Table Of ContentPhysical 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)
TheNavier-Stokesequationcontainstwotermswhichhavebeensubjectedtoslightmodification:
(a) the viscosity term depends of time (the viscosity in average on time is zero, but its variance
6 is non-zero); (b) the pressure gradient contains an added term describing the quantum entropy
1 gradient multiplied by the pressure. Owing to these modifications, the Navier-Stokes equation
0 can be reduced to the Schro¨dinger equation describing behavior of a particle into the vacuum
2 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
g
with zero-point energy of the vacuum. Radius of the vortex trembles around some average value.
u
This observation sheds the light to the Zitterbewegung phenomenon. The long-lived vortex has a
A
non-zero core where the vortex velocity vanishes.
8
Keywords: Navier-Stokes; Schro¨dinger; zero-point fluctuations; superfluid vacuum; vortex;
] Bohmian trajectory; interference
h
p
t- I. INTRODUCTION registered. Instead, the wave function represents
n
it existence within an experimental scene [13].
a
u Another interpretation was proposed by Louis
A dramatic situation in physical understand-
q
de Broglie [18], which permits to explain such an
[ ing of the nature emerged in the late of 19th cen-
experiment. In de Broglie’s wave mechanics and
tury. Observedphenomena on microscales came
4
the double solution theory there are two waves.
v into contradiction with the general positions of
3 There is the wave function that is a mathemat-
classical physics. It was a time of the origina-
6 ical construct. It does not physically exist and
7 tion of new physical ideas explaining these phe-
is used to determine the probabilistic results of
6 nomena. Actually, in a very short period, pos-
0 experiments. There is also a physical wave guid-
tulates of the new science, quantum mechanics
.
1 ing the particle from its creation to detection.
were formulated. The Copenhagen interpreta-
0
As the particle moves from a source to a detec-
5 tion was first who proposed an ontological ba-
1 tor, the particle perturbs the wave field and gets
sis of quantum mechanics [32]. These positions
: a reverse effect from it. As a result, the physi-
v can be stated in the following points: (a) the
i cal wave guides the particle along some optimal
X description of nature is essentially probabilistic;
trajectory, Bohmian trajectory [9, 10], up to its
r (b)aquantumsystemiscompletelydescribedby
a detection.
a wave function; (c) the system manifests wave–
A question arises, what is the de Broglie phys-
particle duality; (d) it is not possible to measure
ical wave? Recently, Couder and Fort [15] has
all variables of the system at the same time; (e)
executed the experiment with the classical oil
eachmeasurementofthequantumsystementails
droplets bouncing on the oil surface. A re-
the collapse of the wave function.
markable observation is that an ensemble of the
Canoneimagineapassageofaquantumparti- 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
∗Electronicaddress: [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(cid:126)v
(cid:126)
sult we shall get the Schr¨odinger equation. . In F = M(cid:126)a = M ; (1)
dt
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
nallyusedtoexplainpropagationoflightandthe 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 ofmotionondeformablebodiesthatareassumed
require the existence of a continuous intermedi- asacontinuum[25]. WerewritethesecondNew-
ate environment between a source and a receiver ton’s law for the case of deformed medium. Let
ofthelight-thelight-bearingether. Itisinstruc- us imagine that a volume ∆V contains a fluid
tive to compare here the two opposite doctrines mediumofthemassM.WedivideEq.(1)by∆V
about the nature of light belonging to Sir Isaac and determine the time-dependent mass density
(cid:126)
Newton and Christian Huygens. Newton main- ρ = M/∆V [58]. In this case F is understood
M
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:
anemptyspaceinaccordancewiththelawofthe
dρ (cid:126)v d(cid:126)v dρ
classical mechanics. Christian Huygens (a con- F(cid:126) = M = ρ +(cid:126)v M. (2)
dt M dt dt
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
dρ ∂ρ
imagine all space populated everywhere densely M = M +((cid:126)v ∇)ρ , (3)
dt ∂t M
by Huygens’s vibrators. All vibrators are silent
d(cid:126)v ∂(cid:126)v
until a wave reaches them. As soon as a wave = + ((cid:126)v ∇)(cid:126)v. (4)
dt ∂t
front reaches them, the vibrators begin to radi-
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
ciesofvisiblelight. Thesevibratorspopulatethe
v2
ether facilitating propagation of the light waves
((cid:126)v ∇)(cid:126)v = ∇ −[(cid:126)v ×[∇×(cid:126)v]]. (5)
through the space. 2
In order to come to idea about existence of As follows from this formula the first term, mul-
theintermediatemedium(ether)thatpenetrates 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 fluidelementmovealongsomecurvein3Dspace.
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(cid:126) = [∇×(cid:126)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. Vectorproduct[(cid:126)v×ω(cid:126)]isperpendicular particles form, in any moment of the time, a
to the both vectors (cid:126)v and ω(cid:126). 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(cid:126). (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.
(cid:32) (cid:33)
∂(cid:126)v
ρ +((cid:126)v ·∇)(cid:126)v Assuming that the fluid is a physical vacuum,
M ∂t
which meets the requirements specified earlier,
F(cid:126) (cid:32) P (cid:33) we must say that the viscosity vanishes. In that
= +µ(t)∇2(cid:126)v −ρ ∇ . (6)
case, the vorticity ω(cid:126) is concentrated in the cen-
∆V M ρ
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 ρ ∇(P/ρ ) = ∇P − P∇ln(ρ ). This
that, we hypothesize that even if there is an ar-
M M M
modification will be important for us when we
bitrary small viscosity, because of the zero-point
shall begin to derive the Schro¨dinger equation.
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-
entific team [15, 16, 21, 50]) of behavior of the
II. VORTEX DYNAMICS
droplets moving on the oil surface, on which the
The second term from the right in Eq. (2) wavesFaradayexist. Hereanimportantmoment
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 goustothezero-pointoscillationsofthevacuum.
for the vorticity: Observe that the bouncing droplet simulates
some aspects of quantum mechanics, stimulat-
∂ω(cid:126)
+(ω(cid:126) ·∇)(cid:126)v = ν∇2ω(cid:126). (7) ingtheoreticalinvestigationsinthisarea[12,27–
∂t
31, 48, 63]. It is interesting to note in this place
Here ν = µ/ρ is the kinematic viscosity. Its di- that Gr¨ossing considers a quantum particle as a
M
mensionis[m2/s]. Itcorrespondstothediffusion 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 (cid:32)∂2ω 1∂ω(cid:33) 2
= νg(t) + . (8)
∂t ∂r2 r ∂r
where Ω is an oscillation frequency and φ is the
uncertain phase.
Here we do not write a sign of vector on the top
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 (cid:40) r2 (cid:41)
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
1 (cid:90)r Γ (cid:32) (cid:40) r2 (cid:41)(cid:33)
(cid:126)v = ω(r(cid:48),t)r(cid:48)dr(cid:48) = 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
t t
0 0
0 0
2 4 5 4
4 3 3
6 2 01 2
8 1r 51 1r
1 0 2 0
0 0
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 (cid:28) 1 this solution is close to the Lamb-Oseen vortex solution [65]
Γ Γ (cid:18) (cid:19)
ω(r,t) = e−r2/4νt, v(r,t) = 1−e−r2/4νt . (12)
4πνt 4πr
5
vortex likely, it manifests itself through interaction of
the quantum object with vacuum fluctuations.
AccordingtoEq.(9),therearehalf-periodswhen
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 en-
ergy to the vortex, Fig. 2. On the whole, the
FIG. 2: Periodic energy exchange between the vortex and
viscosity of the fluid medium, within which the
vacuum fluctuations
vortex tube evolves, in the average remains at
y
zero. Itcanmeanthatthismediumissuperfluid.
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
totheelectron-phononinteraction. Thephonons
are thermal excitations of a lattice. In that case,
they play a role of the background medium.
Qualitativeviewofthevortextubeinitscross-
section is shown in Fig. 3. Values of the veloc-
ity 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.
Hereitlooksasasmalllightgreydisk, wherethe
FIG. 3: Cross-section of the vortex tube in the plane (x,y).
velocities have small values. In the very center
Values of the velocity v are shown in grey ranging from light
of the disk, in particular, the velocity vanishes.
grey (minv) to dark grey (maxv). Density of the pixels rep-
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 cen-
ter. 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
dependingonthedistancetothecenterthefunc- 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
demonstratepulsationsonthefrequencyΩ. Am- order to evaluate this radius we equate to zero
plitudeofthepulsationsisthesmaller, thelarger the first derivative by r of equation (11)
value of the parameter n. At n tending to infin- (cid:40) r2 (cid:41)
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)
4π(ν/Ω)(sin(Ωt+φ)+n))
to assumption, that the kinematic viscosity is
a periodic function of time, namely, νg(t) = The radius is a root of this equation
νcos(Ωt + φ). The viscosity in the quantum (cid:113) (cid:114)ν
r = 2 a (n+sin(Ωt+φ))· . (14)
realm is not a good concept, however. Most v 0
Ω
6
Here a ≈ 1.2564312 is a root of the equation wavelength, λ = 2.426 · 10−12 m, in about 12
0 C
ln(2a + 1) − a = 0. One can see, that r is times. So, for choosing n ≈ 31 we find from (14)
0 0 v
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 r at choosing the viscosity ν equal to From the above we see that, on a distance about
v
ν¯ = h¯/2m. Here ν¯ 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 electron ν¯ ≈ 5.79·10−5 m2/s. As for Thisdancinghappensattremblingmotionofthe
Ω, let it be equal to 2mc2/h¯, or approximately electron with the frequency Ω = 2mc2/h¯. That
1.6·1021 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)
(cid:32) (cid:33) (cid:32) (cid:33)
4π (cid:82)tν(τ)dτ +σ2 4π (cid:82)tν(τ)dτ +σ2
0 0
Γ r2
v(r,t) = 1−exp − (cid:32) (cid:33) (16)
2πr 4π (cid:82)tν(τ)dτ +σ2
0
The viscosity function ν(t) is a quasi-periodic out interaction.
function or even is represented by a color noise.
The viscosity function ν(t) is a quasi-periodic
The integral of the viscosity function memo-
function or even is represented by a color noise.
rizes integrally character of the viscosity of the
The integral of the viscosity function memo-
medium. Due to this memory effect, the vortex
rizes integrally character of the viscosity of the
may live a long enough. As for interpretation
medium. Due to this memory effect, the vortex
of these solutions with the quantum-mechanical
may live a long enough. As for interpretation
point of view, we may say that there exists a reg-
of these solutions with the quantum-mechanical
ularexchangebyquantawiththevacuumfluctu-
point of view, we may say that there exists a reg-
ations, Fig. 2. The integral accumulates all cases
ularexchangebyquantawiththevacuumfluctu-
of the exchange with the vacuum. The constant
ations, Fig. 2. The integral accumulates all cases
σ having dimension of length, prevents appear-
of the exchange with the vacuum. The constant
ance of singularities. One can see that even at
σ having dimension of length, prevents appear-
ν = 0, but σ > 0, abundance of long-lived vor-
ance of singularities. One can see that even at
tices can exist in the vacuum. Such vortices are
ν = 0, but σ > 0, abundance of long-lived vor-
”ghosts” in the superfluid being invisible with-
tices can exist in the vacuum. Such vortices are
7
z (pointed in the figure by arrow c) to the cen-
ter 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 ω and ω are
1 2
frequencies of rotation along the arrow a about
the center of the torus (about the axis z) and
a x
y c
rotation along the arrow b about the center of
b
the tube (about the axis pointed by arrow c),
FIG. 4: Helicoidal vortex ring: r = 2, r = 3, ω = 12ω ,
0 1 2 1
respectively. Phases φ and φ have uncertain
φ2 =φ1 =0. 1 2
quantities ranging from 0 to 2π. By choosing
thephaseswithinthisintervalwithasmallincre-
”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 max-
imal along the center of the tube. Whereas the
A. Vortex rings and vortex balls velocityofrotationaboutthiscenterinthevicin-
ity 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
Lettheradiusr inEq.(17)tendstozero. The
onthehelicoidalvortexring[60]intheCartesian 1
helicoidal vortex ring in this case will transform
coordinate system is given by
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 = (r +r cos(ω t+φ ))sin(ω t+φ ), ω = 3ω and φ = φ = 0 drawn by thick curve
1 0 2 2 1 1 2 1 2 1
z = r sin(ω t+φ ). colored in deep green is shown in Fig. 5. Motion
0 2 2
(17) of an elementary vortex clot along the vortex
Here r is the radius of the tube. And r rep- ring(alongthethickcurvecoloredindeepgreen)
0 1
resents the distance from the center of the tube takes place with a velocity
v = −r ω sin(ω t+φ )cos(ω t+φ )−r ω cos(ω t+φ )sin(ω t+φ )
R,x −r10ω10sin(ω10t+φ10), 1 1 0 1 0 0 1 1
(cid:126)v = v = −r ω sin(ω t+φ )sin(ω t+φ )+r ω cos(ω t+φ )cos(ω t+φ ) (18)
R R,y 0 0 0 0 1 1 0 1 0 0 1 1
v = +rr1ωω1ccooss((ωω1tt++φφ1)),,
R,z 0 0 0 0
The velocity of the clot at the initial time is v = −r ω = 3r ω . We designate this veloc-
R,z 0 2 0 1
v = 0, v = r ω , v = r ω = 3r ω ity as (cid:126)v . Sum of the two opposite velocities,
R,x R,y 0 1 R,z 0 2 0 1 −
(the initial point (x,y,z) = (4,0,0) is on the top (cid:126)v and (cid:126)v , gives the velocity (cid:126)v = (0,2r ω ,0).
+ − 0 0 1
of the ball). We designate this velocity as (cid:126)v . During t = (1+3k)π/3ω and t = (2+3k)π/3ω
+ 1 1
Through the time t = πω the elementary vor- (k = 1,2,···) the clot travels through the posi-
1
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 v = 0, v = r ω , ward directions, respectively. In the vicinity of
R,x R,y 0 2
8
x ever, it is slightly differ from the pressure gra-
u-
dient presented in the customary Navier-Stokes
0 u
u+ 0
equation [38, 39]. One can rewrite this term in
detail
(cid:32) (cid:33)
P
ρ ∇ = ∇P −P∇ln(ρ ). (19)
M ρ M
z u- u0 u+ M
2 y The first term, ∇P, is the customary pressure
gradient represented in the Navier-Stokes equa-
1 tion. Whereas, the second term, P∇ln(ρ ), is
u- M
u+ u an extra term describing changing the logarithm
0
of the density along increment of length (the en-
FIG.5: Helicoidalvortexring(coloredindeepgreen)convo-
tropy increment) multiplied by P. It may mean
luted into the vortex ball. The input parameters of the ball
are as follows r =4, r =0.01(cid:28)1, ω =3ω , φ =φ =0. that change of the pressure is induced by change
0 1 2 1 2 1
Theradiusr representsameanradiusoftheball,wherethe
0 of the entropy per length, or else by change of
velocity v reaches a maximal value.
0
the information flow [41, 55] per length. This
x
.... term has signs typical of the osmotic pressure,
0 0 0 mentioned by Nelson [45].
0
0
0 LetusconsiderinthisrespectthepressurePin
0
.... 00 moredetail. Weshallrepresentthepressurecon-
.... sisting of two parts, P1 and P2. We begin from
0 the Fick’s law [28]. The law says that the diffu-
.... 0 0 0 0 0 0 0 sion flux, J(cid:126), is proportional to the negative value
z y of the density gradient J(cid:126) = −D∇ρ . Here D is
M
the diffusion coefficient ν¯ = h¯/2m [45], see Nel-
son’s definition in Appendix. Since the term
FIG.6: Thevortexballrotatingaboutaxiszwiththemax- (cid:126)
ν¯∇J has dimension of the pressure, we define P
imal velocity v0 that is reached on the surface of the ball 1
as the pressure having diffusion nature
thesepointsthevelocities(cid:126)v and(cid:126)v yieldthere- h¯2
+ − P = ν¯∇J(cid:126) = − ∇2ρ . (20)
sulting velocity (cid:126)v0 directed along the circle lying 1 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(cid:126)/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 (cid:126)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 (cid:32) J(cid:126) (cid:33)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 P +P , divided by ρ gives a term
1 2 M
harmonics [20].
h¯2 (cid:32)∇ρ (cid:33)2 h¯2 ∇2ρ
Q = M − M. (22)
M 8m2 ρ 4m2 ρ
M M
III. DERIVATION OF THE SCHRO¨DINGER
EQUATION One can see that accurate to the divisor m this
term represents the quantum potential.
ThethirdtermintherightsideofEq.(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 ρ the probability density interesting to interpret an osmotic nature of the
M
according to the following presentation pressure [45]. The interpretation can be the fol-
lowing (see Appendix): a semipermeable mem-
M mN
ρ = = = mρ. (23) brane where the osmotic pressure manifests it-
M ∆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-
massmbythenumberofthesemasses, N, filling resents the semipermeable membrane in the 4D
the volume ∆V. Then the mass density ρ is world). In other words, the thermalized fluctu-
M
defined as a product of the elementary mass m ating force field described by Gro¨ssing [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 Schro¨dinger equation
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-
(cid:32) (cid:33)
∂(cid:126)v
m +((cid:126)v ·∇)(cid:126)v late to vortex-free and vortex motions of the
∂t medium, respectively. The basis for the latter
F(cid:126) is the Kelvin-Stokes theorem. Scalar and vector
= +ν(t)∇2m(cid:126)v −∇Q. (24)
fieldsunderlieofmanifestationoftheirrotational
N
and solenoidal velocities
(cid:126)
Here F/N is the force per one the quasi-particle.
1
The kinetic viscosity ν(t) = µ(t)/ρ is rep- (cid:126)v =(cid:126)v +(cid:126)v = ∇S +(cid:126)v . (26)
M S R m R
resented through the diffusion coefficient ν¯ =
h¯/2m [45]. Namely, ν(t) = 2ν¯g(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
h¯2 (cid:32)∇ρ(cid:33)2 h¯2 ∇2ρ
Q = − . (25) (cid:26)(∇·(cid:126)v ) (cid:54)= 0, [∇×(cid:126)v ] = 0,
S S (27)
8m ρ 4m ρ (∇·(cid:126)v ) = 0, [∇×(cid:126)v ] = ω(cid:126).
R R
Gro¨ssingnoticedthattheterm∇Q, thegradient
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
thefluctuatingforceisexpressedviathegradient
define the momentum and the kinetic energy
of the pressure divided by the density distribu-
tionofsub-quantumparticleschaoticallymoving p(cid:126) = m(cid:126)v = ∇S +m(cid:126)v ,
in the environment. Perhaps, they are virtual R
(28)
particle-antiparticle pairs. v2 1 v2
m = (∇S)2 +m R.
Sincethepressureprovidesabasisofthequan- 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
(cid:40) (cid:41)
∂ 1
(∇S +m(cid:126)v ) + ((∇S)2 +m2v2)+[ω(cid:126) ×(∇S +m(cid:126)v )]
∂t R 2m R R
10
= −∇U −∇Q+ν(t)∇2(∇S +m(cid:126)v ) (29)
R
The term embraced by the curly brackets in the first line stems from ((cid:126)v∇)(cid:126)v = ∇v2/2 + [ω(cid:126) ×(cid:126)v],
see Eq. (5). As for the external force in the Navier-Stokes equation (24), we take into account that
(cid:126)
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
(cid:32) (cid:33)
∂ 1 m ∂
∇ S+ (∇S)2+ v2+U+Q−ν(t)∇2S = −m (cid:126)v −[ω(cid:126) ×(∇S+m(cid:126)v )]+ν(t)m∇2(cid:126)v . (30)
∂t 2m 2 R ∂t R R R
Weassumethatfluctuationsoftheviscosityaboutzerooccurmuchmorefrequent,thancharacteristic
timeofdisplacementsofthequasi-particles. Forthatreason, weomitthetermν(t)∇2S bysupposing
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 (cid:126)v we find that the right part of this equation
S
vanishes since ((cid:126)v ·(cid:126)v ) = ((cid:126)v ·ω(cid:126)) = 0. On the other part (cid:126)v ·∇ = d/dl represents a derivative along
S R S S
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 (cid:34)(cid:32)∇ρ(cid:33)2 ∇2ρ(cid:35)
S + (∇S)2 + v2 +U((cid:126)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(cid:126) ·∇)(cid:126)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.
and separating on real and imaginary parts we
Both the continuity equation
come to Eqs. (31) and (32). So, the Navier-
∂ρ Stokes equation (6) with the slightly expanded
+((cid:126)v ·∇)ρ = 0, (32)
∂t the pressure gradient term can be reduced to the
Schro¨dinger equation 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 Shro¨dinger wave equation can be resolved
from the following Schr¨odinger equation
by heuristic writting of a solution through us-
ing the Huygens’ principle in its mathematical
∂Ψ 1
ih¯ = (−ih¯∇+m(cid:126)v )2Ψ+U((cid:126)r)Ψ−CΨ. integral presentation
∂t 2m R
(33)
(cid:90)
The kinetic momentum operator (−ih¯∇+m(cid:126)v ) Ψ(x,t ) = G(x,y)Ψ(y,t )dy, t > t . (35)
2 1 2 1
R
contains the term m(cid:126)v describing a contribution
R
of the vortex motion. This term is analogous to The function G(x,y) is called a propagator. It
