We live in an amazing Universe. Its relict radiation or the cosmic microwave background – the radiation filling everywhere the universe with a temperature of about 3 K, poses, in fact, the blackbody radiation [1]. The first question that arises in this regard concerns the essence of space-time containing this blackbody.

What is space, time, and matter motion? This question has accompanied humanity since antiquity [2]. The contrary views about space and time have developed in the form of two dialectical opposite ideas, later known as the Democritus-Newton and conceptions [3,4]. According to Democritus everything is made up of tiny, invisible, indivisible corpuscles, named "atoms". Atoms diffuse in space and can combine by forming complex visible objects. Between atoms there exists an empty space. On the other hand, Aristotle, student of Plato’s Academy in Athens, believed that the Universe is presented in two parts, the terrestrial part and celestial. The terrestrial part comprises combinations of four substances - fire, air, water, and earth (plasma, gas, fluid, solid, in the modern view). Whereas the celestial part (Sun, planets, stars) consists of ether, a special kind of matter that is not found on Earth. According to Aristotle, emptiness does not exist.

These two opposite ideas on the organization of our world, conditionally speaking the corpuscular world and wavy world, compete for centuries.

The Aristotle celestial element has none of the qualities of the four terrestrial classical elements – fire, air, water, and earth. This element, named by the fifth element and also called ether, possesses radically other qualities than the terrestrial elements. The ether attracted the attention of naturalists as a medium capable of transferring light perturbations.

Isaac Newton and Christian Huygens put forward theories in the seventeenth century for the spreading of light through space.

Huygens believed that the space is everywhere densely filled with the ether [5], through which light may spread from point to point due to a secondary re-emission of these elements of ether. According to Huygens’ ideas, light is a stream of waves distributed in an unknown hypothetical medium called ether, which fills the space everywhere tightly [6,7]. This ether penetrates into objects – air, glass, water, and, of course, it fills the entire vast space between the stars, planets and other celestial bodies. Huygens’ principle states [5], as a light wave spreads ahead, points along its wavefront become sources of new waves emitted in all directions. As a result, their mutual interference forms the light wave spreading further.

In turn, Newton adhered to the so-called corpuscular theory of light [8], according to which light is a stream of particles (or in Latin "corpuscles") coming from a light source in all directions. In contrast to Huygens’ light theory this picture of the light corpuscular spreading is clearly explained by three Newtonian laws.

Is the light a flow of corpuscles or is it a wave? Both theories based on the limited understanding of light could not propose accurate experiments for supporting their views.

The understanding of the mechanisms of light propagation through space started with discovering James Maxwell of equations of the electromagnetic field [9] entitled by his name. The question arises here how electric and magnetic fields are transmitted over large distances through the empty space. It is natural to assume that a medium supporting the transmission of the electromagnetic fields at distances, the ether. However, Maxwell’s attempt to build a model of such a medium which would be like liquid, gas, or solid crystal, was not successful. Perhaps the failure was that this model had been built in accordance with the hallmarks of classical mechanics. The classical ether which could decelerate or accelerate light depending on oncoming or accompanying the earth’s rotation (ether dragging) was rejected after the experiment of Michelson-Morley [10] which was specifically performed.

Along with that, investigations were rapidly unfolding at the frontier of the 20th century in understanding the processes taking place on a small scale. The milestone for these events was laid by an ordinary work of Max Planck in the attempt to combine the laws of black body radiation with a single formula [11]. Two known laws were based on the classical principles of thermodynamics. Each of them gives a good description of radiation on the opposite regions of the spectrum. In the long-wavelength region of the spectrum the Rayleigh – Jeans law works. While the second law of Wine radiation gives good results In the short-wavelength region of the spectrum. The attempt to combine these laws led Planck to a somewhat unexpected result – it was necessary to introduce an extra constant to ensure a smooth connection of the aforementioned classical radiation laws. This constant obtained at the tip of the pen, the quantum of action, reads This result says that oscillators comprising the black body could not absorb or emit the radiant energy with any frequencies from the continuous spectrum but only by discrete portions. And re-emit it by discrete portions as well. On December 14, 1900, Planck spoke at a meeting of the German Physics Community, where he presented the formula for the radiation of a blackbody.

The quantum hypothesis of the German scientist, the deep meaning of which was revealed only much later, marked the birth of quantum physics. For a thorough understanding of the new physics, it was necessary to convene the Solvay Congresses - a series of international conferences to discuss fundamental problems of physics and chemistry, held since 1911 in Brussels by the International Solvay Institute of Physics and Chemistry.

At the first Solvay Congress (1911) Planck outlined the arguments that led him to the discovery of the quantum of action. As soon as Max Planck in 1900 suggested that light is emitted only in quantized portions, then Albert Einstein explained in 1905 the photoelectric effect based on the assumption that light with a certain wavelength is emitted and absorbed exclusively in certain portions (photons) with the energy $h\nu$, where $\nu$ is the wave frequency. Thus, it turned out that light exhibits not only waves, but also corpuscular properties.

French scientist Louis de Broglie (1892-1987), developing ideas about the dual wave-particle nature of light, put forward in 1923 a hypothesis about the universality of wave-particle dualism [12]. At the fourth Solvay Congress (1924) Louis de Broglie successfully compared the propagation of waves to the motion of a particle. He argued that not only massless photons, but also electrons and any other massive particles of matter have wave properties, along with corpuscular ones. According to de Broglie, corpuscular characteristics – energy E and momentum $\overrightarrow{p}$ – are associated with each particle, on the one hand, and on the other hand, wave characteristics – frequency $\nu$ and wavelength $\lambda$ This comparison was soon brilliantly confirmed in the experiments of Clinton Davisson and Lester Germer [13] and George Thomson [14]. These experiments showed that electron beams can undergo diffraction when passed through the atomic crystals. From the wave-particle duality principle stated by de Broglie the particle interferometry follows immediately. Now interference patterns even from heavy molecular beams such as fullerene beams [15] can easily be computed [16,17] by applying the Feynman path integral technique [18].

The principle of particle-wave duality received a more concrete and correct embodiment in Schrödinger ’s "wave mechanics", which then turned into modern quantum mechanics. Schrödinger ’s equations gave solutions that were clearly verifiable by experiments. However, one unclear issue was open. The wave function is a complex-valued one. The physical meaning of phase waves remained largely unclear. Therefore, when de Broglie tried to build a theory of a double solution [19], following the principle of wave-particle duality, he attributed to a wave function an additional pilot wave – a "fictitious wave" moving with a particle. A goal of the latter was to accompany the particle along an optimal path. The particle was presented as a kind of singularity, accompanied by the pilot wave from creation of the particle up to its detection. The pilot wave in a sense "probings" the surrounding space and transmits the relevant information to the singularity, directing its movement" [20]. The velocity of a particle being driven by a wave with this approach is a hidden parameter that cannot be measured.

De Broglie failed to convince his colleagues of the validity of his ideas during the fifth Solvay Congress (October 1927), where he made a report on his preliminary theory of the pilot wave. The causal theory of the pilot wave met with a cool reception from the participants of the Solvay Congress. The pilot wave caused confusion by introducing an additional hidden entity into the quantum problem. It remains to be regretted that de Broglie did not know about Madelung’s article "Quantumtheorie in hydrodynamische form" published in German in the journal Zts. f. Phys., 1926 [21] immediately after Schrödinger’s landmark article [22]. Unfortunately, we must admit that Madelung’s article went unnoticed by many physicists of that time. Now interest in the Madelung equations has increased due to the fact that the flow of an ideal fluid gives a fine explanation of complex-valued solutions of the Schrödinger equation - wave mechanics corresponds to the Eulerian picture [23,24,25,26].

Only after the appearance of the works of David Bohm (1917-1992) in 1952 [27,28] de Broglie again rekindled his early ideas about the pilot wave accompanying a particle along an optimal trajectory. Bohm essentially repeated Madelung’s results, but went much further by considering the motion of not only one particle, but also their ensemble. Bohm managed also to go much further than de Broglie in substantiating his views, in particular, to build a theory of measurements. The pilot wave theory, which has since been often called the de Broglie–Bohm theory, provides a consistent description of the optimal trajectories along which the singularity (imitating the particle) moves. Further the theory constructed by Bohm was called the Bohmian mechanics [25,29,30,31,32,33], which contained a new attempt to construct a quantum theory with "hidden parameters".

In the early 1960s, de Broglie formulated the idea of hidden thermodynamics of isolated particles, according to which a random element is introduced into the motion of a single particle due to its interaction with a hidden "subquant medium". It is remarkable that such ideas were further developed by Nelson [34,35,36,37] and Grössing [38,39,40], each with their own peculiar approach. A more detailed historical and philosophical analysis of the scientific activities of Louis de Broglie and David Bam on the deep issues of quantum mechanics (the double solution theory, the pilot-wave dynamics) was carried out by Aurélien Drezet in the article [41].

As for this article it is organized as follows:

Let us begin from the Schrödinger equation describing motion of a particle with mass m in 3D space under the external potential $U(\overrightarrow{r},t)$: Hereinafter we use the Dirac’s writing of the wave function as $\left|\psi \right(\overrightarrow{r},t)\rangle$. Here $\hslash =h/2\pi$ is the reduced Planck constant, which has units of action (energy multiplied by time). We refrain from considering the problem in 3N-D space ($N>1$), so as not to load subsequent calculations of the quantum potential and the variance of Bohmian trajectories.

Since this equation contains the imaginary unit, i, as a multiplier at the partial differentiation by time t, then the wave function $\varphi (\overrightarrow{r},t)$ is obviously a complex-valued function. We write the wave function in the polar form: where $\rho (\overrightarrow{r},t)$ is the density distribution of the carriers N (particles with the mass m) in the unit volume $\Delta V$, and the scalar function $S(\overrightarrow{r},t)$ is the action. The latter has the dimension equal to energy multiplied by time. So, the ratio $S(\overrightarrow{r},t)/\hslash$ is a dimensionless phase. Substituting this function into Eq. (3) and dividing this equation into imaginary and real parts, we come, as a result, to two equations.

The first equation composed of imaginary variables is as follows: Here we define the current velocity proportional accurate to the divisor m, mass of the particle, to the gradient of the scalar function $S(\overrightarrow{r},t)$ The rightmost presentation is taken for completeness from Dürr et al. [42,43]. Substituting this result to Eq. (5), we come to the customary version of the continuity equation: Here ${\rho }_M(\overrightarrow{r},t)=m\rho (\overrightarrow{r},t)=mN/\Delta V=M/\Delta V$ is the mass density within the unit volume $\Delta V$. The continuity equation describes the conservation of mass M, or the amount of carriers (particles), N, of the elementary mass m, as the unit volume $\Delta V$ deforms with the time going on.

The second equation, represented by real variables, is the Hamilton-Jacobi equation, with an added extra term - the quantum potential $Q(\overrightarrow{r},t)$: and the quantum potential reads: where $R=\sqrt{\rho }$ is the amplitude density. Further we will show in subsec. Section 3.2 as the quantum potential follows from the Fick’s laws acting in the medium containing the normal fluid. In Eq. (8) an arbitrary integration constant $C_0$ having the dimension equal to energy stems from the fact that the wave function of the Schrödinger equation (3) is defined accurate to an arbitrary phase $\theta$: $\psi (\overrightarrow{r},t)\leftrightarrow \psi (\overrightarrow{r},t)·exp\left\{i\theta \right\}$. In fact, this constant characterizes the energy level of the medium where the quantum system evolves. It is usually taken as zero - the zero-point energy level.

Two equations, the continuity equation (7) and the modified Hamilton-Jacobi equation (8), returning real solutions replace fully the Schrödinger equation (3) that deals with the complex-valued wave function. The wave function provides simply condensed information regarding both the velocity field of particles and their possible location.

Let us now apply to Eq. (8) the operator ∇. First we note that the gradient of the scalar function $S(\overrightarrow{r},t)$ divided by the mass m returns the current velocity $\overrightarrow{\mathit{v}}(\overrightarrow{r},t)$, as is shown Eq. (6). In that case, Eq. (8) transforms to the following form: Since this equation has no viscous friction term, it describes the flow of an ideal fluid. In hydrodynamics such a fluid is an imaginary (idealized) liquid in which, unlike a real liquid, there is no viscosity [44].

The main highlight in the task with bouncing droplets on the surface of oil in the bath is a constant shaking of the bath in the vertical direction. The shaking occurs with such frequency and intensity that a homogeneous state of the oil surface without any waves on this surface is supported. But this regime of the shaking is in the very immediate vicinity of the beginning of these waves [81,82,84] – Faraday waves. In fact, the point of separation of these states, where there are no waves and where they already exist, is called the subcritical bifurcation point [87]. As for the shaking of the surface of a fluid the subcritical bifurcation associates with the Faraday threshold [78,88,89]. When dropping the droplet on the oil surface it generates near the Faraday threshold the appearance of symmetrical lateral waves. The latter by spreading over the whole oil bath and induce a pattern of the classical Faraday instability. This interaction of the droplet with the oil surface Protiére, Boudaoud, and Couder [78] represent by the following equation: This is Newton’s first law applied to a drop with mass $m\sim 1\mathrm{mg}$ moving in the direction of the x-axis. As for the right side of the equation, here we note only the parameters that are important for the subsequent presentation. A more detailed description of them is given in the mentioned work [78]. Here $V_W=dx/dt$ is the velocity of the drop moving along the x-axis and $V_F^{\varphi }$ is the phase velocity of the Faraday wave induced by the previous collision. The force $F^b\sim {10}^{-6}\mathrm{N}$ is proportional to the amplitude of the vertical acceleration ${\gamma }_m$ and to the slope of the generated surface waves. The term $f^{\nu }\sim {10}^{-6}\mathrm{N}·{\mathrm{m}}^{-1}\mathrm{s}$ is the effective damping based on the viscosity of air.

The steady regime is when the acceleration $d^{2}x/d{t}^2$ is equal to zero, which means to vanish Eq. (42). For computing the steady regime we expand the sine into a power series up to the second degree. As a result, we come to two equations. The first equation gives $V_W=0$ for all values of the control parameter $F^b$. The second equation is quadratic and returns two solutions [78]: Here is the subcritical bifurcation threshold. When the control parameter $F^b$ reaches this threshold, the walker velocity $V_W$ as a function of $F^b$ undergoes a jump of the first derivative by the parameter $F^b$, as shown by the black curve in Figure 15. Here only the positive branch is shown.

Although the authors [78] considered the model of the drop movement in one-dimensional space, these conclusions are easily transferred to its motion on a two-dimensional surface. The oil in a bath subjected to vertical shaking can come from the homogeneous state of rest to the state when the shaky waves begin to cover its surface. It occurs when the critical shaking threshold is reached. These waves named the Faraday waves cover the whole surface reflecting from any obstacles placed in this bath. A complex wave pattern induced over the whole bath surface bears information about all obstacles placed in the bath.

The Faraday wave pattern formation on 2D surface after reaching the bifurcation point $F_C^b$ is the second order phase transition [90,91]. In this case we observe the symmetry breaking at the transition from the homogeneous steady state to the emergence of the high symmetric Faraday wave pattern. The order parameter here can be the amplitude of the Faraday waves.

But a more informative parameter at the second order phase transition is the zero-field susceptibility $\chi$. In our case it comes the differentiation of the drop velocity $V_W$, i.e., Eq. (43) by the control parameter $F^b$ from: In Figure 15 this function is drawn by red. One can see that this function goes to infinity as the control parameter $F^b$ tends to $F_C^b$. It means that the susceptibility of the oil to any extraneous fluctuations, for example, touching the surface of the oil by a bouncing drop, induces long-lived oscillations on the oil surface.

Theoretically, when the control parameter reaches the critical phase transition point then at this point the fluctuations become infinite in lifetime and infinitely long-correlated. Such fluctuations store information about all obstacles immersed in this fluid for a long time. Near the particle the interference fringes from these fluctuations create a pattern mapping the carried information. As for the de Broglie hidden medium [19] (the ether), here the singular particle traveling through this medium experiences each time the influence of it, due to which the particle moves along some optimal trajectory. Following the above remarks one can guess that the hidden medium is at zero-point energy level, and it possesses high susceptibility in relation to external excitations. The latter may come from different devices embedded in this environment.

Further we try to understand in detail what can represent such a hidden fluid medium.

Let us glue the opposite ends of the vortex tube having the radius a, see Figure 3. As a result, we get a torus shown in Figure 17(a). In fact, here the topological transformation of an infinitely long vortex tube was done. As a result of this transformation we get a torus.

The torus shown in Figure 17(a) is calculated at the parameters $a=1$ and $b=2$. On the torus surface a one-dimension string $S_1$ twice covering the torus tube is shown. The parameters defining the torus (18) are as follows ${\omega }_0=1$, ${\omega }_1=1/2$, and ${\varphi }_0={\varphi }_1=0$.

Note that the torus volume and its surface are computed by the formulas At a going to zero the volume and surface vanish. The torus degenerates to a ring of the radius $b=2$ laying in the plane $(x,y)$. When b goes to zero the volume and surface vanish as well. However, the torus in this case transforms to an object twice covering a 2D sphere of the radius $a=1$, Figure 17(b). The string $S_1$ maps to the string $S_1^{\prime }$ laying on the surface of the 3D sphere. The white arrows on these strings point to the orientation of currents flowing along them. On the second figure this orientation is seen to reach its initial state marked by the number 1 after two of its rotations through the bottom pole. It is a characteristic sign of the half-integer spin. The spin rotation on this 2D sphere is represented as a motion of the unit vector in the magnetic field [103,104]: The solution of the spin rotation in this magnetic field looks as These components submit to the condition This four-dimension unit vector describes a complex two-fold curve on the surface of this sphere along the string $S_1^{\prime }$ pointed by the white arrows in Figure 17(b). Here the parameters specify the position and the width of a resonance maximum, where are the Larmor precession frequency and the apex angle of the cone described by the vector $\overrightarrow{B}$, respectively [103]. The solution (77) points indirectly to the double hollow sphere endowed with a fluctuating string enveloping its surface, Figure 17(b) as a possible model of a particle.

The whole cycle of the torus transformation to the 2D sphere twice covering itself is shown in detail in Figure 18. Initially a ring torus having radii $a<b$ shown in Figure 18(a) transforms to the horn torus, Figure 18(b), when its radii become equal, $a=b$. Take note that the string $S_1$ tunnels through the ring, the radius of which is equal to zero, from up to down along a narrow bridge. The following stages are transformations through spindle tori when b becomes less than a, see Figure 18(c), (d), (e). In this case, the string $S_1$ spends part of the time on the outer spindle, and part on the inner. The final stage of the torus transformation is when b turns to zero. The spindle torus transforms to a double-covered sphere, Figure 18(f). In this case, the string spends half of the time on the outside of the 2D sphere, and the other half on the inside. These crossings from the outside to inside and back occur on the top and down poles of the 3D sphere.

Let us glance carefully on Figure 17(b) and Figure 18(f). Arrows near the top pole of the 2D sphere on these figures are marked by numbers 1, 2, 3, 4. They point both the inflow of current to the pole and the outflow from it. Let the arrow mark orientation of a flag attached to a flagpole that comes out from the origin, as shown in Figure 19. Now one can see, when the flagpole revolves from the point 1 to the down pole and further to the top pole, arrow 1 returns to arrow 2. The flag turned 180 degrees. Further the flagpole revolves along the path from point 3 to 4. On this path the flag rotates a 180-degree angle. As a result, the flagpole should twice revolve about the origin in order that the flag completes a full 360-degree rotation.

The above rotation on the 2D sphere is a typical situation of revolution of the half-integer spin 1/2. This spin involving the SU(2) group elements is represented in the following form [104]: Here ${\sigma }_0$ is the unit matrix and ${\sigma }_x$, ${\sigma }_y$. ${\sigma }_z$ are three Pauli matrices. The dimension of space where the spin 1/2 moves is four. Let us compute the square of the vector $\overrightarrow{u}$ at given the two-component spinor $|\psi \rangle \leftarrow {\psi }_{\uparrow },{\psi }_{\downarrow }$: The square of the axial vector $\overrightarrow{b}=(b_x,b_y,b_z)$ is 1. From here it follows that the tip of the spin lies on the 2D sphere of the unit radius, see Figure 17(b). Further we have the squares of sine and cosine is 1. This degree of freedom points that the axial vector $\overrightarrow{b}$ possesses by one more degree of freedom. It is a flag that rotates about this vector, as shown in Figure 19. That is, the rotation of this flag lies in the tangent space of the 2D sphere, Figure 17(b). So, one may take this 2D sphere twice covering itself the radius of which is proportional to $\hslash /2$ as a base for the elementary particle.

White arrows shown on the red string drawn on the surface of this sphere, Figure 17(b) points to motion of the currents flowing along this string. It is a solenoidal motion. This sphere with the solenoidal flows on its surface is immersed in the irrotational flow field. The latter play a role of the guidance wave of this sphere along its own flow. The section along the plane $(x,z)$ of the irrotational flows wrapping around the sphere is shown in Figure 20. The double covered sphere of radius $a=1$ is drawn in this figure in red. It may be the sphere shown, for example, in Figure 18(b).

The flows washing the sphere shown in blue follow from Eq, (19) calculated at the value $\theta =1$. While the flows shown in pink relate to $\theta =0$ in the same formula. In this figure is shown an irrotational filament drawn by alternating blue - pink dashed curve. It marks the boundary separating the flow washing the sphere and not. It is the partition boundary.

Figure 21 shows the flowing of the double covered sphere from all sides by the irrotational flows depicted by yellow and blue filaments. The yellow filaments drawn at $N=1$ lie to the partition boundary. The blue filaments washing the sphere are drawn at $N=32$. All they are drawn by Eq. (19) at ${\omega }_1=0$ and the phase ${\varphi }_1$ with the spacing equal to $\Delta {\varphi }_1=0.125\pi$. The other parameters are ${\omega }_0=1$, ${\varphi }_0=0$.

Note, if we accept in Eq. (19) non-zero ${\omega }_1$ then the outside flow acquires a vorticity about the z axis. Such spiral–shaped vortex filaments are shown in Figure 22 as yellow at $N=1$ and blue at $N\gg 1$ curves converging to the z axis. The phase parameter ${\varphi }_1$ is 0, $\pi /2$, $\pi$, and $3\pi /2$, respectively. While ${\omega }_1=6·{\omega }_0$ and ${\omega }_0=1$, ${\varphi }_0=0$. Such twisting of the irrotational filaments is possible in the presence of electromagnetic fields.