂[˘ψς(r)σ]2+∂2∂[˘ψς(i)σ]2,

where and .

For further it is convenient to represent the general solution of the equation (49) in the following integral form:

 P({˘ψ},¯s;r+,r−,t)=∫Ξ6P0({ψ0})∏ς,σexp{−(ψςσ−˘ψςσ)(¯ψςσ−¯˘ψςσ)2¯sεςσ}dψςσ√2π¯sεςσ,dψςσ=dψς(r)σdψς(i)σ, (50)

where denotes an initial condition of the equation (49) at , before including of interaction with the random environment. Recall that the integration over the 6D Hilbert space , ie by the spectrum, in accordance with the ergodic hypothesis, is equivalent in this case to integration over the full 12D configuration space. Note that the integration in (50) should be understood as an integral operator that acts on each element of the matrix, which, as we shall show below, is the .

On the basis of physical considerations, as an initial condition, we can choose the probability density of the ”ground state” of a scalar field’s particle, which can be formed by two vector bosons with projections of spin and , respectively. It is obvious that the probability density should be normalized on the unit matrix of the size :

 ∫Ξ6P({˘ψ},¯s;r+,r−,t)∏ς,σd˘ψςσ=I3. (51)

Substituting into and integrating over variables within , we get the following condition for the density matrix :

 ∫Ξ6P0({ψ0})∏ς,σdψςσ=I3. (52)

The wave function of a scalar field particle (scalar boson with spin-0) can be represented as the maximally entangled state of two ”ground states” of the vector field particles (EPR state Einst ()):

 ψ↕(r+,r−,0)=1√2{|↑⟩1⊗|↓⟩2−|↓⟩1⊗|↑⟩2}, (53)

where we make the following notation: and , while and . Note that the wave function (53) denotes one of Bell’s entangled states of four possible Bell (). Recall that in the equation (53) the symbol denotes the operation of a direct (tensor) product:

 ¯A=|↓⟩1⊗|↑⟩2=⎡⎢ ⎢ ⎢⎣\vspace1mm¯ϕ+x\vspace1mm¯ϕ+y\vspace1mm¯ϕ+z⎤⎥ ⎥ ⎥⎦[¯ϕ−x¯ϕ−y¯ϕ−z], (54)

where A denotes the third-rank matrix, while its complex conjugation. The wave function of the boson of a scalar field can be represented in the form of a difference of two matrices (see Appendix B):

 ψ↕(r+,r−,0)=1√2{A−¯A}=1√2% B= 1√2⎡⎢⎣\vspace1mmB11B12B13\vspace1mmB21B22B23\vspace1mmB31B32B33⎤⎥⎦=1√2⎡⎢⎣\vspace1mmB1100\vspace1mm0B220\vspace1mm00B33⎤⎥⎦, (55)

where the elements of the matrix B are calculated explicitly:

 B11=2i[ϕ+(i)xϕ−(r)x−ϕ