Physical Vacuum Is a Special Superfluid Medium

Physical vacuum is a special superﬂuid medium

Valeriy I. Sbitnev∗ St. Petersburg B. P. Konstantinov Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina, Leningrad district, 188350, Russia; Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720, USA (Dated: August 9, 2016)

The Navier-Stokes equation contains two terms which have been subjected to slight modification: (a) the viscosity term depends of time (the viscosity in average on time is zero, but its variance is non-zero); (b) the pressure gradient contains an added term describing the quantum entropy gradient multiplied by the pressure. Owing to these modifications, the Navier-Stokes equation can be reduced to the Schrödingerequation describing behavior of a particle into the vacuum being as a superfluid medium. Vortex structures arising in this medium show infinitely long life owing to zeroth average viscosity. The non-zero variance describes exchange of the vortex energy with zero-point energy of the vacuum. Radius of the vortex trembles around some average value. This observation sheds the light to the Zitterbewegung phenomenon. The long-lived vortex has a non-zero core where the vortex velocity vanishes.

Keywords: Navier-Stokes; Schrödinger; zero-point fluctuations; superfluid vacuum; vortex; Bohmian trajectory; interference

I. INTRODUCTION registered. Instead, the wave function represents it existence within an experimental scene [13]. A dramatic situation in physical understand- Another interpretation was proposed by Louis ing of the nature emerged in the late of 19th cen- de Broglie [18], which permits to explain such an tury. Observed phenomena on micro scales came experiment. In de Broglie’s wave mechanics and into contradiction with the general positions of the double solution theory there are two waves. classical physics. It was a time of the origina- There is the wave function that is a mathemat- tion of new physical ideas explaining these phe- ical construct. It does not physically exist and nomena. Actually, in a very short period, pos- is used to determine the probabilistic results of tulates of the new science, quantum mechanics experiments. There is also a physical wave guid- were formulated. The Copenhagen interpreta- ing the particle from its creation to detection. tion was first who proposed an ontological ba- As the particle moves from a source to a detec- sis of quantum mechanics [32]. These positions tor, the particle perturbs the wave field and gets can be stated in the following points: (a) the a reverse effect from it. As a result, the physi- description of nature is essentially probabilistic; cal wave guides the particle along some optimal (b) a quantum system is completely described by trajectory, Bohmian trajectory [9, 10], up to its arXiv:1501.06763v4 [quant-ph] 8 Aug 2016 a wave function; (c) the system manifests wave– detection. particle duality; (d) it is not possible to measure A question arises, what is the de Broglie phys- all variables of the system at the same time; (e) ical wave? Recently, Couder and Fort [15] has each measurement of the quantum system entails executed the experiment with the classical oil the collapse of the wave function. droplets bouncing on the oil surface. A re- markable observation is that an ensemble of the Can one imagine a passage of a quantum parti- droplets passing through the barrier having two cle (the heavy fullerene molecule [35], for exam- gates shows the interference fringes typical for ple) through all slits, in once, at the interference the two slit experiment. Their explanation is experiment? Following the Copenhagen inter- that the droplet while moving on the surface in- pretation, the particle does not exists until it is duces on this surface the weak Faraday waves. The latter provide the guidance conditions for the droplets. In this perspective, we can draw ∗Electronic address: [email protected] conclusion that the de Broglie physical wave 2 can be represented by perturbations of the ether velocity; (b) the second law states: the net force when the particle moves through it. In order applied to a body with a mass M is equal to to describe behavior of such an unusual medium the rate of change of its linear momentum in an we shall use the Navier-Stokes equation with inertial reference frame slightly modified some terms. As the final re- d~v sult we shall get the Schrödinger equation. . In F~ = M~a = M ; (1) d t physical science of the New time the assumption for the existence of the ether medium was origi- (c) the third law states: for every action there is nally used to explain propagation of light and the an equal and opposite reaction. long-range interactions. As for the propagation Leonard Euler had generalized Newton’s laws of light, the wave ideas of Huygens and Fresnel of motion on deformable bodies that are assumed require the existence of a continuous intermedi- as a continuum [25]. We rewrite the second New- ate environment between a source and a receiver ton’s law for the case of deformed medium. Let of the light - the light-bearing ether. It is instruc- us imagine that a volume ∆V contains a fluid tive to compare here the two opposite doctrines medium of the mass M. We divide Eq. (1) by ∆V about the nature of light belonging to Sir Isaac and determine the time-dependent mass density ~ Newton and Christian Huygens. Newton main- ρM = M/∆V [58]. In this case F is understood tained the theory that the light was made up of as the force per volume. Then the second law in tiny particles, corpuscles. They spread through this case takes a form: an empty space in accordance with the law of the dρ ~v d~v dρ classical mechanics. Christian Huygens (a con- F~ = M = ρ + ~v M . (2) d t M d t d t temporary of Newton), believed that the light was made up of waves vibrating up and down The total derivatives in the right side can be perpendicularly to the direction of its propa- written down through partial derivatives: gation, as waves on a water surface. One can dρM ∂ρM imagine all space populated everywhere densely = + (~v ∇)ρM , (3) by Huygens’s vibrators. All vibrators are silent d t ∂ t d~v ∂~v until a wave reaches them. As soon as a wave = + (~v ∇)~v. (4) front reaches them, the vibrators begin to radi- d t ∂ t ate waves on the frequency of the incident wave. Eq. (3) equated to zero is seen to be the conti- So, the infinitesimal volume δV is populated by nuity equation. As for Eq. (4) we may rewrite infinite amount of the vibrators with frequen- the rightmost term in detail cies of visible light. These vibrators populate the v 2 ether facilitating propagation of the light waves (~v ∇)~v = ∇ − [~v × [∇ × ~v]]. (5) through the space. 2 In order to come to idea about existence of As follows from this formula the first term, mul- the intermediate medium (ether) that penetrates tiplied by the mass, is gradient of the kinetic overall material world, we begin from the funda- energy. It is a force applied to the fluid element mental laws of classical physics. Three Newton’s for its shifting on the unit of length, δs. The laws first published in Mathematical Principles second term is acceleration of the fluid element of Natural Philosophy in 1687 [3] we recognize as directed perpendicularly to the velocity. Let the basic laws of physics. Namely: (a) the first law fluid element move along some curve in 3D space. postulates existence of inertial reference frames: Tangent to the curve in each point refers to ori- an object that is at rest will stay at rest unless entation of the body motion. In turn, the vector an external force acts upon it; an object that ~ω = [∇ × ~v] is orientated perpendicularly to the is in motion will not change its velocity unless plane, where an arbitrarily small segment of the an external force acts upon it. The inertia is a curve lies. This vector characterizes a quantita- property of the bodies to resist to changing their tive measure of the vortex motion and it is called 3

vorticity. Vector product [~v ×~ω] is perpendicular particles form, in any moment of the time, a to the both vectors ~v and ~ω. It represents the vortex line, then the same particles support the Coriolis acceleration of the body under rotating vortex line both in the past and in the future; it around the vector ~ω. (ii) ensemble of the vortex lines traced through The term (5) entering in the Navier-Stokes a closed contour forms a vortex tube. Intensity equation [38, 39] is responsible for emergence of the vortex tube is constant along its length of vortex structures. The Navier-Stokes equa- and does not change in time. The vortex tube tion stems from Eq. (2) if we omit the rightmost (a) either goes to infinity by both endings; (b) or term, representing the continuity equation, and these endings lean on walls of bath containing specify forces in this equation in detail [57]: the fluid; (c) or these endings are locked to each on other forming a vortex ring. ∂~v ! Assuming that the fluid is a physical vacuum, ρM + (~v · ∇)~v ∂ t which meets the requirements specified earlier, F~ P ! we must say that the viscosity vanishes. In that = + µ(t) ∇ 2~v − ρ ∇ . (6) ∆V M ρ case, the vorticity ~ω is concentrated in the cen- M ter of the vortex, i.e., in the point. Mathemati- This equation contains two modifications repre- cal representation of the vorticity is δ-function. sented in the two last terms from the right side: Such singularity can be a source of possible di- the dynamic viscosity µ depends on time and vergences of computations in further. the rightmost term has a slightly modified view, We shall not remove the viscosity. Instead of namely ρM ∇(P/ρM ) = ∇P − P ∇ ln(ρM ). This that, we hypothesize that even if there is an ar- modification will be important for us when we bitrary small viscosity, because of the zero-point shall begin to derive the Schrödingerequation. oscillations in the vacuum, the vortex does not However first we shall examine the Helmholtz disappear completely. The vortex can be a long- vortices with time dependent viscosity. lived object. The foundation for that hypothe- sis is the observation (performed by French sci- II. VORTEX DYNAMICS entific team [15, 16, 21, 50]) of behavior of the droplets moving on the oil surface, on which the The second term from the right in Eq. (2) waves Faraday exist. Here an important moment represents the viscosity of the fluid (µ is the is that the Faraday waves are supported slightly dynamic viscosity, its units are N·s/m2 = below the super-critical threshold. Due to this kg/(m·s)). Let us suppose that the fluid is ideal, trick the droplets can live on the oil surface ar- barotropic, and the mass forces are conserva- bitrary long, before they disappear in the oil. tive [38]. At assuming that the external force The Faraday waves that are supported near the is conservative, we apply to this equation the super-critical threshold may play a role analo- operator curl. We get right away the equation gous to the zero-point oscillations of the vacuum. for the vorticity: Observe that the bouncing droplet simulates some aspects of quantum mechanics, stimulat- ∂ ~ω + (~ω · ∇)~v = ν∇ 2~ω. (7) ing theoretical investigations in this area [12, 27– ∂ t 31, 48, 63]. It is interesting to note in this place

Here ν = µ/ρM is the kinematic viscosity. Its di- that Grössingconsiders a quantum particle as a mension is [m2/s]. It corresponds to the diffusion dissipative phase-locked steady state, where an coefficient. For that reason the energy stored in amount of zero-point energy of the wave-like en- the vortex will dissipate in thermal energy. As vironment is absorbed by the particle, and then a result, the vortex with the lapse of time will during a characteristic relaxation time is dissi- disappear. pated into the environment again [28]. With omitted the term from the right (i.e., Here we shall give a simple model of such a ν = 0) the Helmholtz theorem reads: (i) if fluid picture. Let us look on the vortex tube in its 4

cross-section which is oriented along the axis z viscosity. For the sake of simplicity, let it be and its center is placed in the coordinate origin looked as of the plane (x, y). Eq. (7), written down in the cross-section of the vortex, is as follows ei(Ωt+φ) + e−i(Ωt+φ) g(t) = cos(Ωt + φ) = (9) ∂ω ∂ 2ω 1 ∂ω ! 2 = νg(t) 2 + . (8) ∂ t ∂r r ∂r where Ω is an oscillation frequency and φ is the Here we do not write a sign of vector on the top uncertain phase. of ω since ω is oriented strictly along the axis Solution of the equation (8) in this case is as z. We introduce time-dependent the kinematic follows

Γ ( r2 ) ω(r, t) = exp − . (10) 4π(ν/Ω)(sin(Ωt + φ) + n)) 4π(ν/Ω)(sin(Ωt + φ) + n)) Here Γ is the integration constant having dimension m2/s. An extra number n > 1. It prevents appearance of singularity in the cases when sin(Ωt + φ) tends to −1. This function at choosing the parameters Γ = 1, ν = 1 , Ω = π and n = 16 is shown in Fig. 1(a). The velocity of the ﬂuid matter around the vortex results from the integration of the vorticity function r 1 Z Γ ( r2 )! ~v = ω(r0, t)r0dr0 = 1 − exp − . (11) r 2πr 4π(ν/Ω)(sin(Ωt + φ) + n)) 0 Fig. 1(b) shows behavior of this function at the same input parameters.

(a) (b)

0.03 0.03

0.02 0.02

0.01 0.01

0 t 0 t

0 0

4 4

2 5

3 3 4

2 10 2

1 15 1 8 20 10 0 r 0 r

FIG. 1: Vorticity ω(r, t) and velocity v(r, t) as functions of r and t for Γ = 1, ν = 1, Ω = π, and n = 16. These parameters are conditional in order to show clearly oscillations of the vortex in time. The solution does not decay with time.

In particular, for n = 0 and Ωt 1 this solution is close to the Lamb-Oseen vortex solution [65]

Γ 2 Γ 2 ω(r, t) = e−r /4νt, v(r, t) = 1 − e−r /4νt . (12) 4πνt 4πr 5

vortex likely, it manifests itself through interaction of the quantum object with vacuum fluctuations. According to Eq. (9), there are half-periods when the energy of the vortex is lost at scattering on the vacuum fluctuations, and there are other vacuum fluctuations half-periods when the vacuum returns this energy to the vortex, Fig. 2. On the whole, the FIG. 2: Periodic energy exchange between the vortex and vacuum fluctuations viscosity of the fluid medium, within which the vortex tube evolves, in the average remains at y zero. It can mean that this medium is superfluid. Such a scenario is not unusual. For example, at transition of helium to the superfluid phase [64] max coherent Cooper pairs of electrons arise through the exchange by phonons. This attraction is due to the electron-phonon interaction. The phonons are thermal excitations of a lattice. In that case, they play a role of the background medium. Qualitative view of the vortex tube in its cross- section is shown in Fig. 3. Values of the velocity v are shown by grey color ranging from light grey (minimal velocities) to dark grey (maximal min ones). A visual image of this picture can be a hurricane (tropical cyclone [1]) shown from the x top. In the center of the vortex, a so-called eye of the hurricane (the vortex core) is well viewed. Here it looks as a small light grey disk, where the FIG. 3: Cross-section of the vortex tube in the plane (x, y). Values of the velocity v are shown in grey ranging from light velocities have small values. In the very center grey (min v) to dark grey (max v). Density of the pixels rep- of the disk, in particular, the velocity vanishes. resents magnitude of the vorticity . Core of the vortex ω is Observe that in the region of the hurricane eye a well visible in the center. wind is really very weak, especially near the center. This is in stark contrast to conditions in the It is seen that the Lamb-Oseen vortex solution region of the eyewall, where the strongest winds decays with time the faster, the more ν. exist (in Fig. 3 it looks as a dark grey annular One can see from the solutions (10) and (11), region enclosing the light grey inner area). The depending on the distance to the center the func- eyewall of the vortex tube (a zone where the ve- tions ω(r, t) and v(r, t) show typical behavior locity reaches maximal values) has the nonzero for the vortices. The both functions do not de- radius. cay with time, however. Instead of that, they Let us find the radius of the vortex core. In demonstrate pulsations on the frequency Ω. Am- order to evaluate this radius we equate to zero plitude of the pulsations is the smaller, the larger the first derivative by r of equation (11) value of the parameter n. At n tending to infin- ( r2 ) exp ity the amplitude of the pulsations tends to zero. 4π(ν/Ω)(sin(Ωt + φ) + n)) At the same time the vortex disappears entirely. r2 The undamped solution was obtained thanks = 2 + 1. (13) to assumption, that the kinematic viscosity is 4π(ν/Ω)(sin(Ωt + φ) + n)) a periodic function of time, namely, νg(t) = The radius is a root of this equation ν cos(Ωt + φ). The viscosity in the quantum q r ν r = 2 a (n + sin(Ωt + φ)) · . (14) realm is not a good concept, however. Most v 0 Ω 6

−12 Here a0 ≈ 1.2564312 is a root of the equation wavelength, λC = 2.426 · 10 m, in about 12 ln(2a0 + 1) − a0 = 0. One can see, that rv is times. So, for choosing n ≈ 31 we find from (14) an oscillating function. The larger Ω, the more that the radius of the vortex is about the Comp- quickly the vortex trembles. As Ω increases, the ton wavelength. This number is quite enough vortex radius decreases. However it grows with to prevent any fatal catastrophe of the vortex. increasing the number n. Let us evaluate the That is, the vortex is a long-lived robust object. radius rv at choosing the viscosity ν equal to From the above we see that, on a distance about ν¯ =h/ ¯ 2m. Hereν ¯ is the diffusion coefficient of the Compton wavelength, virtual particles can the Brownian sub-quantum particles wandering be involved into a vorticity dancing around the in the Nelson’s aether [45], see Appendix. In electron core, by polarizing the electron charge. the case of electronν ¯ ≈ 5.79 · 10−5 m2/s. As for This dancing happens at trembling motion of the Ω, let it be equal to 2mc2/h¯, or approximately electron with the frequency Ω = 2mc2/h¯. That 1.6 · 10 21 radians per second for electron. Here c oscillating motion has a deep relation to the so- is the speed of light and m is the electron mass. called ”Zitterbewegung” [4, 33]. Then we have (ν/Ω)1/2 ≈ 1.93 · 10−13 m. This One can give a general solution of Eq. (8) length is seen to be smaller then the Compton which has the following presentation

      Γ  r2  ω(r, t) = ! exp − ! , (15) t  t  R 2  4π R ν(τ)dτ + σ2  4π ν(τ)dτ + σ   0 0

       2  Γ   r  v(r, t) = 1 − exp −  (16)  t !  2πr   R 2   4π ν(τ)dτ + σ   0 

The viscosity function ν(t) is a quasi-periodic out interaction. function or even is represented by a color noise. The integral of the viscosity function memo- The viscosity function ν(t) is a quasi-periodic rizes integrally character of the viscosity of the function or even is represented by a color noise. medium. Due to this memory effect, the vortex The integral of the viscosity function memo- may live a long enough. As for interpretation rizes integrally character of the viscosity of the of these solutions with the quantum-mechanical medium. Due to this memory effect, the vortex point of view, we may say that there exists a reg- may live a long enough. As for interpretation ular exchange by quanta with the vacuum fluctu- of these solutions with the quantum-mechanical ations, Fig. 2. The integral accumulates all cases point of view, we may say that there exists a reg- of the exchange with the vacuum. The constant ular exchange by quanta with the vacuum fluctu- σ having dimension of length, prevents appear- ations, Fig. 2. The integral accumulates all cases ance of singularities. One can see that even at of the exchange with the vacuum. The constant ν = 0, but σ > 0, abundance of long-lived vor- σ having dimension of length, prevents appear- tices can exist in the vacuum. Such vortices are ance of singularities. One can see that even at ”ghosts” in the superfluid being invisible with- ν = 0, but σ > 0, abundance of long-lived vortices can exist in the vacuum. Such vortices are 7

z (pointed in the figure by arrow c) to the center of the torus located in the origin of coor- dinates (x, y, z). A body of the tube, for the sake of visualization is colored in cyan. Eq. (17) parametrized by t gives a helicoidal vortex ring shown in this figure. Parameters ω1 and ω2 are frequencies of rotation along the arrow a about a the center of the torus (about the axis z) and y c x b rotation along the arrow b about the center of the tube (about the axis pointed by arrow c), FIG. 4: Helicoidal vortex ring: r0 = 2, r1 = 3, ω2 = 12ω1, φ2 = φ1 = 0. respectively. Phases φ1 and φ2 have uncertain quantities ranging from 0 to 2π. By choosing the phases within this interval with a small incre- ”ghosts” in the superfluid being invisible with- ment, we may fill the torus by the helicoidal vor- out interaction. tices everywhere densely. The vorticity is maximal along the center of the tube. Whereas the A. Vortex rings and vortex balls velocity of rotation about this center in the vicinity of it is minimal. However, the velocity grows If we roll up the vortex tube in a ring and glue as a distance from the center increases. After together its opposite ends we obtain a vortex reaching of some maximal value the velocity fur- ring. A result of such an operation put into the ther begins to decrease. (x, y) plane is shown in Fig 4. Position of points on the helicoidal vortex ring [60] in the Cartesian Let the radius r1 in Eq. (17) tends to zero. The coordinate system is given by helicoidal vortex ring in this case will transform into a vortex ring enveloping a spherical ball.  x = (r + r cos(ω t + φ )) cos(ω t + φ ), The vortex ring for the case r = 4, r ≈ 0,  1 0 2 2 1 1 0 1 y = (r1 + r0 cos(ω2t + φ2)) sin(ω1t + φ1), ω2 = 3ω1 and φ2 = φ1 = 0 drawn by thick curve   z = r0 sin(ω2t + φ2). colored in deep green is shown in Fig. 5. Motion (17) of an elementary vortex clot along the vortex Here r0 is the radius of the tube. And r1 rep- ring (along the thick curve colored in deep green) resents the distance from the center of the tube takes place with a velocity

 v = −r0ω 0 sin(ω 0t + φ 0) cos(ω1t + φ1) − r0ω1 cos(ω 0t + φ 0) sin(ω1t + φ1)  R,x  −r ω sin(ω t + φ ),  1 1 1 1 ~vR = vR,y = −r0ω 0 sin(ω 0t + φ 0) sin(ω1t + φ1) + r0ω1 cos(ω 0t + φ 0) cos(ω1t + φ1) (18)   +r1ω1 cos(ω1t + φ1),  vR,z = r0ω 0 cos(ω 0t + φ 0),

The velocity of the clot at the initial time is vR,z = −r0ω2 = 3r0ω1. We designate this veloc-

vR,x = 0, vR,y = r0ω1, vR,z = r0ω2 = 3r0ω1 ity as ~v−. Sum of the two opposite velocities, (the initial point (x, y, z) = (4, 0, 0) is on the top ~v+ and ~v−, gives the velocity ~v0 = (0, 2r0ω1, 0). of the ball). We designate this velocity as ~v+. During t = (1+ 3k)π/3ω1 and t = (2+ 3k)π/3ω1 Through the time t = πω1 the elementary vor- (k = 1, 2, ···) the clot travels through the posi- tex clot returns to the top position. The veloc- tions 1 and 2 both in the forward and in back-

ity in this case is equal to vR,x = 0, vR,y = r0ω2, ward directions, respectively. In the vicinity of 8

x ever, it is slightly differ from the pressure gra- u- 0 u0 dient presented in the customary Navier-Stokes u+ equation [38, 39]. One can rewrite this term in detail P ! ρM ∇ = ∇P − P ∇ ln(ρM ). (19) u0 ρM z u- u+ 2 y The first term, ∇P , is the customary pressure gradient represented in the Navier-Stokes equa- 1 u- tion. Whereas, the second term, P ∇ ln(ρM ), is u+ u0 an extra term describing changing the logarithm FIG. 5: Helicoidal vortex ring (colored in deep green) convo- of the density along increment of length (the en- luted into the vortex ball. The input parameters of the ball tropy increment) multiplied by P . It may mean are as follows r0 = 4, r1 = 0.01 1, ω2 = 3ω1, φ2 = φ1 = 0. that change of the pressure is induced by change The radius r0 represents a mean radius of the ball, where the of the entropy per length, or else by change of velocity v0 reaches a maximal value. x the information flow [41, 55] per length. This .... term has signs typical of the osmotic pressure, 0 0 0 0 mentioned by Nelson [45]. 0 0 0 Let us consider in this respect the pressure P in . 0 . 0 .. more detail. We shall represent the pressure con- .. .. sisting of two parts, P1 and P2. We begin from

0 0 0 the Fick’s law [28]. The law says that the diffu- 0 0 .... 0 0 ~ 0 sion flux, J, is proportional to the negative value z y ~ of the density gradient J = −D∇ρM . Here D is the diffusion coefficientν ¯ =h/ ¯ 2m [45], see Nel- son’s definition in Appendix. Since the term FIG. 6: The vortex ball rotating about axis z with the max- ~ imal velocity v0 that is reached on the surface of the ball ν¯∇J has dimension of the pressure, we define P1 as the pressure having diffusion nature

2 these points the velocities ~v+ and ~v− yield the re- ~ h¯ 2 P1 =ν ¯∇J = − ∇ ρ . (20) sulting velocity ~v0 directed along the circle lying 4m2 M in the plane (x, y). Observe that the kinetic energy of the diffusion The ball can be filled everywhere densely by flux is (m/2)(J/ρ~ )2 . It means that there ex- other rings at adding them with other phases φ M 1 ists one more pressure as the average momentum and φ ranging from 0 to 2π The velocity ~v for 2 0 transfer per unit area per unit time: any ring will lie on the same circles centered on the axis z. We see a dense ball that rolls along ρ J~ !2 h¯2 (∇ρ )2 P = M = M . (21) the axis y, Fig. 6. Observe that the ball pulsates 2 2 ρ 8m2 ρ on the frequency Ω as it rolls along its path, as M M it follows from the above computations. Perfect Now we can see that sum of the two pressures, modes describing the rolling ball are spherical P1 + P2, divided by ρM gives a term harmonics [20]. 2 !2 2 2 h¯ ∇ρM h¯ ∇ ρM QM = 2 − 2 . (22) 8m ρM 4m ρM III. DERIVATION OF THE SCHRODINGER¨ EQUATION One can see that accurate to the divisor m this term represents the quantum potential. The third term in the right side of Eq. (6) deals To bring the expression (22) to a form of the with the pressure gradient. One can see, how- quantum potential, we need to introduce instead 9

of the mass density ρM the probability density interesting to interpret an osmotic nature of the according to the following presentation pressure [45]. The interpretation can be the following (see Appendix): a semipermeable mem- M mN brane where the osmotic pressure manifests it- ρM = = = mρ. (23) ∆V ∆V self is an instant, which divides the past and the Here the mass M is a product of an elementary future (that is, the 3D brane of our being rep- mass m by the number of these masses, N, filling resents the semipermeable membrane in the 4D world). In other words, the thermalized fluctu- the volume ∆V . Then the mass density ρM is defined as a product of the elementary mass m ating force field described by Grössing[27–30] is by the density of quasi-particles ρ = N/∆V . We asymmetric with respect to the time arrow. may imagine such a set of the quasi-particles as a collection of particles which comes from the realm of classical Lagrange fluids [34]. A. Transition to the Schrödingerequation Let us divide the Navier-Stokes equation Eq. (6) by the probability density ρ. We obtain The current velocity v contains two compo- nent – irrotational and solenoidal [38] that re- ∂~v ! m + (~v · ∇)~v late to vortex-free and vortex motions of the ∂ t medium, respectively. The basis for the latter F~ is the Kelvin-Stokes theorem. Scalar and vector = + ν(t) ∇ 2m~v − ∇Q. (24) N fields underlie of manifestation of the irrotational and solenoidal velocities Here F~ /N is the force per one the quasi-particle. The kinetic viscosity ν(t) = µ(t)/ρ is rep- 1 M ~v = ~vS + ~vR = ∇S + ~vR . (26) resented through the diffusion coefficientν ¯ = m h/¯ 2m [45]. Namely, ν(t) = 2¯νg(t) = νg(t), Here subscripts S and R hint to scalar and vec- where ν =h/m ¯ and g(t) is the dimensionless tor (rotational) potentials underlying emergence time dependent function. The function Q here of these two components of the velocity. These is the real quantum potential velocities satisfy the following equations 2 !2 2 2 h¯ ∇ρ h¯ ∇ ρ Q = − . (25) (∇ · ~vS ) 6= 0, [∇ × ~vS ] = 0, 8m ρ 4m ρ (27) (∇ · ~vR ) = 0, [∇ × ~vR ] = ~ω. Grössingnoticed that the term ∇Q, the gradient The scalar field is represented by the scalar func- of the quantum potential, describes a completely tion S - action in classical mechanics. Both ve- thermalized fluctuating force field [27, 28]. Here locities are perpendicular to each other. We may the fluctuating force is expressed via the gradient define the momentum and the kinetic energy of the pressure divided by the density distribution of sub-quantum particles chaotically moving  ~p = m~v = ∇S + m~v ,  R in the environment. Perhaps, they are virtual  2 v 2 (28) particle-antiparticle pairs.  v 1 2 R  m = (∇S) + m . Since the pressure provides a basis of the quan- 2 2m 2 tum potential, as was shown above, it would be

Now we may rewrite the Navier-Stokes equation (24) in the more detailed form

∂ ( 1 ) (∇S + m~v ) + ((∇S)2 + m2v 2) + [~ω × (∇S + m~v )] ∂ t R 2m R R 10

2 = −∇U − ∇Q + ν(t)∇ (∇S + m~vR ) (29) The term embraced by the curly brackets in the first line stems from (~v∇)~v = ∇v 2/2 + [~ω × ~v], see Eq. (5). As for the external force in the Navier-Stokes equation (24), we take into account that it is conservative, i.e., F~ /N = −∇U, where U is the potential energy relating to the single sub- particle. The terms ∇U and ∇Q are gradients of the potential energy and of the quantum potential, respectively. The third term in the second line describes the viscosity of the medium. As was said above the viscosity coefficient in the average is equal to zero. Let us rewrite Eq. (29) by regrouping the terms ∂ 1 m ! ∂ ∇ S + (∇S)2 + v 2 +U +Q−ν(t)∇2S = −m ~v −[~ω ×(∇S +m~v )]+ν(t)m∇2~v . (30) ∂ t 2m 2 R ∂ t R R R We assume that fluctuations of the viscosity about zero occur much more frequent, than characteristic time of displacements of the quasi-particles. For that reason, we omit the term ν(t)∇2S by supposing in the first approximation, that the medium is absolutely superfluid - there are no energy sources and

sinks. By multiplying this equation from the left by ~vS we ﬁnd that the right part of this equation

vanishes since (~vS · ~vR ) = (~vS · ~ω) = 0. On the other part ~vS · ∇ = d/dl represents a derivative along the streamline [37] - the left part vanishes along this stream line. The expression under the brackets should be a constant. As a result, we come to the following modiﬁed Hamilton-Jacobi equation

∂ 1 m h¯2 " ∇ρ!2 ∇2ρ# S + (∇S)2 + v 2 + U(~r) + − = C. (31) ∂ t 2m 2 R 2m 2ρ 2ρ

The modification is due to adding the quantum the vector potential multiplied by the ratio of potential (25). In this equation, C is an integra- the charge to the light speed, which appears in tion constant. We see that the third term in this quantum electrodynamics [44]. Appearance of equation represents energy of the vortex. On this term in this equation is conditioned by the the other hand, we can see that the vortex given Helmholtz theorem. by Eq. (7) is replenished by the kinetic energy By substituting into Eq (33) the wave function coming from the scalar field S, namely via the Ψ represented in a polar form term (~ω · ∇)~v. Solutions of these two equations, √ Eq. (7) and Eq. (31), describing dynamics of the Ψ = ρ exp{iS/h¯} (34) vortex and scalar fields, depend on each other. Both the continuity equation and separating on real and imaginary parts we come to Eqs. (31) and (32). So, the Navier- ∂ ρ Stokes equation (6) with the slightly expanded + (~v · ∇)ρ = 0, (32) ∂ t the pressure gradient term can be reduced to the Schrödingerequation if we take into considera- which stems from Eq. (3), and the quantum tion also the continuity equation. Hamilton-Jacobi equation (31) can be extracted The Shrödingerwave equation can be resolved from the following Schrödingerequation by heuristic writting of a solution through us- ∂Ψ 1 ing the Huygens’ principle in its mathematical ih¯ = (−ih¯∇ + m~v )2Ψ + U(~r)Ψ − CΨ. ∂ t 2m R integral presentation (33) Z Ψ(x, t ) = G(x, y)Ψ(y, t )dy, t > t . (35) The kinetic momentum operator (−ih¯∇ + m~vR ) 2 1 2 1 contains the term m~vR describing a contribution of the vortex motion. This term is analogous to The function G(x, y) is called a propagator. It 11

p(y,z) 0.1 0.1 Number of slits = 9 m = 1.204E-24 kg 0.075 λ = 5E-12 m d = 2.5E-7 m 0.05 0.05 the Talbot length: 2 ___d 0.025 yT = 2 λ 0 0

2yT z 4yT

6yT

8yT

10yT

FIG. 7: Probability density distribution from scattering the fullerene molecules (60 carbon atoms packed on the sphere like a football) on the grating containing 9 slits: de Broglie wavelength is 5 pm and the distance between slits is 250 nm. bears information about a physical scene con- tion can be achieved by applying the Feynman taining physical equipment such as sources, de- path integral technique [19, 24]. By applying tectors, collimators, gratings, etc. This scene is this technique for describing the wave propaga- simulated by the potential U(~r) embedded in the tion through a lattice consisting of N slits we get Schr¨odingerequation and by the boundary con- the solution [54] ditions. So, a solution of the Schr¨odingerequa-

 ! !2   N − 1  N−1  z − n − d  1 X  2  |Ψ(y, z)i = s · exp − ! (36) λy n=0  2 λy  N 1 + i  2b 1 + i  2πb2  2πb2  12

yT 2yT 3yT 4yT 5yT 6yT FIG. 8: Interference pattern of the coherent ﬂow of the fullerene molecules with the de Broglie wavelength λ = 5 pm within a zone y ≤ 6yT from the grating containing 9 slits. Lilac curves against the grey background represent the Bohmian trajectories.

√ Here i = −1 is imaginary unit, λ is the de This length bears name of Henry Fox Talbot who Broglie wavelength, b is the slit width, and d discovered in 1836 [61] a beautiful interference is the distance between slits. In this calcula- pattern of light, named further as the Talbot tion we have used λ = 5 pm, b = 5 · 103λ, and carpet [6, 7]. d = 5·104λ = 10b = 250 nm. By choosing N = 9 The particles, incident on the slit grating, slits, for example, we find the interference pat- come from a distant coherent source. The de tern shown in Fig. 7 as the density distribution Broglie wavelength of the particle, λ = h/p (h function [54]. This function is a scalar product is the Planck constant, and p is the particle mo- of the wave function |Ψ(y, z)i, namely: mentum) is a main characteristics binding the corpuscular Newtonian physics with the wave p(y, z) = hΨ(y, z)|Ψ(y, z)i. (37) Huygens’ physics. It is that we call now the A useful unit of length at observation of the in- wave-particle dualism. The de Broglie pilot wave terference patterns is the Talbot length: being represented by the complex-valued wave function |Ψi fills all ambient space, except of d2 opaque objects, which determine the boundary y = 2 . (38) T λ conditions. The particle passes from the source 13 to a place of detection along the optimal trajec- 80 tory, Bohmian trajectory [55]. The equation de- eV scribing motion of the Bohmian particle can be 60 found, for example, in [5]. There is the unique (p) trajectory for each the particle, the vortex ball in 40 our case. However, an attempt to measure exact position of the ball along the trajectory together with its velocity fails. Namely, there is no way to 20 measure simultaneously the complementary parameters, such as coordinate and velocity, what 0 10 10 10 0 2.10 4.10 -1 6.10 follows from the uncertainly principle [56]. p⁄h-, m Figure 8 shows in lilac color Bohmian trajec- FIG. 9: The dispersion relation vs. p. The dotted curve tories divergent from the slit grating. The prob- shows the non-relativistic square dispersion relation ∼ p2 . ability density distribution is shown here in grey The hump on the curve is a contribution of the roton compo- nent p f(p − p ), p /¯h ≈ 1.89 · 1010 m−1, and σ = 0.5p . color ranging from white for p = 0 to light R R R R grey for max p. Bundle of the Bohmian trajectories imitates a fluid flow through the ob- and the viscosity equal to zero. Nearest analogue stacle, containing slits, relatively well. One can of such a medium is the superfluid helium [64], see that characteristic streamlets are formed in which will serve us as an example for further the flow, along which particles move. Such a consideration of this medium. The vacuum is de- vision of hydrodynamical behaviors of quantum fined as a state with the lowest possible energy. systems is typical for many scientists since the We shall consider a simple vacuum consisting formation of the quantum mechanics up to our of electron-positron pairs. The pairs fluctuate days [11, 17, 26, 36, 43, 62, 66]. Principal mo- within the first Bohr orbit having energy about ment is that the Schrödingerequation describes 13.6 · 2 eV ≈ 27 eV. Bohr radius of this orbit −11 the expiration of the superfluid medium, which is r 1 ≈ 5.29 · 10 m. These fluctuations occur depend on the boundary conditions and other about the center of their masses. The total mass devices perturbing it (as, for example, the slit of the pair, mp, is equal to doubled mass of the gratings, collimators and others). The vortex electron, m. The charge of the pair is zero. The balls move along optimal directions of the flows total spin of the pair is equal to 0. The angu- - along the Bohmian paths. lar momentum, L, is nonzero, however. For the first Bohr orbit L =h ¯. The velocity of rotating about this orbit is L/(r m) ≈ 2.192 · 10 6 m/s. IV. PHYSICAL VACUUM AS A SUPERFLUID 1 MEDIUM It means that there exist an elementary vortex. Ensemble of such vortices forms a vortex line. The Schrödingerequation (33) describes a flow We may evaluate the dispersion relation be- of the peculiar fluid that is the physical vac- tween the energy, (p) =hω ¯ , and wave number, uum. The vacuum contains pairs of particle- p =hk ¯ , as it done in [40]. As follows from the antiparticles. The pair, in itself, is the Bose par- Schrödingerequation (33) we have: ticle that stays at a temperature close to zero. 1 2 In aggregate, the pairs make up Bose-Einstein (p) = (p + pR f(p − pR )) . (39) 2m condensate. It means that the vacuum repre- p sents a superfluid medium [59]. A ’fluidic’ na- Here pR = L/r 1 = mpvR is the momentum of the ture of the space itself is exhibited through this rotation. The function f(p−pR ) is a form-factor medium. Another name of such an ’ideal fluid’ relating to the electron-positron pairs rotating is the ether [44]. about the center of their mass mp. The form- The physical vacuum is a strongly correlated factor describes dispersion of the momentum p system with dominating collective effects [51] around pR conditioned by fluctuations about the 14

vortex-free state 2 B R

vortex front

Ω

twisted vortices

1 FIG. 11: Rotation of the superfluid fluid is not uniform but takes place via a lattice of quantized vortices, whose cores 2R (colored in yellow) are parallel to the axis of rotation [22, 42]. solid wall Green arrows are the vorticity ω. Small black arrows indi- cate the circulation of the velocity vR around the cores. The vacuum is supported between two non-ferromagnetic disks, A FIG. 10: The formation of twisted vortex state [22]. The and B, fixed on center shafts, 1 and 2, of electric motors [52], vortices have their propagating ends bent to the side wall see Fig. 12. Radiuses of the both disks are R = 82.5 mm and of the rotating cylinder. As they expand upwards into the distance between them can vary from 1 to 3 mm and more. vortex-free state, the ends of the vortex lines rotate around The vortex bundle rotates rigidly with the disk A. As soon as the cylinder axis. The twist is nonuniform because boundary the vortex bundle reaches the top disk B it begins rotation. conditions allow it to unwind at the bottom solid wall. The figure gives a snapshot (at time t = 25Ω−1, where Ω is the angular velocity.) of a numerical simulation of 23 vortices shift of the wave function, and most possible this initially generated near the bottom end (t = 0). Courtesy kindly by Erkki Thuneberg. phase relates to the chemical potential of a boson (the electron-positron pair) [40]. We shall not ground state with the lowest energy. The form- take into account contribution of this term in factor is similar to the Gaussian curve the dispersion diagram because of its smallness. As follows from the above consideration of the ( (p − p )2 ) f(p − p ) = exp − R . (40) Navier-Stokes equation, Eq. (41) can be reduced R 2σ2 to the Euler equation Here σ is the variance of this form-factor. It ∂~vR ∇P is smaller or close to p . The dispersion rela- + [~ω × ~vR ] = − , (42) R ∂ t m ρ tion (39) is shown in Fig. 9. The hump on the p curve is due to the contribution of the rotating that describes a flow of the inviscid incompress- electron-positron pair about the center of their ible fluid under the pressure field P . One can masses. These rotating objects are named ro- see from here that the Coriolis force appears tons [40]. as a restoring force, forcing the displaced fluid Rotons are ubiquitous in vacuum because of a particles to move in circles. The Coriolis force huge availability of pairs of particle-antiparticle. is the generating force of waves called inertial The movement of the roton in the free space is waves [40]. The Euler equation admits a sta- described by the Schrödingerequation tionary solution for uniform swirling flow under ∂Ψ 1 the pressure gradient along z. i = (−ih¯∇ + m ~v )2Ψ − CΨ. (41) Formation of the swirling flow, the twisted ∂ t 2m p R p vortex state, has been studied in the super- The constant C determines an uncertain phase fluid 3He-B [22]. These observations give us a 15

FIG. 12: Basic diagram (a) and general view of the device (b) for researching mass dynamics effects [52]: 1 and 2 are shafts with mounted on them electric motors; 3 and 4 are steel plates with mounted on them electromagnetic brakes; 5 and 6 (A and B, see Fig. 11) are disks rigidly fixed on flanges of the rotors of the electric motors. Courtesy kindly by Vladimir Samokhvalov. possibility to suppose the existence of such phe- is the radius of the first Bohr orbit. Really, the nomena in the physical vacuum. The twisted number of the vortices situated on the square, vortex states observed in the superfluid 3He-B N, is considerably smaller. It can be evaluated are closely related to the inertial waves in rotat- by multiplying Nmax by a factor δ. This factor is ing classical fluids. The superfluid initially is at equal to the ratio of the geometric mean of the 6 rest [22]. The vortices are nucleated at a bottom velocities vR =h/ ¯ (r 1m) ≈ 2.192 · 10 m/s and disk platform rotating with the angular veloc- VD = RΩ to their arithmetic mean. Here Ω is ity Ω about axis z. As the platform rotates they an angular rate of the disk A. So, we have propagate upward by creating the twisted vortex state spontaneously, Fig. 10. The Coriolis forces q v · V s take part in this twisting. The twisted vortices R D VD 15 grow upward along the cylinder axis [42]. N = Nmax = Nmax ≈ 6 · 10 (43) vR + VD vR Analogous experiment with nucleating vortices can be realized when the lower disk A rotates in the vacuum, Fig. 11. In this case, the vortices are at the angular rate Ω = 160 1/s [52] the disk velocity V = 13.2 m/s. Now we can evaluate viewed as the dancing electron-positron pairs on D the first Bohr orbit. As the vortices grow upward the kinetic energy of the vortex bundle induced the spontaneous twisted vortex states arise. The by the rotating disk A.This kinetic energy is E = N · m v 2/2 ≈ 0.026 J. This energy is sufficient latter by reaching upper fixed disk B can capture p R it into rotation. for transfer of the moment of force to the disk B. Measured in the experiment [52] the torque is Pr. V. Samohvalov has shown through the ex- about 0.01 N·m. So, the disk B can be captured periment [52], that the vortex bundle induced by the twisted vortex. by rotating the bottom non-ferromagnetic disk A leads to rotation of the upper fixed initially The formation of the growing twisted vortices non-ferromagnetic disk B, Fig. 12. Both disks can be confirmed with attracting modern meth- at room temperature have been placed in the ods of interference of light rays passing through container with technical vacuum at 0.02 Torr, the gap between the disks. Light traveling along The utmost number of the vortices that may be two paths through the space between the disks placed on the square of the disk A is Nmax = undergoes a phase shift manifested in the inter- 2 2 18 (2πR )/(2πr 1) = 2 · 10 , where R = 82.5 mm ference pattern [8, 44] as it was shown in the −11 is the radius of the disk and r 1 ≈ 5.29 · 10 m famous experiment of Aharonov and Bohm [2]. 16

V. CONCLUSION particles of matter (”droplets”), which exist in it and move through it [44, 49, 53]. The particle The Schrödinger equation is deduced from traveling through this medium perturbs virtual two equations, the continuity equation and the particle-antiparticle pairs, which, in turn, create Navier-Stokes equation. At that, the latter both constructive and destructive interference at contains slightly modified the gradient pressure the forefront of the particle [24]. Thus, the virtual pairs interfering each other provide an opti- term, namely, ∇P → ρM ∇(P/ρM ) − P ∇ ln(ρM ). mal, Bohmian, path for the particle. The extra term P ∇ ln(ρM ) describes change of the pressure induced by change of the entropy Assume next, that the baryonic matter is similar, say, on ”hydrophobic” fluid, whereas the ln(ρM ) per length. In this case, the modified gradient pressure term can be reduced to the dark matter, say, is similar to ”hydrophilic” quantum potential through using the Fick’s law. fluid. Then the baryonic matter will diverge each In the law we replace also the diffusion coeffi- from other on cosmological scale owing to repul- cient by the factorh/ ¯ 2m, whereh ¯ is the reduced sive properties of the dark matter, like soap spots Planck constant and m is mass of the particle. diverge on the water surface. Observe that this We have shown that a vortex arising in a fluid phenomenon exhibits itself through existence of can exist infinitely long if the viscosity under- the short-range repulsive gravitational force that goes periodic oscillations between positive and maintains the incompatibility between the dark negative values. At that, the viscosity, in aver- matter and the baryonic matter [14, 23]. At that, age on time, stays equal to zero. It can mean the dark matter stays invisible. One can imag- that the fluid is superfluid. In our case, the su- ine that the zero-point vacuum fluctuations are perfluid consists of pairs of particle-antiparticle nothing as weak ripples on a surface of the dark representing the Bose-Einstein condensate. matter. As for the quantum reality, such a periodic regime can be interpreted as exchange of the Acknowledgments energy quanta of the vortex with the vacuum through the zero-point vacuum fluctuations. In The author thanks Mike Cavedon for useful reality, these fluctuations are random, covering and valuable remarks and offers. The author a wide range of frequencies from zero to infinity. thanks also Miss Pipa (Quantum Portal admin- Based on this observation we have assumed that istrator) for preparing a program drawing Fig. 8. the fluctuations of the vacuum ground state can support long-lived existence of vortex quantum Appendix: Nelson’s derivation of the Schrödinger objects. The core of such a vortex has nonzero equation radius inside of which the velocity tends to zero. In the center of the vortex, the velocity vanishes. Nelson proclaim that the medium through The velocity reaches maximal values on bound- which a particle moves contains myriad sub- ary of the core, and then it decreases to zero as particles that accomplish Brownian motions by the distance to the vortex goes to infinity. colliding with each other chaotically. The Brow- The experimental observations of the Couder’s nian motions is described by the Wiener process team [15, 16, 21, 50] can have far-reaching on- with the diffusion coefficient tological perspectives in regard of studying our h¯ universe. Really, we can imagine that our world ν¯ = . (44) is represented by myriad of baryonic and lepton 2m ”droplets” bouncing on a super-surface of some Here m is mass of the particle andh ¯ = h/2π unknown dark matter. A layer that divides these is the reduced Planck constant. Here we use ν ”droplets”, i.e., particles, and the dark matter is with the upper bar in order to avoid confusion the superfluid vacuum medium. This medium, with the kinematic viscosity adopted in hydro- called also the ether [45], is populated by the dynamics. As seen this motion has a quantum 17 nature [45] in contrast to the macroscopic Brow- There is a one more velocity, which is represented nian motions where the diffusion coefficient has via difference of b(x(t), t) and b∗(x(t), t): a viewν ¯ = kT/mβ−1; here k is Boltzmann con- 1 stant, T is a temperature, and is the relaxation ~u(t) = b(x(t), t) − b∗(x(t), t) . (51) time. 2 Two equations are main in the article [45]. The According to Einstein’s theory of Brownian mo- position x(t) of the Brownian particle, being sub- tion, ~u(t) is the velocity acquired by a Brownian jected either by external forces or by currents in particle, in equilibrium with respect to an exter- the medium, can be written by two equivalent nal force, to balance the osmotic force [46]. For equations this reason, this velocity is named the osmotic velocity. It can be expressed in the following dx(t) = b(x(t), t)dt + dw(t), (45) form dx(t) = b (x(t), t)dt + dw (t). (46) ∗ ∗ h¯ ∇R(t) ~u(t) =ν ¯∇(ln(ρ(t)) = , (52) Here w(t) and w∗(t) are the Wiener processes, m R(t) both have equivalent properties. Variables b and where ρ(t) is the probability density of disclos- b are vector-valued forward and backward func- ∗ ing a particle in the vicinity of x(t) and R(t) = tions on space-time, respectively. In fact, they ρ(t)1/2 is the probability density amplitude. In are the mean forward and mean backward mea- turn, the current velocity is expressed through sured quantities gradient of a scalar field S called the action x(t + ∆t) − x(t) h¯ b(x(t), t) = lim E t , (47) ~v = − ∇S(t). (53) ∆t→0+ ∆t m x(t) − x(t − ∆t) b∗(x(t), t) = lim E t . (48) The both equations, Eqs. (45) and (46), in- ∆t→0+ ∆t troduced above are important for derivation of

Here E t denotes the conditional expectation (av- the Schrödingerequation. The derivation of the erage) given the state of the system at time equation is provided by the use of the wave func- t, and 0+ means that ∆t tends to 0 through tion presented in the polar form positive values. Thus b(x(t), t) and b∗(x(t), t) are again stochastic variables [46, 47]. It is in- Ψ = R exp{iS/h¯}, (54) structive to compare calculus (47) and (48) with by replacing the velocities ~v(t) and ~u(t) in the classical calculations of infinitesimal small incre- initial equations. It should be noted that Nelson ments departs from two equations describing directed x(t + ∆t) − x(t) the forward and backward Brownian motions v(t) = lim which are written down for real-valued func- ∆t→0+ ∆t tions. In order to come to the Schrödinger equa- x(t) − x(t − ∆t) = lim . (49) tion he has used a complex-valued wave function ∆t→0+ ∆t exp{R + iS} instead of the generally accepted These calculations are seen to be symmetrical R exp{iS/h¯}. Obviously, this discrepancy are with respect to the time arrow, whereas (47) eliminated by replacing exp{R} → R. and (48) are not. Observe that the wave function represented in It should be noted that b(x(t), t) and the polar form (54) is used for getting equations underlying the Bohmian mechanics [5]. These b∗(x(t), t) are not real velocities. The real current velocity of the particle is calculated as two equations are the continuity equation and the Hamilton-Jacobi equation containing an ex- 1 tra term known as the Bohmian quantum poten- ~v(t) = b(x(t), t) + b (x(t), t) . (50) 2 ∗ tial. The quantum potential has the following 18 view: cata and Fiscaletti [41], who have shown that the quantum potential has relation to the Bell h¯2 ∇2R h¯ (∇R~u) Q = − = − length indicating a non-local correlation, one can 2m R 2 R add that the non-local correlation exists also be- m h¯ = − u2 − (∇~u). (55) tween the past and the future. E. Nelson as one 2 2 can see has considered a particle motion through One can see that the quantum potential depends the ether populated by sub-particles experienc- only on the osmotic velocity, which is expressed ing accidental collisions with each other. The through difference of the forward and backward Brownian motions of the sub-particles submits averaged quantities (47) and (48). These for- to the Wiener process with the diffusion coef- ward and backward quantities can be interpreted ficient ν proportional to the Plank constant as as uncompensated flows through a ’semiperme- shown in Eq. (44). The ether behaves itself as a able membrane’ which represents an instant di- free-friction fluid. viding the past and the future. Following to Li-

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