Quantum-Phase-Field: from De Broglie--Bohm Double Solution

Quantum-phase-ﬁeld: from de Broglie - Bohm double solution program to doublon networks

J. Kundin∗ and I. Steinbach† Ruhr-University Bochum, ICAMS, Universitaetsstrasse 150, 44801 Bochum, Germany (Dated: November 27, 2019) We study different forms of linear and non-linear field equations, so-called ‘phase-field’ equations, in relation to the de Broglie-Bohm double solution program. This defines a framework in which elementary particles are described by localized non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schrödinger type equation. First, we consider the phase-field order parameter as the phase for the linear pilot wave, second as the pilot wave itself and third as a moving soliton interpreted as a massive particle. In the last case, we introduce the equation for a superwave, the amplitude of which can be considered as a particle moving in accordance to the de Broglie-Bohm theory. Lax pairs for the coupled problems are constructed in order to discover possible non-linear equations which can describe the moving particle and to propose a framework for investigating coupled solutions. Finally, doublons in 1+1 dimensions are constructed as self similar solutions of a non-linear phase-field equation forming a finite space-object. Vacuum quantum oscillations within the doublon determine the evolution of the coupled system. Applying a conservation constraint and using general symmetry considerations, the doublons are arranged as a network in 1+1+2 dimensions where nodes are interpreted as elementary particles. A canonical procedure is proposed to treat charge and electromagnetic exchange.

PACS numbers: 04.20.Cv, 04.50.Kd, 05.70.Fh

I. INTRODUCTION to ‘local, realistic’ classical theories, Newton’s mechanics and General Relativity (GR), and the ‘non-local, proba- bilistic’ Copenhagen interpretation of quantum mechan- Since the emergence of quantum theory physicists try ics. to understand the physical world as a dual phenomenon De Broglie, the father of the wave interpretation of el- of waves and particles. A very promising concept, but to- ementary particles, presented his ideas, how to construct day almost forgotten, is the double solution program of stable elementary particles and their interaction as wave de Broglie and Bohm (dBB) [1–5]. The concept is based phenomena, first at the Solvay conference 1927. For a on two coupled wave equations. The so-called ‘pilot wave’ recent review and details see [8]. The solution of a linear corresponds to the probability wave, ψ, in the Copen- Schrödingertype wave equation couples to the solution hagen interpretation of quantum mechanics and evolves of a non-linear wave equation which has localized ampli- according to a linear Schrödingertype equation. This tudes interpreted as massive particles, the u wave. Both pilot wave couples to the u wave (in de Broglies nota- waves are connected by their phases, which is why the tion [3]), which shall describe physical ‘particles’. Doing pilot wave can guide the particle. According to Bohm’s this, it is possible to interpret e.g. the famous ‘double work, the Schrödingerequation for the ψ wave can be slit’ experiment in a picture where the particle remains decomposed into two equations, for the amplitude (R in as such from the moment of emission until the moment Bohm’s notation) and for the phase (S in Bohm’s no- of absorption, but is ‘informed’ about the probability of tation). The probability of finding a particle (P = R2) different paths by the pilot wave, i.e. it is guided by the relates to the probability in the Copenhagen interpreta- pilot wave [6]. The path of the particle within the pe- tion. Using the phase S, the velocity of the particle can 2 riod between emission and absorption is unknown (if no be found as vp = −c ∇S/∂tS. This can be seen in anal- additional measurement is performed which would spoil ogy to the particles in water waves which make loops that the experiment), that is why the particle itself is termed gradually advance in the direction of wave propagation ‘hidden’ during the experiment. The theory is ‘non-local, [9]. arXiv:1911.11571v1 [physics.gen-ph] 23 Nov 2019 realistic’ along the classification of Bell [7]. Non-local Why treat matter as coupled waves? Why not sim- means that physical observables are represented by gradi- ply accept the existence of elementary particles as point ent operators acting on fields in space and time. Realistic masses? The most prominent opponent of the dualism means that physical observables have well defined values of mass and space was Einstein, who worked for long on (not necessarily ‘exact’, but subject to Heisenberg’s un- a ‘unitary’ or ‘monistic’ view of the physical world, as certainty) independent of a measurement. This contrasts reported in [8]: ”[...] Einstein [...] had a very similar objective in the framework of his theory of general relativity in the 20’s: ∗Electronic address: [email protected] the postulate of geodesics would not be an extra hypothe- †Electronic address: [email protected] sis, but would be obeyed, de facto, by peaked solutions of 2

Einstein’s nonlinear equations moving on a weakly vary- a wave, although we will show that indeed it can have a ing metric background. This idea presents many deep similar formal meaning if we speak about localized wave similarities with de Broglie’s guidance equation which lies solutions, i.e. solitons. at the heart of the de Broglie-Bohm hidden variable the- One distinguishes primarily two sets of ’solitons’, as de- ory [1, 2]. In fact, Bohm-de Broglie trajectories are the picted in Figure 1 with the eigen-coordinate ξ, the space counterpart of geodesic trajectories in Einstein’s unitar- coordinate x, time coordinate t, the size η and the veloc- ian version of general relativity ...” ity v: Today it is almost fully accepted that elementary particles cannot be considered as point masses and a non- local theory, as the dBB double solution program, is needed. An appropriate quantum field theory, which is based on elementary space quanta with finite dimensions, e.g. strings [10] or branes [11], had not been worked out at that time, and is still missing in a concise form today. The double solution program offers an elegant solution to this problem because it combines a linear Schrödinger type wave equation for a ψ-wave with a non-linear wave FIG. 1: The soliton in 1 + 1 dimensions. a) Symmetric equation for a u-wave describing the particle. soliton, b) Half-sided soliton as an integral form of the The main difficulty, why the theory did not prevail, symmetric soliton. η is the characteristic size of the was the construction of the u wave representing the par- wave. ticle, or other quantum-mechanical objects. Although de Broglie’s primary idea had been the description of Figure 1 a) depicts the classical soliton as an excitation matter as a wave phenomenon, the double solution pro- against a homogeneous background. Besides numerous gram is based on the traditional understanding of ‘parti- application in wave mechanics (see [15] for an exhaustive cles’ and ‘space’ as separate elements of matter. Recently, review at its time), it has been proposed as a template the second author developed a monistic view of mat- to describe elementary particles [16]. The original soli- ter, called ‘quantum-phase-field concept’ [12, 13], treat- ton is a 1-dimensional, or planar 2-dimensional solution ing space-time and mass as two manifestations of matter of a spacial non-linear wave equation. True 3-dimensional with dualistic character. In the quantum-phase-field con- realizations have been shown to be mathematically im- cept, a special field solution, which we will call ‘doublon’ possible [21]. The construction of ‘particles’ then is only in the following, forms the elementary building block of possible in a spherical symmetric approximation (see e.g. matter. It unifies the aspects of particles and space. Only [22]), which by construction, prevents investigating scat- after publication, his attention was called to the similar- tering events. It has also been realized early on, that the ity with the dBB double solution program, that shall be intrinsic length scale of the soliton, its size η, leads to a elaborated in the present article. ‘natural mode of quantization’ [23]. The latter statement Here we have to link to another development, which leads back to the de Broglie-Bohm program. This con- began in the 1960s in the analysis of non-linear wave nection has been little addressed in the literature, but is equations that are related to a completely different phe- clearly pointed out by Colin et al. [8]. nomenon: shallow water waves in channels and ocean Figure 1 b) shows a solution of an integral form of shores. The problem has been described by Korteweg the classical Korteweg-de Vries type equation, which and de Vries (KdV) in 1896 [14]. Solutions of this type introduces parity in the space coordinate. We will later, of non-linear wave equations are characterized by a self- chapter VI, combine two of these ‘half-sided solitons’, similar shape of wave packets, nowadays known as ‘soli- a ’right moving soliton’ and a ‘left moving soliton’, to tons’, for review see [15, 16]. The soliton solutions had an antisymmetric pair, which we call ‘doublon’ [24] The been unknown to de Broglie and Bohm in the beginning doublon will serve i) to define massive particles with of the 1960s. positive energy by gradient operators and ii) define ‘space’ as the distance between the particles, attributed Today the ‘soliton’ appears in a new application, the by negative energy. phase-field (PF) theory [17–19] which bases on a soliton solution of a special non-linear wave equation derived We will proceed as follows: from a Ginzburg-Landau functional. The PF theory is applied to investigate pattern formation in condensed • Describe the relation of the ‘phase-field‘ equation matter physics and materials science. The pattern here to the dBB double solution program. is the morphology of a ‘phase’ in real space and its evo- • Present Lax-pairs which couple two wave functions lution, therefrom the name. A phase is identified by an similar to the dBB program. order parameter φ, called ‘phase-field’ variable. It is a state of matter with a discontinuous property relation to • Construct ‘doublons’ from an antisymmetric pair the other phases in the sense of Landau [20]. We shall not of half-sided left /right moving solitons from the confuse this interpretation of a ‘phase’ with the phase of ‘phase-field‘ equation. 3

• Construct a ‘doublon network’ in 1+1+2 dimen- With these relations, the following form of the evolu- sions which embeds massive particles. tion equation for ψ can be suggested:

2 • Propose a generalization of the concept to charged τ∂tψ = η ∂xxψ + iφψ + iηe0(∂xφ)ψ, (4) particles and ﬁelds. where e0 is a parameter to be deﬁned. By the substi- • Discuss limitations and perspectives of the current tution of the solution (1) in this equation and collecting approach. the necessary imaginary terms, we obtain the equation for the phase (2) with e = e0. The equation for the amplitude is the collection of the II. PF AS A TOY PROBLEM FOR THE DBB remaining terms, i.e., DOUBLE SOLUTION PROGRAM 2 ∂t(aψ) 2 2 2 ∂xxaψ 2 ∂xaψ A. PF equation for a phase of a wave function τ 2 = −η (∂xφ) + η + 2iη (∂xφ). (5) 2aψ aψ aψ

2 2 The phase-field order parameter φ can be considered in ∂xxaψ ∂xx(aψ) (∂xaψ) analogy to the phase S in the dBB double solution pro- After substitution = 2 − 2 , we aψ 2aψ aψ gram [5]. De Broglie and Bohm derived equations for the obtain phase and for the amplitude taking the pilot wave in the form ψ = ReiS and substituting it into the Schrödinger 2 2 2 2 2 2 2 2 τ∂t(aψ) = η ∂xx(aψ) − 2η (∂xφ) aψ − 2η (∂xaψ) equation. Here we solve the inverse problem and seek for 2 an equation for a ψ wave, whose phase evolves according + 4iη aψ(∂xaψ)(∂xφ). (6) to a PF equation. In order to be consistent, we define φ Since this equation can be considered as a PF-like as the phase of a wave function ψ with an amplitude aψ equation, it can be divided into two equations as 2 2 2 τ∂t(a ) = 2iη (∂xφ)∂x(a ), (7) ψ = a eiφ. (1) ψ ψ ψ 2 2 2 2 2 2 2 ∂xaψ 2 η ∂xx(aψ) = 2η (∂xφ) aψ + 2η aψ. (8) Then we consider the linear PF equation as an evolu- aψ tion equation for the phase: Equation (7) is the advection equation 2 τ∂tφ = η ∂xxφ + φ + ηe∂xφ, (2) 2 2 ∂t(aψ) = vaψ ∂x(aψ) (9) 2 where ∂ = ∂ , ∂ = ∂ and ∂ = ∂ . t ∂t x ∂x xx ∂x2 with the velocity Here τ is a kinetic parameter with dimension of time, η is a characteristic length to be identified with the size 2iη2 of the resulting soliton solution (see Figure 1) which is va = ∂xφ. (10) ψ τ also called ‘interface width’ in the PF model, and e is a dimensionless parameter related to the velocity of trans- Equation (8) defines the self-similar shape and can be port. The simple solution of this equation is φ = eiθ written as x + vφt ηe with θ = and the velocity vφ = . For this η2 ∂ (a2 ) = −a2 (11) η τ aψ xx ψ ψ kind of solution, the PF equation can be separated into with two equations: the advection equation ∂tφ = vφ∂xφ and the equation for the self-similar shape of the soliton. !−1 1 ∂ a 2 From the equation for the phase, we can reconstruct η2 = − (∂ φ)2 + x ψ . (12) aψ x the equation for ψ. The following relations can be useful 2 aψ as parts of the sought equation: 2 ix/ηaψ A solution of this equation is aψ(x, 0) = e or iφ ∂taψ ix/(2η ) ∂tψ = ∂t(aψe ) = ψ + i(∂tφ)ψ a (x, 0) = e aψ . a ψ ψ Using the advection equation (7), we obtain a to- 2 ∂t(a ) 2 iθa ψ tal solution for the amplitude aψ(x, t) = e with = 2 ψ + i(∂tφ)ψ, 2a x + va t ψ θ = ψ . a η ∂xaψ 2 2 ∂xxaψ aψ ∂xxψ = 2i (∂xφ)ψ + i (∂xφ) ψ + i(∂xxφ)ψ + ψ Assuming that the velocity of the amplitude aψ aψ 2 2iη 2 ∂xxaψ ∂xaψ vaψ = ∂xφ is equal to the particle velocity = −(∂xφ) + ψ + i 2 (∂xφ) + ∂xxφ ψ. τ aψ aψ 2 ∂xφ ieφ vp = − c , and substituting ∂tφ = , we can (3) ∂tφ τ 4

2 2 c τ iS x + vψt define the characteristic length η from η2 = . For and define ψ = aψe with S = and 2eφ η 1 cτ ∂tS eη the mean value φ¯ = , η ≈ √ . The particle velocity is vψ = = . Then we assume that the pilot 2 e ∂xS τ 2φc c wave ψ is the phase of a superwave then defined as va = √ ≈ √ = vp, and the velocity ψ e e iψ √ Φ = aΦe . (16) of the phase φ is defined as vφ = c e. Moreover, by ∂ a i iφ¯ From eq. (15), we can reconstruct the equation for Φ. substitution in (12) x ψ = and ∂ φ = , we get a 2η x η Using the terms (3) (where we replace ψ by Φ, aψ by aΦ, ψ aψ and φ by ψ), i.e., ηaψ = η. According to our results, the particle velocity decreases ∂taΦ with increasing e whereas the phase velocity increases. ∂tΦ = Φ + i(∂tψ)Φ, Both velocities approach c when e decreases, moreover, aΦ 2 ∂ a va vφ = c in accordance with the dBB theory. 2 xx Φ ψ ∂xxΦ = −(∂xψ) + Φ In quantum mechanics, the momentum (i.e., the par- aΦ ticle velocity) should be inverse proportional to the wave ∂xaΦ length, identified with the characteristic length p ∼ 1/η, + i 2 (∂xψ) + ∂xxψ Φ, (17) aΦ and direct proportional to the particle velosity.√ Hence, the following form of the evolution equation for Φ can be if we assume τ = τ0e, η will be proportional√ to e and suggested: p as well as vaψ will be proportional to 1/ e according to expectations. Based on that result, the characteristic 2 √ η ∂xxΦ ψΦ f(aΦ)Φ length can be defined as η = cτ0 e. τ∂tΦ = + + In summary, the particle is described by the amplitude 2i 2 2i function a (x, t) as a one-dimensional singularity of size a ∂xaΦ ψ + iηe0(∂xψ)Φ + ηe0 Φ, (18) aΦ η moving with the velocity vaψ , which is the solution of the PF-like equation (6). This amplitude corresponds to a where e0 and e0 are parameters responsible for the mo- the u-wave in the dBB theory and φ is the pilot wave for tion of ψ and aΦ, respectively, which have to be defined. the amplitude. The dynamics of φ on the other hand is f(aΦ)Φ The term i is added by analogy to ψΦ. By sub- governed by eq. (2). stituting the Φ wave (16) in this equation and collecting Finally, we show that eq. (2) can be transformed to the the imaginary terms in (17), the equation for ψ (15) can Schrödingerequation with a constant potential. Dividing be reconstructed with the two first terms on the right hand side of eq. (2) by 2i (this is allowed because the PF equation can be separated ∂xaΦ e = e0 − iη . (19) into two separated equations), we get aΦ The equation for the amplitude is the collection of the η2 φ τ∂ φ = ∂ φ + + ηe∂ φ. (13) real terms in (17), i.e., t 2i xx 2i x 2 2 ∂taΦ η ∂xxaΦ f(aΦ) η 2 a ∂xaΦ ~ iφ τ = + − (∂xψ) + ηe0 . aΦ 2i aΦ 2i 2i aΦ Then by substitution τ0 = 2 and ∂xφ = , we mc η (20) obtain the Schrödinger-type equation This equation can be treated as a set of two equations ~ iU0 ∂tφ = ∂xxφ + φ (14) 2 2im ~ a iη 2 aΦ τ∂taΦ = ηe0∂xaΦ + (∂xψ) ∂xaΦ, (21) 2 ∂xaΦ 2 1 2 with U0 = mc 1 − . Here, m is the mass of the η ∂xxaΦ = −aΦf(aΦ), (22) 2e aΦ particle, it the Plank’s constant. ~ where ηaΦ = η. By assumption f(aΦ) = 1, a solu- ix/ηa tion of eq. (22) reads aΦ(x, 0) = e Φ . Substituting ∂xaΦ i B. PF equation for the pilot wave = , we can rewrite the advection equation (21) aΦ η in the form In this section, we consider the transformed PF equa- ∂ a = v ∂ a , (23) tion (14) as a Schrödinger-type equation with a given t Φ aΦ x Φ potential. The phase-field order parameter φ will be now ηea considered as the pilot ψ wave. We rewrite eq. (13) as with va = − and Φ τ η2 ψ η2 τ∂ ψ = ∂ ψ + + ηe∂ ψ (15) ea = ea + (∂ ψ)2. (24) t 2i xx 2i x 0 2 x 5

iθa The full solution for the amplitude is aΦ(x, t) = e ψ is coupled to the function φ by the Lax pair. Hence x − vaΦ t the Lax method can describe the coupling of two wave with θa = . The amplitude can be interpreted as ηaΦ functions in the dBB double solution theory [5]. 2 ∂xS In order to find the solution of the PDE, one should a particle moving with the velocity vaΦ = vp = −c . ∂tS solve the forward and inverse scattering problems. The From this, we can define the characteristic length as forward problem is solved by the first and second Lax c2τ 2 η2 = . The particle velocity is then defined as equations with the eigenvalue λ: eea rea v = c , and the velocity of the wave ψ is defined Lψ = λψ, (28) aΦ e ∂tψ = Aψ. (29) r e as vψ = c . Here we have restriction for the particle ea The inverse scattering problem is solved by calculating ∂ a i velocity ea ≤ e. By substituting x Φ = in (19), we the Gelfand-Levitan-Marchenko integral (31). The func- aΦ ηaΦ tion φ(x, t) is constructed from the formula obtain e = e0 + η/ηaΦ = e0 + 1. Note that the phase and the amplitude of the wave ∂ iS φ(x, t) = −2 K(x, x, t), (30) ψ = aψe evolve according to dBB equations [5], which ∂x are the Hamilton-Jacobi equation for the phase and the continuous equation for the amplitude. The difference where K(x, y, t) is the solution of the linear integral equa- between the dBB program and our solution is that in tion the present variant the amplitude of the superwave aΦ is +∞ representing the particle which was previously considered Z as the u-wave. The guidance occurs as before by the dBB K(x, y) + F (x + y) + K(x, z)F (y + z)dz = 0 (31) quantum force, which acts on the phase S, which then x changes the particle velocity. Hence the function ψ is the pilot wave for aΦ. defined for a fixed time t and F is the function which is Finally, we can rewrite eq. (15) in physical units. By based on the solution of the Lax equations for ψ. substitution τ = τ e, τ = ~ , and η = τ v , we obtain 0 0 mc2 0 ψ

2 2 ~vψ mc A. Lax pair for the phase and amplitude equations ∂tψ = ∂xxψ + ψ + vψ∂xψ. (25) mc2 ~ Now we show that the phase equation (2) can be re- Then, multiplying the two ﬁrst terms on the right c2 ψ constructed by the Lax method applied to a function ψ hand side by and using ∂ ψ = i , we get the whose evolution equation is described by the second Lax (2iv2 ) x η ψ equation. Schr¨odinger-type equation We choose the Lax operators as ~ U0 ∂tψ = ∂xxψ + i ψ (26) L = τη∂ + τφ + τηi∂ + τiφ, 2im ~ x x η2 φ ηe 2 ! 0 c A = ∂xx + i + i ∂xφ. (32) 2 τ τ τ with U0 = mc 1 − 2 . In the limit vψ → ∞, U0 → 2vψ mc2. The non-zero components of the Lax equation (27) have the terms

III. LAX METHOD FOR THE SOLUTION OF [φ, ∂xx]ψ = φψxx − ∂xx(φψ) WAVE EQUATIONS = φψxx − ∂x(φxψ + φψx) = φψ − (φ ψ + φ ψ + φ ψ + φψ ) The solution of a scattering problem in quantum me- xx xx x x x x xx chanics is associated with the linear Lax operators L and = −φxxψ − 2φxψx, A, called Lax pair, which satisfy the Lax equation [25], [∂x, φ]ψ = ∂x(φψ) − φψx = φxψ,

[∂x, φx]ψ = φxxψ. (33) Lt + [L, A] = 0. (27)

iθ iθ This equation should reproduce a partial diﬀerential Using φ = φ0e and ψ = ψ0e , we can substi- equation (PDE) for a function φ, whereas the Lax oper- tute φx = (log φ)xφ and ψx = (log ψ)xψ and get ators are applied to a function ψ, which is the solution of φxxψ = iθxφxψ, φxψ = iθxφψ, and φxψx = iθxφxψ. the Schr¨odinger-type equations. Therefore, the function Then we assume θx = 1/η. The components of the Lax 6 equation including real and imaginary terms become ∂xaψ Here we used ψx = ψ + iφxψ and normalized aψ 2 2 2 η [φ, ∂xx]ψ = −η φxxψ − 2η φxψx the aψ-operator by aψ. We also do not write the similar = −η2φ ψ − 2iηφ ψ, complex conjugate terms to be compact. xx x Substituting all non-zero components in eq. (27) and 2 2 2 η [iφ, ∂xx]ψ = −iη φxxψ − 2iη φxψx collecting imaginary and real terms, we obtain two equa- 2 tions = −iη φxxψ + 2ηφxψ, η[∂ , iφ]ψ = iηφ ψ = −φψ, 2 x x τ∂tφ = η ∂xxφ + φ + e0η(∂xφ), η[i∂x, iφ]ψ = −ηφxψ = −iφψ, 2 2 2 2 2 2 2 2 τ∂taψ = η ∂xxaψ − 2η (∂xφ) aψ − 2η (∂xaψ) 2 2 η e0[∂x, iφx]ψ = iη e0φxxψ = −ηe0φxψ, 2 2 ∂xaψ 2 2 + 4iη aψ (∂xφ), (40) η e0[i∂x, iφx]ψ = −η e0φxxψ = −iηe0φxψ. (34) aψ

After substitution in eq. (27), we obtain which for e = e0 lead to equations (2) and (6) for the phase and the amplitude, whereas the evolution equation 2 τ∂tφ = η ∂xxφ + φ + (e0 − 2)η∂xφ, for the wave ψ is in the same form as before (see eq. (37)) iτ∂ φ = iη2∂ φ + iφ + i(e + 2)η∂ φ. (35) i.e., recovers the equation (4) for the ψ wave. Hence, we t xx 0 x have shown that the Lax method can be used also for the These equations recover the PF equation (2) for coupling of three functions, one of which is the function of two others. Note that in this case, φ is the pilot wave e = e0 ± 2. The ﬁrst Lax equation for the ψ functionwith the eigen- for R. The Lax pair for eqs. (15)-(20) can be found in the sim- values λ1 and λ2 is ilar manner by adding the necessary terms in the opera- iS iψ iη∂xψ + φψ + η∂xψ + iφψ = λ1ψ + iλ2ψ, (36) tors L and A. Thus for the ansatz ψ = e , Φ = aΦe , and ∂xS = 1/η, the Lax pair for eqs. (15)-(20) have the and second Lax equation is form:

2 2aΦ(x, t) 2aΦ(x, t) τ∂tψ = η ∂xxψ + iφψ + iηe0(∂xφ)ψ, (37) L = τη∂x + τψ + τ + τηi∂x + τiψ + τi , aΦ aΦ 2 a which recovers eq. (4) for the ψ wave one to one. η iψ ηe0 f(aΦ) ηe0 ∂xaΦ The Lax pair can be found also for the waves in section A = ∂xx + + i ∂xψ + + . 2iτ 2iτ τ 2iτ τ aΦ iθ iφ II A in the form φ = e and ψ = aψe , where φ is (41) the phase of the pilot wave ψ. Doing so, we deviate from the Lax method, which requires that the phase of Finally, the Lax equation (27) recover eqs. (15) and (20) both functions should be equal and time independent. for ψ and aΦ, and ﬁrst and second Lax equations (28)- However, we use again the assumption that θx = 1/η. (29) recover eq. (18) for Φ. Note that in this case, the We show now that the following Lax pair can reproduce function ψ is the pilot wave for aΦ. equations (2) and (6):

2aψ(x, t) 2aψ(x, t) B. Lax pairs for the non-linear PF equation L = τη∂x + τφ + τ + τηi∂x + τiφ + τi , aψ aψ 2 We consider a non-linear wave equation for a wave φ, η φ ηe0 A = ∂xx + i + i ∂xφ. (38) also known in material physics as the ‘phase-ﬁeld equa- τ τ τ tion’ with double obstacle (DO) potential [18] The non-zero components of the Lax equation have the 2 terms τ∂tφ = η ∂xxφ − fφ(φ) + ηe∂xφ. (42)

η2[φ, ∂ ]ψ = −η2φ ψ − 2η2φ ψ fφ(φ) is a non-linear scalar function in φ which is the ﬁrst xx xx x x derivative of the double obstacle potential 2 2 2 2 ∂xaψ = −η φxxψ − 2iη (φx) ψ − 2η φxψ, 1 aψ f(φ) = |φ(1 − φ)|. (43) 2 η[∂x, iφ]ψ = iηφxψ = −φψ, 2 2 The non-linearity is hidden in the break points φ = 0 η e0[∂x, iφx]ψ = iη e0φxxψ = −ηe0φxψ, and φ = 1. We will argue later that this ansatz is respon- 2 η 2 ∂xxaψ 2 ∂xaψ sible for the localization of the wave solution. One can [iaψ(x), ∂xx]ψ = −iη ψ − 2iη ψx aψ aψ aψ use here, without loss of generality, the functional form ∂ a (∂ a )2 ∂ a 2 xx ψ 2 x ψ 2 x ψ 1 = −iη ψ − 2iη 2 ψ + 2η φxψ. (39) a a a fφ(φ) = − φ signum(φ(1 − φ)), (44) ψ ψ ψ 2 7

1 that means fφ(φ) = ( 2 − φ) for 0 ≤ φ ≤ 1 and The solution of this equation reads 1 fφ(φ) = (φ − ) otherwise. kx 2 ψ = ψ0(k, t)e , (52) A solution of eq. (42) with the potential (43) is, e.g., where the amplitude ψ0(k, t) is a function of wave number 1 x − vt 1 and time. φ(x, t) = sin + , (45) 2 η 2 Substituting ψ in the second Lax equation (29), we η η obtain for − 2 −vt < x < 2 −vt for a half-wave traveling with ve- 2 ηe η V (x) e0 locity v = (see also [12]). We see that the non-linear ∂tψ = ∂xxψ + ψ + φψ. (53) τ τ τη τ absolute operator, |.|, cuts out one localized half wave This equation describes the time evolution of the scat- from the periodic sinus wave. Although the solution is tering amplitude ψ0(k, t). Using the asymptotic V (x) → known, we will investigate the possibility to find a solu- 0 at x → −∞, we obtain tion of this PDE by the ‘inverse scattering method’ and 2 demonstrate connections between two wave functions φ η 2 ∂tψ0(k, t) = k ψ0(k, t) (54) and ψ. For this aim, we will find the Lax pairs and show τ that eq. (42) can be reproduced by the Lax representa- and 2 2 tion. k η t ψ0(k, t) = ψ0(k, 0)e τ . (55) Hence the solution of the forward problem has the form 1. Variant I kx νt ψ(x, t) = ψ0(k, 0)e e , (56) We choose the Lax operators in the form k2η2 where ν = τ . L = τη∂ + τφ, Using the solution (56), we solve the inverse scattering x problem. First, we define the function F (x, t) as 2 η V (x) e0 A = ∂ + + φ, (46) F (x, t) = F ekxe2νt (57) τ xx τη τ 0 and search the solution of eq. (31) in the form K(x, y, t) = R x where V (x) = fφ(φ)dx, e0 is a constant to be defined. M(x, t)eky. After substitution we get This type of equations was suggested first by Zakharov ky k(x+y) 2νt and Shabat [26]. M(x, t)e + F0e e The non-zero components of the Lax equation are +∞ Z kz k(y+z)+2νt + M(x, t)e F0e dz = 0, [∂x,V (x)]ψ = ∂x(V (x)ψ) − V (x)ψx x = ψ∂xV (x) + V (x)ψx − V (x)ψx = fφ(φ)ψ; then [φ, ∂xx]ψ = −φxxψ − 2φxψx; M(x, t) + F ekx+2νt [∂x, φ]ψ = ∂x(φψ) − φψx = φxψ. (47) 0 x Z After substitution in eq. (27), we obtain 2νt 2kz + M(x, t)F0e e dz = 0, 2 2 τ∂tφψ + fφ(φ)ψ − η ∂xxφψ − 2η ∂xφ∂xψ + e0η∂xφψ = 0. −∞ (48) and finally 2kx Then we use ψx = (log ψ)xψ, define a wave number kx+2νt 2νt e M(x, t) + F0e + M(x, t)F0e = 0. k = (log ψ)x, and obtain the phase-field equation in the 2k form Hence τ∂ φ = η2∂ φ − f (φ) + (2ηk − e )η∂ φ, (49) −2kF ekx+2νt t xx φ 0 x M(x, t) = 0 2k + F e2kx+2νt which recovers the phase-field equation (42) with e = 0 and from eq. (30) we obtain (2ηk − e0). Now we solve the forward scattering problem. The φ(x, t) = −2k2sech2(−kx − νt − δ), (58) first Lax equation (28) after substituting the operators and assuming λ = τηk becomes where δ = 1/2 log(F0/2k). This solution has the form of the known solution of KdV equation. The difference is ∂xψ + φψ/η = kψ. (50) in the dependency of ν on k, which is quadratic. The 1 1 comparison with the solution (45) gives k = , ν = , Assuming the asymptotic behavior φ → 0 at x → −∞, η τ we get η v = , e = 1, e0 = 1. The functions are compared in τ √ ∂xψ = kψ. (51) Figure 2 for η = 2 and t = 0. 8

Here we obtain a linear dispersion relation, i.e. the kηe frequency is proportional to k, ν = τ . The procedure of the solution of the inverse scattering problem is similar to Variant I. Finally, the solution is

φ(x, t) = −2ksech2(−kx − νt − δ). (67) 1 Comparison with the sin-solution (45) gives k = , η e ν ηe ν = and v = = . τ k τ It can be shown that eq. (67) is the solution of PF equation (42) with the potential f˜(φ) = 2φ2(1 − φ). It is also interesting to mention here that with this potential the PF equation can be easily transformed to the KdV equation by taking the space derivatives of the two first FIG. 2: Comparison of the functions 1 sin( x ) + 1 (top, 2 η 2 terms on the right hand side of the PF equation. This blue) and 2k2sech2(−kx − π ) (bottom, red) for k = √1 2 √ 2 kind of potential is not stable, because it is unbounded and η = 2. from below and offers an infinite energy for a very large φ. To overcome this difficulty, we suggest to use the absolute value |1 − φ|, which produces the second minimum and 2. Variant II allows to get a stable soliton. Figure 3 shows the moving soliton solved numerically ˜ 2 The second variant of the Lax operators reads by the PF equation (42) with f(φ) = 2φ |1 − φ|, the derivative of which is L = τη∂x + τφ, ˜ φ(φ − 1) ηe V (x) η fφ(φ) = 2φ 2|φ − 1| + . (68) A = ∂ + − ∂ φ. (59) |φ − 1| τ x τη τ x The parameters of the model are: η = 20, e = 1, The non-zero components of the Lax equation are τ = 0.5, ∆t = 1, ∆x = 1. The numerical solution in Figure 3 sketches the analytical traveling wave solution [∂x,V (x)]ψ = fφ(φ)ψ; [φ, ∂x]ψ = φψx − φxψ − φψx = −φxψ; x t φ(x, t) = sech2 − − . (69) [∂x, φx]ψ = −φxxψ. (60) η τ After substitution in eq. (27), we recover the phase- field equation (42) with arbitrary e:

2 τ∂tφ = η ∂xxφ − fφ(φ) + ηe∂xφ. (61) The ﬁrst Lax equation has the same form as eq. (50)

η∂xψ = −φψ + kηψ. (62) with the solution (52). The second Lax equation reads ηe V (x) η ∂ ψ = ∂ ψ + ψ − (∂ φ)ψ. (63) t τ x τη τ x

Using the asymptotic V (x) → 0 and φx → 0 at x → −∞, we obtain ηe ψ (k, t) = kψ (k, t), (64) 0 t τ 0 which has the solution FIG. 3: Traveling wave solution for the functions 2 kηe t sech (−x/η − t/τ) with periodic boundary conditions. ψ0(k, t) = ψ0(k, 0)e τ . (65) Hence the solution of forward problem in the second Note that the solution (69) deﬁnes a symmetric soliton. variant is Such kind of solitons are typically derived from a KdV

kx kηe t type of equation which is 3rd order in the space deriva- ψ(x, t) = ψ0(k, 0)e e τ . (66) tive, while (69) is derived from a 2nd order PF equation. 9

The important diﬀerence in the type of soliton solutions, After substituting it in eq. (31), results in symmetric or antisymmetric, thereby seems to lie in the −iky −ik(x+y) 2iνt parity asymmetry of one operator, either the diﬀerential M(x, t)e + iF0e e +∞ operator or the potential. Here more future work is nec- Z essary. The form (68) opens a new class of symmetric −ikz −i(k(y+z)−2νt) + i M(x, t)e F0e dz = 0, soliton solutions. x −i(kx−2νt) M(x, t) + F0e 3. Variant III +∞ Z iνt −2ikz + M(x, t)F0e e dz = 0. (79) The third variant of the Lax operators is x

L = τη∂x + τφ, The last term cancels by adding the complex conjugate ηe 1 η γ (c. c.) function. Hence A = ∂x + V (x) − ∂xφ + , (70) τ τη τ τ −i(kx−2νt) M(x, t) = −F0e , where γ is added to Variant II for the normalization of K(x, x, t) = −F e−i(2kx−2νt). the scattering data, see [27]. 0 Now we consider the imaginary forms of Lax equations. As a result, we obtain from (30) Substituting the operators and taking λ = τηk, we get 2F0k −i(2kx−2νt) F0 −i (x−vt) the ﬁrst equation φ(x, t) = − e = − e η , (80) i iη i∂ ψ + iφψ/η = kψ. (71) x where Assuming an asymptotic behavior φ → 0 at x → ∞, we 1 e ν ηe k = , ν = , v = = . (81) obtain 2η 2τ k τ

i∂xψ = kψ (72) By adding the c. c. function and normalizing, a real solution can be found in the form (45). with the solution

−ikx ikx ψ = e + b(k, t)e for x → ∞, C. Lax pair for the relativistic case ψ = a(k, t)e−ikx for x → −∞. (73) The phase-field equation for the relativistic singularity This solution describes scattering from the right of the can be written as −ikx incident wave e on the potential φ, b(k, t) represents 2 1 a reflection coefficient and a(k, t) is a transmission coef- τ∂tφ = η ∂xxφ − ∂ttφ + fφ(φ) + ηe∂xφ. (82) ficient. The second Lax equation with the same asymp- c2 totic becomes This equation has the form of the Klein-Gordon equa- ηe i∂ ψ = ∂ ψ + γψ. (74) tion with advection. Using the solution that the singu- t τ x larity moves with the velocity v = ηe/τ according to advection equation, we can replace φ by v2φ [12]. Then By substitution of γ = ikηe and a(k, 0) = 0, we obtain tt xx the equation becomes

i2kηe τ t 2 b(k, t) = b(k, 0)e . (75) τ∂tφ = ηv∂xxφ + fφ(φ) + ηe∂xφ, (83)

p 2 2 Hence the solution of the forward problem has the form where ηv = η 1 − v /c . The Lax pair can be chosen based on the second variant (59) ψ(x, t) = b(k, 0)e−ikxe2iνt, (76) L = τηv∂x + τφ, kηe where ν = τ . ηe 1 ηv To solve the inverse scattering problem, we deﬁne A = ∂x + V (x) − φx. (84) τ τηv τ F (x, t) as follows With this choice the equations for the pilot-wave func- −i(kx−2νt) F (x, t) = F0e . (77) tion become

We search for a solution in the form ηv∂xψx = −φψ + ψ, (85)

−iky V (x) K(x, y, t) ∼ M(x, t)e . (78) τ∂tψ = ηeψx + ψ − ηv∂xφψ. (86) ηv 10

R x IV. QUANTUM PHASE FIELD CONCEPT where V (x) = − fφ(φ) dx. The non-zero components of the Lax equation are Let φ be a scalar quantum field which acts on a quantum state ψ = ae−iθ of a vacuum oscillation. The Hamil- [∂x,V (x)]ψ = −fφ(φ)ψ; tonian of such a system, which formally corresponds to [φ, ∂xx]ψ = −φxxψ − 2φx∂xψ; a free energy of a closed thermodynamic system, can be [∂x, φ]ψ = φxψ. (94) expanded in the non-relativistic case (for the general case see [12]) This is valid also for ψ†. Then we can reproduce the Z PF equation (92) for the field φ by the Lax equation † ˆ iL + [L, A] = 0. H = U0 hψ |h|ψi dx, (87) t Ω The corresponding Lax equation for the complex func- 2 † η tion ψ (or ψ ) have the form iψt = Aψ, i. e. hˆ(φ) = (∂ φ)2 + f(φ) 2 x V (x) iτ∂ ψ = η2∂ ψ + ψ + iφψ. (95) as the integral over the domain Ω of the free energy den- t xx η sity, which is defined as the expectation value of the non- linear soliton operator hˆ(φ) acting on the pilot wave ψ. An alternative method to obtain the evolution equa- The first term can be calculated as tion for a state function is the relaxation dynamics:

† † 1 δH hψ |∂xφ∂xφ|ψi = ∂x(φψ )∂x(φψ) iτ∂ ψ = − t U δψ† = ψ†∂ φ + φ∂ ψ† (ψ∂ φ + φ∂ ψ) 0 x x x x η2 η2 † 2 2 † = φ2∂ ψ − (∂ φ)2ψ − f(φ)ψ. (96) = ψψ (∂xφ) + φ ∂xψ∂xψ 2 xx 2 x + φ∂ φ ψ∂ ψ† + ψ†∂ ψ . (88) x x x From this equation we can see that the ψ function ex- The integration of the last term gives 0. Then by the ists only in the region where φ > 0 and it is bounded by relaxation dynamics the potential f(φ), which tends to 0 at φ → 0, 1. There- fore, the problem is similar to the Schr¨odingerequation

† 1 δH for a particle in a box potential (see below, Figure 4 ). ψψ τ∂tφ = − , (89) U0 δφ we obtain V. THE DOUBLON IN 1+1 DIMENSIONS

† 2 † 2 † † ψψ τ∂tφ = η ψψ ∂xxφ − η φ∂xψ ∂xψ + ψψ fφ(φ). Having the solution for the wave φ in the form of the so- (90) called half sided ‘soliton’ with spinor character regarding parity in space, we easily construct a double sided ‘soli- † 2 † 1 Substituting ∂xψ ∂xψ = (θx) ψψ and fφ = φ − ton’, which we will call ‘doublon’ as an antisymmetric 2 pair of two half sided solitons and the ‘space’ in between as defined in (44), we get the PF equation with the DO them: potential  1 1 π(x − x1 + vt) τ∂ φ = η2∂ φ + f − η2(θ )2φ, (91)  + sin t xx φ x 2 2 η  η η 2 2  for − ≤ x − x1 + vt < where η (θx) is a quantum driving force. Using the com-  2 2 −iθ(x) 1 1 π(x − x2 − vt) plex function φ = e with θx = 1/η, eq. (91) can be φ = − sin 2 2 η rewritten as   η η  for − 2 ≤ x − x2 − vt < 2 2  iτ∂tφ = η ∂xxφ + fφ + iη∂xφ. (92) 1 for x − vt + η ≤ x < x + vt − η  1 2 2 2  Equation (90) has the similar meaning as eq. (48) 0 otherwise. which is obtained by means of the Lax pair method. The (97) function ψψ† is an analogy of the function ψ in section The solution is depicted in Figure 4 in 1 dimension III. Hence, using the Lax method, we can find the Lax neglecting time. The doublon consists of an antisym- equations for ψ and ψ†. For eq. (92) we can choose the metric pair of two half-sided solitons and a finite space, following Lax pair attributed by to a 1-dimensional line coordinate x, where L = τη∂ + τφ, φ(x) ≡ 1. This space is bounded by the non-local tran- x ∂ 2 sition regions of width η where ∂x φ(x) 6= 0, the right- η V (x) i and left moving soliton, which we identify with parti- A = ∂xx + + φ, (93) τ τη τ cles moving with relative velocity v = vp. Thereby the 11

• define a set of doublons φI ,I = 1...N with a number N > 6 for a 3-dimensional space filling network, • define the interaction between doublons and • embed the description in 3+1 dimensional space time.

The second item will follow canonically from the pos- FIG. 4: The doublon φ as a combination of a left- and a tulate that the set of doublons is closed in itself, i.e. right-moving soliton deﬁnes the box for vacuum ﬂuctuations ψ which determine the spacial energy of the X φI = 1. (99) doublon. I=1...N

Using expression (87) for the free energy of an indi- doublon forms a 1-dimensional box for vacuum fluctua- vidual field I, H → HI , we define the total free energy tions ψ. Regions outside the doublon, i.e. regions on the P I H = I=1...N H , then conserving (99) (see [28] for de- line coordinate x with φ(x) ≡ 0 have no physical mean- tails) we end up with the equation of motion for the dou- ing. The doublon forms the elementary building block blons of the physical world in the present concept. It unifies space-time and mass in a monistic structure. ‘Mass’, attributed by positive energy, is related to η2 while space I 1 X δ δ H τ∂tφ = − { I − J } is attributed by negative energy e. The latter is easily N δφ δφ U0 J=1...N calculated from quantum fluctuations with discrete spec- 1 X trum p and frequency ω = πcp , where Ω = |x − x | = − ΦIJ . (100) p 2Ω 1 2 N is the size of the doublon. According to Casimir [29], J=1...N this has to be compared to a continuous spectrum. This yields the negative energy e of space: The last expression defines an antisymmetric object of dual character, the pair-exchange operator ΦIJ be- " ∞ # tween two doublon fields I and J. This beautiful result hc X Z ∞ hc e = α p − pdp = −α , (98) is a mere consequence of the system being closed in itself 4Ω 48Ω p=1 1 and leads to a natural decomposition of the multi-body interaction between the doublons into pair-wise contri- where α is a positive, dimensionless coupling coefficient. butions. Knowing the structure of doublons consisting of We have used Euler–MacLaurin formula in the limit two antisymmetric solitons and the space between them, → 0 after renormalization p → pe−p. The quantum it is obvious that the interaction of doublons only hap- fluctuations are the Schroedinger type solutions of the pens in the soliton regions where 0 < φI < 1 with size ψ wave in the dBB picture. Here we have to note one η. This size is estimated in [12] to be below 10−15 m, important difference in the approaches (see also the dis- i.e. in the range of elementary particles in the classical cussions in chapter VIII). In the dBB program the pilot sense. We will call this region ‘quasi-local’, meaning that wave is the primary object which lives in a given envi- we have a non-local theory with highly localized states. ronment and drives the u wave in the form of a soliton as Within the quasi-local position of an elementary particle the object of investigation. In the present concept, the all N doublon fields may interact, the particle is under- φ wave in the form of the doublon defines the environ- stood as a junction between doublons. Although doublon ment for the ψ wave. Up to now only the ‘quasi-static’ fields are constructed along a 1-dimensional line coordi- limit has been worked out, i.e. that the doublon solution nate, we may argue that the junctions form 0-dimensional is kept fixed for the quantum solution (98). The direct object. Later, when introducing charges in section VII, coupling of both waves by their phase in the transition we will also discuss a finite width of the doublons to al- region 0 < φ < 1 may be investigated using the Lax low for optical excitations. Each pair-exchange operator formalism. We leave a closer investigation of the cou- ΦIJ carries an orientation from the half sided soliton at pling between the φ and ψ waves and a dynamic coupled the endpoint of the doublon φI , considered as a spinor. solution to future work. Due to the isomorphism of spinors with the 3-dimensional SU(2) group, we therefore may embed the doublons into a 3-dimensional spacial environment within the quasi-local VI. THE DOUBLON NETWORK IN 1+1+2 environment of a particle and a small surrounding. Here DIMENSIONS we define the 3+1 dimensional field variable φ˜I (~x,t) for which a standard phase-field description in 3-dimensional To proceed towards a multi-dimensional (in the space space is applicable. Outside this local environment the coordinate) case we have to field collapses to the space like part of a doublon. We 12 formally define the doublon as the trajectory, or path, of a classical flow between the endpoints of the doublon in space time. The advancing flow corresponding to dou- I blon I,ΞAdv and its time reversed equivalent, the re- I tarded flow ΞRet, connects space-time events (~x1, t1) and (~x2, t2). This path is determined from a minimum action condition E(φ˜I ) = 0 (see [30, 31]):

˜I I ˜I ˜I I ˜I φ (~x,t) = ΞAdvφ (~x1, t1); φ (t, ~x) = ΞRetφ (~x2, t2) (101) The doublon then follows the probability P to ﬁnd non-zero values of the ﬁeld φ˜I (~x,t) at space point ~x and time t in 3+1 dimensions :

h i Z h i P φ˜I (~x,t), φ˜I (~x , t ) = Dφ˜I δ E(φ˜I ) , (102) 1 1 D FIG. 6: Doublon network for 6 interacting doublons φ˜1...φ˜6 forming a 3-dimensional space ﬁlling where δD is the Dirac functional delta function. environment with 4 particles P1...P4. The junctions of Any event (~x,t) laying of the doublon connecting the size η are stretched out by a minimum of 3 doublons φI events (~x1, t1) and (~x2, t2) can be reached either via the deﬁned on linear independent directions in action of advancing operator on (~x1, t1) or equivalently 3-dimensional space. the retarding operator on (~x2, t2).

centering VII. PHASE-FIELD EQUATION IN ELECTROMAGNETIC FIELD

1.0

0.8 We will proceed here in the canonical way along the

0.6 Ginzburg-Landau free energy functional for a superfluid Φ ˜I 0.4 phase defined by a doublon φ in electromagnetic field.

0.2 The energy functional (87) now reads:

0.0 Z I I ˜I H = U0 h (φ , A) dx, (103) y Ω x where

FIG. 5: Sketch of a doublon represented by the " ˜I 2 3+1-dimensional field φ (~x,t) I I 1 2 qA ˜I h (φ , A) = η ns ∇ − φ 2 ~ The doublon network is schematically depicted in Fig- 1 (∇n )2 B2 + n |φ˜I (1 − φ˜I )| + s (φ˜I )2 + . (104) ure 6 for 6 interacting doublons forming a 3-dimensional s 4 n µU space filling environment with 4 particles. As indicated, s 0 we define around the particles a 3-dimensional environment of size η, corresponding to the (inverse) gradient of Here A is the vector potential, B is the magnetic field, the fields φI which interact in this quasi-local junction. µ is the magnetic permeability, x is the space vector, q I 2 Since each doublon φ carries a different orientation in is the charge, U0 = mc is the energy of a superfluid, the 3-dimensional space it is embedded in, we define here n = ψψ† is the concentration of a superfluid, η = ~ is the 3-dimensional field φ˜I (~x,t). As indicated by the solid s mc end-segments of the doublons, the volume of the junc- B2 the interface width, and is the energy density of the tions can uniquely be divided into volume segments as- 2µ signed to the volumetric fields φ˜I . Outside the junctions magnetic field. these volume fields collapse to tube-like objects along the n Using the relaxation dynamic (89) with τ = ~ s , we definition (102), as indicated by the dashed lines repre- mc2 senting the space part of the doublon. obtain the evolution equation for a phase in an electro- 13 magnetic field form the interference pattern. In the dBB picture the pilot wave replaces the probability wave in the Copenhagen 2 ˜I 2 qA ˜I ˜I 1 interpretation. From a rational point of view these waves τ∂tφ = η ∇ − φ + (φ − ) ~ 2 can be seen as kind of a Fourier transformed snapshot of 2 the experimental set up. Time needed for transport is not 1 (∇ns) ˜I + 2 φ . (105) included in this interpretation. Also the scale on which 4 ns interference is observed only depends on the relation be- In order to derive the kinetic equation for the super- tween the distance of the slits and the characteristic wave fluid, we minimize the functional HI with respect to A. length of the pilot wave, i.e. the dominating Fourier com- First, we rewrite functional to collect all terms depending ponent. This solution is a classical wave solution negat- on A as ing the existence of discrete objects, the particles. It can be obtained by solving the differential wave-mechanical qn hI (φ˜I , A) = hI (φ˜I ) − ~ s (φ˜I )2∇ · A equation for given boundary conditions, which represent 0 m the environment: emitter, absorber screen and double slit qn q2n A2 B2 barrier. So it is not astonishing that the slit experiment − ~ s A · ∇(φ˜I )2 + s (φ˜I )2 + . (106) m 2m 2µ can be repeated with water waves and small cork particles in the classroom, or in a more sophisticates set up Since ∇·A = 0, the term with the divergence of A can with oil droplets in the laboratory [33, 34]. Compare also be neglected. By substitution ∇ × A = B, the variation the hydrodynamic interpretation of quantum-mechanics of the last term with respect to A gives: by Madelung in the 1920th [35]. Now we must recall one major criticism of Einstein ∂|∇ × A|2 δB2 = δ|∇ × A|2 = δ(∇ × A) (and others) on these interpretations of quantum me- ∂(∇ × A) chanics that they anticipate an instantaneous knowledge = 2(∇ × A)δ(∇ × A) (107) of the situation in the experiment. The time needed for = −2∇((∇ × A) × δA) + 2(∇ × ∇ × A)δA, transport is not considered. An elegant solution to this problem is given by Feynman’s path integral theory [36], where the integral of the first term is 0 by the Gauss’ Figure 7 c). The probability to detect a particle at a spe- theorem. cial position at the absorber screen is determined by the Finally, the minimization of HI gives action of the particle along different paths. A global wave within the whole experimental setup does not exist and δHI qn q2n A ∇ × B = −~ s φ˜I ∇φ˜I + s φ˜2 + = 0. (108) the time of transport is consistently considered. Also the δA m m I µ scale, on which quantum interference can be observed, is determined by the action in comparison to Plancks quan- Substituting ∇ × B = µ js, we obtain the second Lon- tum. So one can clearly distinguish between quantum don equation effects and classical hydrodynamic effects [33, 34]. q2n The path integral picture leads us directly to the j = − s A (109) s m doublon-network model in section VI, where the doublon represents a kind of elementary space quantum connect- for the case of the uniform φ˜I = 1. ing emitter and absorber. The doublon connects these points by a flow of quantum fluctuations. Elementary particles by themselves are defined from gradient con- VIII. DISCUSSION AND CONCLUSION tributions, half-sided solitons, on both ends of the doublon. The concept is consistent with the approach of The de Broglie-Bohm double solution program sets a dBB, but distinct in several aspects, the most important framework of coupled wave equations to represent ‘par- is the topology of space and the definition of particles. ticles’ which are guided by a probability wave function As we can see from section II, the PF formalism allows in order to be consistent with observations of quantum to treat both forms of the particle representation: the statistical behavior of these particles. Let us recall the fa- symmetric solution and the half-sided solution (Figure mous double slit experiment: particles (photons or elec- 1). Due to their parity one may identify the symmetric trons or even fullerenes [32]) emitted on one side of a solution with a bosonic particle and the antisymmetric, double slit form an interference pattern at the absorber or half-sided solution with a fermionic particle. We state screen. It must be stated clearly that an individual par- that fermionic particles always have to come in pairs and ticle has only one strike on hte absorber screen. The that there is a conservation constraint. For bosonic parti- interference pattern, however, also appears if the objects cles, on the other hand, no conservation is expected since are recorded with some time delay, i.e. that the events their creation or annihilation does not change the global can be treated uncorrelated. The situation is depicted in wave solution apart from the individual position of the Figure 7 a) and b). An individual particle only has one bosons. We may speculate that the space-doublons ψ˜I strike at the absorber screen, while a number of particles possess a transverse width which may be determined by 14

a bosonic wave solution in transverse direction according to Eq. (68) and (69). In this transverse direction optical excitations are possible corresponding to electromagnetic waves. More future work is needed to investigate such solutions. In section II, we have considered several variants of PF equations within the framework of the dBB double solution program. Special equations are found which define particle-like wave solutions. More technically, we have separated the real parts of the PF equation, which are responsible for the phase evolution, and the imaginary parts for the amplitude evolution. A promising (a) ——————————————————- variant seems to be the definition of the particle velocity as the velocity of an amplitude of a superwave whose phase is the probability wave function (see section II B). Both, the amplitude and the phase, are solutions of Schrödinger-type equations, which have the form of the transformed PF equation. Hence, the particle is defined as a wave function which is coupled to the probability wave function through the superwave. If the state function changes due to a quantum or other forces, then the particle will change its velocity which depends on the phase of the probability function, whereas the quantum force is defined by the amplitude of the probability function. Therefore, the probability function works as the pilot wave in analogy the de Broglie-Bohm theory. Finally we have demonstrated the potential of the Lax (b) ——————————————————- formalism with the given explicit forms of the Lax operators to trace the coupling of the the particle wave ψ˜ and the probability wave φ in section II, as well as the coupling of the the quantum oscillations ψ˜ and the doublon φI in section IV. In conclusion, our findings provide a new interpretation of particles in quantum mechanics and new oppor- tunities for the formulation of the quantum wave equations. Our treatment goes significantly beyond the original dBB program by establishing a consistent set of wave equations derived by variational principles. Furtheron it allows predictions (see [12]) regarding structure formation in the universe and its accelerating expansion [37].

FIG. 7: Double slit experiment. a) probability wave Acknowledgement with one individual particle, b) probability wave with many particles, c) path integral for a single particle The author would like to thank Fathollah Varnik, trajectory Bochum, for support with discussions and suggestions and Dmitry Medvedev, Bochum/Novosibirsk, for critical reading of the manuscript.

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