An Analogy Between the Properties of Light and Properties of Vortex-Wave Process in the Medium Similar to Superfluid <SUP>3</SUP>Не-В

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Since in quantum mechanics any measurement is a reduction of wave function, two different measurements of photon a may result in two different states of photon b, that is, the properties of photon b will depend on the interaction of photon a with the detector.

According to experimental data, quantum correlations take place:

1. Independent of the distance between the detectors; this was established over distances of 10 km ^[5].

2. Not only between specially prepared photons obtained from the same light source, but also between photon beams of the same frequency, emitted by different sources. This is supported by the fact that overlapping photon beams of the same frequency emitted by different lasers can interfere with each other and thus be coherent ^[1].

3. In the case of different times of traveling of the photons from their sources to the detectors, that is, when one of the photons has reached its detector and the other one is still en route ^[6].

Note: this property does not allow the interpretation to be applied to quantum correlations. The interpretation rejects the possibility of endowing quantum entities with a priory properties; the following statement is ascribed ^[1] to Niels Bohr: “A quantum phenomenon may be considered a phenomenon if only it is a measurement event.” However, with different optical paths of photons, in the interpretation a quantum correlation takes place between the detected photon and something which has not yet been detected, i.e. something nonexistent.

4. The features of the process determining quantum correlations can be seen using the example of two-photon interference ^{[1, 7]}. Let us consider the interference of two beams of light. Figure 2 shows a diagram of experimental setup, where both beams undergo phase delays. Fields a₁ and a₂ of frequency and also fields b₁ and b₂ of frequency are mixed by respective beam splitters.

Figure 2. Diagram of a four-mode intensity interferometer:

are input beams of light;

and

are phases of delay of light beams. The detectors and coincidence circuit measure the correlation of intensities

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The detectors and coincidence circuit measure the correlation of intensities. The observed correlation and anticorrelation of intensities of output fields a and b depend periodically on phase delays and . Depending on the statistics of incident beams of light, two types of interference may take place: one with phase , the other with phase . The first interference is referred to as classical interference (the Hanbury Brown and Twiss intensity interference ^[7]), the second one is quantum, the so called two-photon interference. Under definite conditions both types of interference are a result of conversion by beam splitters of fluctuations of phases of input fields into fluctuations of intensities of output fields. What are the conditions? Let fields and (k = 1, 2) have constant unit amplitudes and respective phases and which drift with time t: , . The condition of the Hanbury Brown and Twiss interference of intensities is determined by equalities: and . The condition for two-photon interference is as follows: , that is, for the changes in the phase of field a, , and field b, , the following is valid:

(1)

These two conditions can be respectively called the condition of correlation and the condition of anticorrelation of phases.

5. Theoretically, the results of experiments on quantum correlations can be interpreted without ignoring the cause and effect relationships by introducing unknown “superluminal” forces between the correlated photons.

It is shown in this paper that if the physical vacuum had the properties of superfluid ³He-B (hereafter referred to as the superfluid physical vacuum – SPV), a vortex-wave process whose properties are similar to the properties of photon given above could propagate in such a vacuum.

The validity of endowing the physical vacuum with the properties of superfluid ³НеВ is substantiated in a number of works. For example the superfluid properties of physical vacuum explain the dissipation free motion of celestial bodies, such as the planets of the solar system ^[8]. In Refs. [9-11]^[9] an analogy is shown between some properties of superfluid ³Не-В and gravitational properties of space, and in Refs. [12-17]^[12] an analogy is shown also between some properties of superfluid ³Не-В and the effects of biologically active substances in ultra-low doses and low-intensity physical factors (including the cavity structure effect) on biological systems.

2. Some Properties of ³He-B

(For properties 1–11 the references used are ^{[18, 19]}, for properties 12–13 the references are ^{[20, 21, 22]}.)

1. In the unperturbed state superfluid ³He-B is a homogeneous and isotropic medium.

2. In superfluid ³He-B, as in any other superfluid, there is no shear viscosity.

3. Quantized vortex lines are created in superfluid ³He-B under rotation, i.e. vortex lines in which the angular momentum of paired ³He atoms equals , where is Planck’s constant, n = 1, 2…

4. The quantized vortices produced in ³He-B do not diffuse in the stationary state.

5. The magnetization of the vortex core along the vortex axis takes place in superfluid ³He-B, i.e. spin polarization of superfluid arises in the vortex core. This means that the Barnett effect takes place in the superfluid: the angular momentum of paired ³He atoms that constitute the vortex can be transferred to spins of the atoms, which process is more pronounced in the core. The existence of the Barnett effect in superfluid ³He-B agrees also with the experimental findings that the vortices having oppositely oriented spins of the constituent atoms are characterized by mutually opposite velocity circulation directions (in other words, the spin-up condensate circles in the direction opposite to that of the spin-down condensate).

6. In superfluid ³He-B the vortices which terminate in the bulk superfluid due to the complete transfer of the angular momenta of paired ³He atoms, which constitute such vortices, to the orbital angular momenta and spins of the atoms may be produced. Examples of such vortices in superfluid ³He-B are vortices of the “hedgehog” type.

7. In vortex cores of superfluid ³He-B, the precession of spins of ³He atoms takes place. This follows from the experiments on magnetic resonance conducted with rotating superfluid ³He-B.

8. The Einstein–de Haas effect (or the effect) takes place: in the case of ³He-B this is a rotation of a volume of superfluid ³He-B caused by magnetization of the superfluid. Since the magnetization of ³Не atoms means as well their spin polarization, then the Einstein-de Haas effect is in fact the rotation of a superfluid volume provided that , where is the total spin of the volume under consideration.

9. The vortex core in superfluid ³He-B is an electric dipole whose electric dipole moment may be aligned with the vortex axis.

10. In the vortex core of superfluid ³He-B, due to unidirectional orientation of spins of ³He atoms the phase transition occurs (the superfluid A phase or spontaneously ferromagnetic -phase may be produced). The phase transition leads to a change in the inertial properties of the superfluid in the vortex core in contrast with the inertial properties of the superfluid in the rest of the volume.

11. The mutually relative motion of particles constituting a Cooper pair in superfluid ³He-B corresponds to the р-wave state. In this state, attractive forces act between like-charged particles with spins aligned in the same direction.

12. In superfluid ³He-B, there may exist regions with identically oriented and coherently precessing spins of ³He atoms, such regions being referred to as homogeneously precessing domains.

13. In superfluid ³He-B the nutation and precession angles of precessing spins of atoms are angles of orientation of the order parameter, and there are processes that tend to equalize both the values of precession angles and the values of nutation angles throughout the liquid volume. Such processes in superfluid ³НеВ are spin supercurrents. Let us consider two homogeneously precessing domains (HPD₁ and HPD₂) having respective precession frequencies and , precession angles and , nutation angles and and the same values of spins S. Frequencies and are aligned with axis z (see Figure 3).

Figure 3. Spin structures

and

are precession frequencies,

and

are precession angles,

and

are nutation angles, S is spin; r.l. is the reference line,

is spin supercurrent

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Figure 4. Normalized spin supercurrent

as a function of precession phase difference

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At the experimentally obtained dependency of spin supercurrent between the two of homogeneously precessing domains on the difference is shown in Figure 4.

Within a definite accuracy the experimentally obtained dependency can be approximated by the linear function:

(2)

where k is a proportionality factor. Theoretically, the spin supercurrent is dissipation free and does not depend on the distance.

3. The Vortex-Wave Process in the Superfluid Physical Vacuum

3.1. The Equations of Vortex-Wave Process

As the properties of the SPV are like those of superfluid ³He-B, then according to properties 1–5 and 8 (Section 2) the dynamics of the SPV in the nonstationary case in a vorticity region can be described by a model of an ideal spin liquid, i.e. the liquid with intrinsic degrees of freedom and without shear (linear) viscosity. The intrinsic degrees of freedom are characterized by both vector A of total spin of infinitesimal volume element of the liquid and the rotational viscosity determining transformation of macrorotations in the medium into microrotations.

The equation for the motion of liquid, if we neglect the momentum flux through the boundary, will be ^[23]:

(3)

where ρ is the liquid density, u is the velocity of the moving liquid, is the basis vector, and are components of the stress tensor in this basis vector (i, j = 1, 2, 3,…). In the Cartesian coordinate system

(4)

Using the formalism developed in mechanics for writing equations of motion of continuum with intrinsic degrees of freedom, we shall express the stress tensor components as follows ^[23]:

(5)

where are the components of the antisymmetric tensor of intrinsic rotations. In the SPV model the tensor characterizes the total spin of infinitesimal volume element of the liquid. We define as:

(6)

The coefficients are defined as follows:

can be expressed through Kronecker’s delta δ, then equation (6) will take the form:

(7)

The coefficients depend on the properties of the liquid; thus they must form a rank IV tensor, and they must not vary with any orthogonal transformation of coordinates. As it was mentioned above, the medium modeling the properties of the SPV is characterized by intrinsic degrees of freedom and rotational viscosity, which manifests itself in nonstationary cases; there being no shear viscosity there. For such a medium, in the linear case, the tensor in question, provided the space is homogeneous and isotropic (property 1, Section 2), has the following components in the Cartesian coordinate system ^{[17, 24]}:

(8)

where , , are constants. Substituting the right-hand members of equations (7) and (8) in equation (5), we obtain the following expression for the stress tensor components:

(9)

Taking into account (6) in (9) we have:

(10)

Further we shall rewrite equation (3), using (4) and (10):

(11)

According to property 8 (Section 2), the Einstein – de Haas effect takes place in the SPV and the following can be written:

(12)

where is the proportionality coefficient

(13)

Let us introduce the constant :

(14)

According to (14) and taking into account (13), the necessary condition for the quantity to have a physical meaning is the following: .

Using equations (14) in (11) and (12) we obtain:

(15)

(16)

These equations describe a spin-wave process propagating at a speed in the SPV. According to property 5 (Section 2) spin arises in vortices, thus the process described by equations (15) and (16) is a vortex-wave process.

In the conventional, i.e. not superfluid, medium the intrinsic rotations cannot propagate as a wave process because the diffusion of the vortices is significant. However, under property 4 (Section 2) quantized vortices do not diffuse in the SPV. Hence if in the SPV there is a mechanism that suppresses a vortex in a point of space and simultaneously transports the vortex energy to adjacent points, then the vortex process will propagate through the SPV.

Let us denote:

(17)

Let us consider the physical sense of variable Y, which according to (13), (14) and (17) is directed oppositely to A, is proportional to the magnitude A and has the dimension of velocity. Let us address the property 11 of superfluid ³He-B (Section 2). Under this property, in a vortex core produced in the SPV the motion of electrically unlike particles occurs in opposite directions, causing the electric polarization of the vortex, thus the vortex can be regarded as an electric dipole (property 9, Section 2). Consequently, the variable Y may denote the speed of this motion relative to the SPV:

Using (17) in (15) and (16), we obtain:

(18)

(19)

According to (18) and (19), in the vortex-wave process discussed there is a transfer of energy associated with the motion of the SPV outside the vortex (velocity u) to the energy associated with the motion of the SPV in the vortex core (velocity Y).

3.2. The Components of the Energy of the Vortex-Wave Process in the SPV
3.2.1. The specific kinetic energy of the SPV, associated with the speed Y

The specific kinetic energy of the SPV, , is associated with the speed Y of motion of electrically charged particles that constitute the vortex core: .( and are the density of positively and negatively charged particles in the medium respectively). Taking into account that , we obtain:

(20)

The mutually relative motion of unlike-charged particles in the vortex core results in electric polarization of the SPV in the vortex core. The specific energy of the produced electric field E ^[25] is determined as:

(21)

Assuming that all the energy is converted into the energy of the produced electric field E, assuming that and using Eqs. (20) and (21) we have:

(22)

Using (22) in (18) and (19) we obtain:

(23)

(24)

Equations (23) and (24) describe the process of propagation of the electric polarization of vortex in the vortex-wave process in the SPV at speed . It is noteworthy that in equation (23) characterizes not a vortex motion in an electric field, but a circulation of E, which circulation is a result of electric polarization of the SPV in the vortex cores.

3.2.2. The specific kinetic energy of the SPV, associated with the velocity u

The specific kinetic energy of the SPV, , associated with the speed of motion of the SPV, u, is:

(25)

Analyzing Eqs. (23) and (24), we shall introduce the following denotation:

(26)

Using (25) in (26), we have:

(27)

Using (26) in (23) and (24), we obtain:

(28)

(29)

Let us discuss the character of dependence of pressure p in the SPV on speed u. Hydrodynamically, the stationary motion of a superfluid is described ^[26] by the following equation: , where is the chemical potential, is the density of the superfluid. The potential may not reveal itself inside the vortex because of a phase transition inside the vortex core (property 10, Section 2). In this connection, the action of the superfluid on the vortex core can be described mathematically by introducing pressure p on the boundary of the vortex core in the SPV ^[24]. The sign of the pressure depends on the nature of internal stresses in the medium. If the internal stresses are like “omniradial tensions” ^[23], the pressure will be negative. Since a “break” of particles constituting the vortices into electrically unlike parts takes place in the vortices of the SPV, it is possible to assume that in the vorticity region of the SPV the internal stresses are like “omniradial tensions”, thus it may be assumed:

(30)

That is, the pressure in the vorticity region may be negative.

In favor of validity of Eq. (30) there are results of experiments with rotating superfluid ³He-B: the so called vortex molecules consisting of two vortices with angular momenta oriented in parallel ^[19] have been detected. This is characteristic of the medium with negative pressure.

3.2.3. The energy associated with the precessing vortex spin

According to property 7 (Section 2), the precession of spins of particles that constitute the vortex cores of the SPV takes place. The frequency of precession, , determines the energy associated with precessing spins of microparticles, ^[23]:

(31)

where is the total value of precessing spins.

Since, under property 5 (Section 2), in the SPV there is a transfer of angular momentum of the pairs of particles that constitute the vortex to the spins of the particles, then, according to property 3 (Section 2), the total spin of the particles in the vortex may be equal to nћ, n = 1, 2, 3,… If n = 1, we have:

(32)

Then equation (31) will be expressed as:

(33)

3.2.4. The energy associated with the inertial properties of vortex

Properties 5, 7, 10 and 11 from Section 2 (the spin orientation, spin precession and nonzero forces between the particles that constitute the vortex, the forces being aligned with the spin of the particles) allow one to introduce the mass of vortex, and also the motion of the mass inside the vortex (Figure 5).

Figure 5. Diagram of precession motion of mass

inside a vortex.

is the vortex precession frequency,

is the total vortex spin;

is the radius of the circular motion of mass

inside the vortex

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Thus the energy of the vortex as an entity with mass can be represented by two terms: the kinetic energy of the center of mass, in which all the vortex mass is assumed to be contained, and the energy associated with the precession of vortex spin. The energy is defined as ^[23]: , where is the moment of inertia associated with mass . The quantity determines the angular momentum of mass and it must be equal to the angular momentum of vortex, , or according to (23):

(34)

Thus the total energy of mass moving relative to the SPV is determined as:

(35)

Being a characteristic of the same entity, i.e. the vortex, the energy associated with mass must be equal to the energy of the vortex, contained in its spin system, ; being the sum of energy (see Eq. (31)) and energy associated with the motion of spin in space at speed . Denote the latter energy as , thus:

(36)

If the motion of spin in space is inertia free, kinetic energy cannot be associated with the motion, that is:

(37)

Then, using Eqs. (35) and (36), we obtain the following expression for the vortex mass:

(38)

Because of creation of mass in vortices, the wave-vortex process in the physical vacuum having the properties of superfluid ³Не-В is at the same time a process in a gravitational field. Therefore, the speed of wave-vortex process will be determined by the properties of gravitational field and will tend to become equal to a definite value in the vicinity of the sources of the field, relative to the sources (i.e. the entities possessing a mass).

Note. In the simplest case of circular motion (with radius , see Figure 5) of mass inside the vortex the quantity is determined as . Using this equation in Eqs. (34) and (38), we obtain the following expression:

(39)

That is, the length of the circle where the mass moves is equal to the wavelength of the vortex-wave process that is described by equations (15) and (16) (or (18)-(19), (23)-(24), (28)-(29)) and propagates at speed .

4. The Interaction of Vortices in the SPV

According to properties 5, 7 and 12 of superfluid ³He-B, the vortices in this fluid can be considered to be homogeneously precessing domains; thus according to property 13 (Section 2) there may exist a process between the vortices that makes equal the respective angles of precession and angles of nutation of spins of particles that constitute the vortices. This process is spin supercurrent. Let us discuss the features of variation of characteristics of vortices using as an example the interaction of and shown in Figure 3. The precession angles and are related to the precession frequencies and (in the special case of independence of precession frequencies on time t), by equations:

(40)

(41)

where and are respectively the values of the precession angles for and at . If spin supercurrent causes changes in precession angles and (these changes are denoted as and ), then according to Eqs. (40) and (41) this results in changes in frequencies and (the changes are denoted as and ). Under Eq. (31), a change in the energy, , and a change in the energy, , are determined as:

(42)

(43)

Since energy conservation in superfluid ³Не-В is valid and the spin supercurrent is dissipation free, the equalization of respective characteristics of and results in:

(44)

Then according to Eqs. (42) and (43):

(45)

According to Eqs. (40), (41) and (45):

(45)

The spin supercurrent in superfluid ³He-B does not depend on the distance between homogeneously precessing domains. Interaction between homogeneously precessing domains is most effective, that is, the respective precession and nutation angles of the domains tend to become equal, provided the difference between their precession frequencies, , satisfies the following condition:

(46)

5. Comparison of Properties of Photons and Properties of Vortex-Wave Process in the SPV

Let us compare the properties of photon discussed in Introduction and the properties of vortex-wave process discussed in Sections 3 and 4, which propagates in the SPV.

С-photon.

1. The representation of photon by a classical wave packet described by Maxwell’s equations is in agreement with the conclusions of Section 3: a vortex-wave process which is described by the equations of Maxwell’s type may propagate in the SPV (see Eqs. 15-16, 18-19, 23-24, 28-29).

2. The electric properties of photon are similar to those of vortices in the vortex-wave process in the SPV (see Eqs. 23 and 24).

3. The preservation of the plane of electric component of photon is similar to the preservation of the plane of electric component of vortex in the vortex-wave process in the SPV. The vortices of the vortex-wave process are characterized by the precession of spins of the particles that constitute the vortex; this precession determines the gyroscopic properties of the vortex-wave process.

4. The equalization of the light speed in any inertial frame of reference to the absolute constant c is similar to the analogous property of vortex-wave process in the SPV: due to creation of mass in the vortex, the vortex-wave process is also a process in a gravitational field. Therefore, the speed of the vortex-wave process is determined by the properties of gravitational field and may tend to become equal to a definite value in the vicinity of the sources of the field, relative to the sources (i.e. the entities possessing a mass).

5. The photon has a magnetic component.

Equations (28) and (29) describing the vortex-wave process imply that variable X may be a candidate for the role of magnetic component. According to Eqs. (25)-(27), variable X is associated with the velocity of motion of the SPV in the vorticity region. It is shown in ^[23] that there is a complete analogy between the structures of formulas describing the magnetic interactions of current-carrying wires and the structures of formulas describing the interaction of vortices in an ideal incompressible liquid having negative pressure. (Note that according to (30) the SPV in the vorticity region is such a medium.) In addition, the field’s magnetic induction B is related to the velocity u of the liquid of density by equation that is similar to equation (26) which relates variable Х to the velocity of the SPV, .

M-photon. The characteristics of M-photon’s mass (“kinetic” mass) , defined as , angular nomentum , energy are similar, according to (38), (32) and (33), to the respective characteristics of vortices in the vortex-wave process: ( is the precession frequency of vortex spin, is the speed of the vortex-wave process).

Note. The energy of M-photon is not equal to the kinetic energy of photon’s mass, , which suggests the existence of intrinsic motions of the mass in a photon. In ^[21] the expression for the energy of photon’s mass, , was derived:

(47)

where is the speed of photon with respect to the reference frame. It is shown in Appendix that on the basis of Eq. (47) and also the Galilean addition of velocities it is possible to derive equations for the description of the transverse and longitudinal Doppler effects, which coincide to a high accuracy with similar equations derived in special relativity ^[27].

The expression (47) is similar to expression (35) that determines the energy associated with inertial properties of a vortex in the vortex-wave process, the SPV.

Q-photon. The properties of quantum correlations discussed in Introduction are similar to the properties (discussed in Section 4) of interaction of vortices in superfluid ³He-B through spin supercurrents.

1. The independence of quantum correlations of the distance between the correlated photons is similar to the analogous property of spin supercurrent in superfluid ³He-B: the independence of the distance between the spin structures, between which it arises.

2. That quantum correlations between photons of the same frequency, emitted by different sources, may take place does not contradict the condition of effective interaction of homogeneously precessing domains in the SPV. According to Eq. (46), the efficacy of spin supercurrents is maximum if the interacting homogeneously precessing domains have equal precession frequencies.

3. That quantum correlations of photons may take place not only in the process of simultaneous detection of both photons is in agreement with the property of interaction of homogeneously precessing domains through spin supercurrents in the SPV. Spin supercurrents arise between such domains under the only condition that there are nonzero gradients of nutation or/and precession angles between them.

4. The relation between the changes in phases of the light waves in the two-photon interference (Eq. (1)), which is a manifestation of quantum nonlocality, is similar, according to Eqs. (43)-(45), to the features of variation of energy, frequency and precession angle in interaction of vortices in the SPV through spin supercurrents.

5. The quantum correlations may be due to “superluminal” forces between the correlated photons. This property of quantum correlations is similar to the property of spin supercurrent arising between homogeneously precessing domains in the SPV. Indeed, the spin supercurrent can be considered a transfer of spin projection gradient onto a preferential direction and, according to Eqs. (36) and (37), can be considered an inertia free process. Therefore, spin supercurrent is not accompanied by the birth of a mass and consequently is not a process in the gravitational field. Thus the speed of spin supercurrent in the SPV may have any value with respect to singularities in the gravitational field, and therefore may be greater than the speed of vortex-wave process in the SPV, which is accompanied by the birth of a mass in vortices (see Eq. (38)) ^{[28, 29]}.

6. Conclusion

Endowing the physical vacuum with the properties of superfluid ³Не-В allows one to represent the main properties of light, i.e. electric, magnetic, the spin, “kinetic mass”, quantum correlation of the photons, the constancy of the speed of light in any inertial frame of reference as properties of a vortex-wave process in the medium similar to superfluid ³Не-В.

7. Appendix. The Doppler Effect for Light

Consider an inertial frame of reference linked to the detector, where the source of light of mass M is moving at velocity . The source of light is assumed to be at rest with respect to the Earth, the speed of light relative to the source is equal to c. We shall derive the formulas describing the longitudinal and transverse Doppler effects on the basis of the Galilean addition of velocities.

It is experimentally established (for example, in the photoelectric effect) that the absorption of light occurs in quanta of energy , where is the frequency of the light detected. If the mass of the detector as well as the mass of the source are great, the motion of the detector due to the pressure of light is negligible and thus the energy loss for the photon may be neglected too. Then in the interaction of the photon and the detector all the energy of the photon in the inertial frame is equal to the detected energy . Let us apply Eq. (47) to the photon energy. Provided we have:

(A.1)

where is the photon frequency in the frame of the source of photons. Let us introduce the vector w directed from the source to the detector (Figure 6):

(A.2)

Figure 6. Ray path diagram: с is the light velocity relative to the source,

is the source velocity relative to the detector, w is the resulting light velocity directed from the source to detector,

is the angle between w and

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