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Original Research ARTICLE

Front. Phys., 17 July 2020 | https://doi.org/10.3389/fphy.2020.00213

Electron Mass Predicted From Substructure Stability in Electrodynamical Model

  • 1CNRS, Univ Rennes, IGDR-UMR 6290, Rennes, France
  • 2COSYS-GRETTIA, Univ Gustave Eiffel, IFSTTAR, Marne-la-Vallée, France

Modern physics has characterized spacetime, the interactions between particles, but not the nature of the particles themselves. Previous models of the electron have not specified its substance nor justified its cohesion. Here we present a relativistic electrodynamical model of the electron at rest, founded on natural interpretations of observables. Essentially intertwined positively and negatively charged subparticles revolve at light velocity in coplanar circular orbits, forming some coherent “envelope” and “nucleus”, possibly responsible for its wavelike and corpuscular behaviors, respectively. We show that the model can provide interpretations of fundamental constants, satisfy the Virial theorem, and exhibit cohesion and stability without invoking Poincaré stresses. Remarkably, the stability condition allows predicting electron mass, regarded as being a manifestation of its total (kinetic and potential) electromagnetic cohesion energy, and muon mass, directly from the substructure. Our study illustrates the possibility of constructing causal and objectively realist models of particles beneath the Compton scale. Finally, wave-corpuscle duality and the relation to quantum mechanics are discussed in the light of our electron model.

Introduction

Depending on the experiment, the most emblematic subatomic particle, the electron, has been found to interact as a point-like corpuscle in scattering experiments [1], or to behave as an extensible wave [2]. Elaborating on Bohr's interpretation of Quantum Mechanics [3], Heisenberg concluded that particles could neither be represented nor even apprehended by the human mind, and that only their abstract mathematical description existed [4]. For de Broglie however, “abstract presentations have no physical reality. Only the movement of elements localized in space, in the course of time, has physical reality” [5]. Hence, modern physics has identified with unprecedented precision the interactions and their underlying principles, has successfully described its environment, spacetime, but still lacks a characterization of the nature of its “objects,” the particles themselves.

Consequently, several kinds of electron models have been proposed: extended models [6], point-like models, and mixed models in which a point-like corpuscle follows an extended trajectory [7]. Early attempts included the spherical models of Abraham [8] and Lorentz [9], which led to theories of electromagnetic mass [1013]. Spherical models soon evolved into the so-called ring models of Parson [14], Webster [15], Allen [16], and Compton [17], constituted of rotating infinitesimal charges and verifying the properties of classical magnetic moment and Compton scattering. Essential constraints however, such as electron cohesion and stability, could not be satisfied: new putative forces, denoted Poincaré stresses [18], were suggested to maintain the cohesion of the negatively charged electron. The abstract descriptions of quantum mechanical theories [19, 20] then successfully accounted for the wave-like behavior of the electron and probabilistically predicted [21] the values of most observables by considering a point-like particle, yet failed at interpreting fundamental constants or explaining how a point-like corpuscle could have spin or a finite energy density. Paradoxically, quantum mechanics revived geometric models when Schrödinger noticed within the Dirac equation itself a rapid oscillatory trembling motion, which he called Zitterbewegung (zbw) [22], exhibiting microcurrents arising at light velocity c. Surprisingly, the electron seemed to follow a helical trajectory of radius ƛc, the reduced Compton wavelength, surrounding the average travel direction (Figure 1A). Several such zbw models, identifying spin with orbital angular momentum, were interpreted classically [2729]. Subsequent electrodynamical or hydrodynamical models involved fluids with spin [30], current loops of a certain thickness [31], Dirac-like Equations [32, 33], moving charged membranes [34], plasmoid fibers [35], or toroidal geometry [35, 36]. Wondering whether zbw could be a real phenomenon, Hestenes emphasized the need to investigate the electron substructure, suggested zbw could originate in the electron self-interaction [37], and showed zbw was compatible with the ring models [38].

FIGURE 1
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Figure 1. Triolets and the helical trajectory. (A) In Schrödinger's Zitterbewegung (zbw) model, the wavefunction associated to the electron seems to revolve at light velocity along a helical trajectory of radius ƛc, the reduced Compton wavelength, surrounding the average travel direction, and to exhibit microcurrents. Quantum mechanics does not specify which forces could cause the electron, which is assumed to be point-like, to follow such a peculiar helical trajectory. (B) Triolets are colorless particles composed of three sparks, each bearing electric charge ±e/6 and a specific strong interaction color charge. Thus triolets bear electric charge ±e/6 or ±e/2 depending on their combination of sparks. They travel at light velocity c, possess angular momentum Ltrlt, and triolets will thereafter be represented as upward or downward, filled or hollow triangles depending on their electric charge, as depicted here. (C) In our model, the electron is composed of triolets forming a nucleus and an envelope. It is conceivable that, in the absence of perturbation, the nucleus of the moving electron attracts envelope triolets and maintains them bound, thus explaining their helical trajectory. Conversely, envelope triolets would revolve at light velocity on an orbit of radius ƛc around the nucleus, exhibiting the zbw microcurrents, and guide the nucleus, as in pilot-wave theories [2326], sensing the electromagnetic fields generated by the envelopes of other particles.

With the development of realist models of the electron emerged theories of electromagnetic mass. At first, the spherical models of Abraham [8], and Lorentz [9] seemed to fail to recover Einstein's relation E = mc2 due to the appearance of a factor 4/3, but later proved to be compatible, once relativistic corrections were accounted for [12]. Stability of the sphere however still relied on Poincaré stresses or unknown surface tension [34], and electron mass could not be predicted from an objective criterion, but depended on the value taken by an arbitrary parameter, whose value is unconstrained, i.e., the radius of the sphere. Of note, the mass of subatomic particles is not predicted by quantum theories, and their values need to be inserted in calculations [12]. Most ring models [1417] are prior to the discoveries of the spin, anomalous magnetic moment, and quantum mechanics. The ring model of Bergman and Wesley [31] exhibited cohesion and stability, but the expression for mass still involved an arbitrary parameter (i.e., width of current loop), and the substance constituting the electron remained indeterminate. More recently, Consa proposed a point-like electron following a toroidal trajectory [36], recovered mass independently of any arbitrary parameter, but did not specify how the trajectory developed nor demonstrated its stability. To our knowledge, the Virial theorem, which should be satisfied since the electron is a bound system, has not been considered in electron models. Potential energy is often equated to +mc2, although cohesion potential energy should be negative for a bound system [as it is for the atom for instance [19]]. Kinetic energy is not usually accounted for, even though Lorentz [9], Hestenes [38] and others [e.g., [32]] noted the existence of a rotating motion and wondered whether kinetic energy did not contribute to rest mass. For Barut and Bracken, rest mass energy of the particle is the energy of the internal motion in the rest frame [29].

Hence, several issues remain to be addressed regarding the electron: for instance, which forces could cause the puzzling helical trajectory? What could be the nature of the substance constituting the electron? Could an electrodynamical description account for electron cohesion and stability? And could Lorentz' hypothesis advocating the electromagnetic origin of mass be simultaneously implemented from an objective criterion, instead of an arbitrary parameter?

In this study, we present a relativistic electrodynamical model of the electron at rest, in which charged subparticles follow definite trajectories. The model is based on two main hypotheses: (i) the existence of charged colorless subparticles called triolets, (ii) the assumption that triolets revolve at light velocity on coplanar circular orbits, constituting an envelope and nucleus, depending on their electromagnetic charges. As the electron is coherent, it is assumed that the model satisfies the Virial theorem. Constraints capturing the measured values of several observables (classical and anomalous magnetic moments, spin, Compton wavelength, kinetic energy) are formulated. Using Liénard-Wichert potentials, we then determine the specific kinds and numbers of triolets satisfying envelope and nucleus stability. Remarkably, we find that these kinds and numbers are precisely those that allow predicting electron mass and muon mass electromagnetically directly from the substructure, thus implementing Lorentz' hypothesis. Electron mass is effectively derived from an expression of substructure stability, which constitutes an objective criterion in our view. Our system also illustrates the possibility of constructing causal, local, objective, and realist models of particles beneath the Compton scale. Finally, we discuss novel perspectives suggested by the model, relative to the understanding of wave-corpuscle duality and to its relation to quantum theory.

Description of the Model and Hypotheses

In a previous study, we proposed that just six kinds of indestructible elementary subparticles denoted sparks, bearing electric charge ±e/6 and a specific strong interaction color charge, are necessary and sufficient to reconstruct all subatomic particles, so that sparks are conserved and reorganized across particle decays and annihilations [Avner, Boillot, Richard, submitted]. Since sparks are subject to both the strong and electromagnetic interactions, with the former dominating at short distances [20], groups of three sparks could presumably assemble beforehand to form composite colorless particles, thereafter called triolets, bearing charge +e/6, –e/6, +e/2, or –e/2 (Figure 1B). Henceforth, we shall suppose that the electron is exclusively composed of triolets, which travel at light velocity [7], exhibit some intrinsic angular momentum Ltrlt, and being colorless, are submitted to electromagnetic and centrifugal forces only (hypothesis A).

Following de Broglie's proposition, we aim at constructing a plausible electrodynamical model of the electron at rest, in which positive and negative triolets form an electromagnetically bound system, exhibit the zbw microcurrents, and account for all experimentally measured observables. The electron is considered here as a particle of a certain extension, composed of revolving charged subparticles, the triolets, thereby exhibiting magnetic moment and intrinsic angular momentum (its spin) sensed by other particles. We know that the measured value of the electron magnetic moment is the sum of Bohr magneton μB = –eℏ/2m, predicted by both classical physics and quantum mechanics, where ℏ is reduced Planck constant, m the electron mass, and e the elementary charge, and an anomalous magnetic moment [39], which accounts for a small fraction aanml ≃ 0.001159 of the previous and is only predicted by quantum electrodynamics [20]. Remarkably, the value of the classical magnetic moment of the electron can be derived by considering a charge (–e) revolving on a circular orbit of radius ƛc [19]. Hence, we reckoned the classical and anomalous magnetic moments could, respectively, be produced by two different components of the electron, namely a negatively charged envelope and a neutrally charged nucleus, also possibly responsible for the electron's wavelike and corpuscular behaviors, respectively. The peculiar helical trajectory of the electron predicted by zbw could then be naturally apprehended by considering that zbw describes the dynamics of envelope triolets, which are attracted and bound to the nucleus (Figure 1C). Electron spin could correspond to the sum of angular momenta of envelope triolets. Moreover, we shall regard electron mass as being a manifestation of the total electromagnetic cohesion energy E of the particle, as Lorentz hypothesized [9], through Einstein's formula m = E/c2. The latter interpretation of the mass is naturally suggested by the observation that the muon possesses a mass ~206.77 times bigger than that of the electron, while its Compton wavelength is ~206.77 times smaller, as would be the case for a mass of electromagnetic origin, presenting a potential proportional to inverse distance.

The net electromagnetic forces acting on any particular envelope triolet should mostly depend on its surrounding triolets. The envelope could be organized into a complex structure, with triolets irregularly distributed along the orbits, or revolving at various radii, or experiencing fluctuations. To facilitate calculations however, we chose to make approximations and consider triolets at radial equilibrium rotating in the same direction on four coplanar circular orbits of different radii depending on their four different electromagnetic charges (hypothesis B, Figure 2). In our model, positive and negative nucleus triolets are intertwined to maintain their cohesion and could rotate along two close yet separate orbits due to the charged envelope. This could cause in turn a similar arrangement in the envelope, which would exhibit predominantly intertwined triolets, in spite of the excess of negative triolets. We are aware our model is only an approximation, even if we reckon that a collection of fluctuating ±e/6 and ±e/2 triolets traveling at light velocity could possibly converge toward such a configuration. Because of their stronger charges, ±e/2 triolets could be more tightly bound and form a condensed nucleus, while ±e/6 triolets would be bound more loosely and constitute the envelope.

FIGURE 2
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Figure 2. Model of the electron at rest. In our simplified model, triolets rotate at light velocity in the same direction along four different coplanar circular orbits depending on their electric charge, constituting an envelope, and nucleus. Negative triolets are more numerous at the envelope, while the nucleus is neutrally charged. Due to the charged envelope, nucleus triolets are separated into two close orbits depending on their charge. Envelope triolets similarly revolve on separated orbits whose radii are close to the reduced Compton wavelength. Possible triolet configurations (triolet kinds and numbers, angular distributions, orbital radii) must fulfill constraints expressing radial stability and the measured values of charge, spin, magnetic moments, and mass. Due to consecutive negative triolets in the envelope, intertwined envelope triolets assemble into stretches separated by a distance denv.

In addition, as the electron is a bound system whose inner potentials allegedly depend on position coordinates only and not velocities (justification is given below), the Virial theorem should be verified [40]: for inverse square law electromagnetic interactions, one typically has E = U/2 and E = –T, where T is the total internal kinetic energy and U the internal potential energy. Therefore, T and U should, respectively, amount to +mc2 and −2mc2, resulting in total internal energy E = T + U = –mc2 corresponding to electron mass, the minus sign being indicative of a bound system. Finally, we shall admit that, for the electron at rest, envelope triolets approximately follow a circular trajectory of radius ƛc = ℏ/mc, as suggested by the classical derivation of Bohr's magneton, and by zbw-like models. Interpretations of fundamental constants associated to the electron, such as reduced Planck constant ℏ and fine-structure constant α = e2/4πε0c (where ε0 designates vacuum permittivity), should also emerge from the model.

Formulation of the Model

Our system captures the measured values of charge, magnetic moments, spin and kinetic energy, and will be validated by showing that cohesion and stability can be satisfied, and potential energy (and thus electron mass) can be recovered. Let us here mathematically formulate the constraints: (i) a charge –e carried by Nenv = Nenv+ + Nenv triolets of charge ±e/nenv at the envelope; (ii) a classical magnetic moment μB generated by envelope triolets rotating at radii ρenv+ = ηenv+ƛc, ρenv = ηenvƛc, and producing currents Ienv+, Ienv; (iii) an anomalous magnetic moment aanml·μB generated by Nnuc nucleus triolets (Nnuc+ = Nnuc) of charge ±e/nnuc rotating in the same direction as envelope triolets at radii ρnuc+ = ηnuc+ƛc, ρnuc = ηnucƛc, with momentum pnuc+pnuc = pnuc and producing currents Inuc+, Inuc; (iv) an internal kinetic energy T = ∑ipic = +mc2; (v) a spin Senv = +ℏ/2 generated by envelope triolets of momentum penv+, penv:

e=e[(Nenv+nenv)(Nenvnenv)+(Nnuc+nnuc)      (Nnucnnuc)],    (1)
-e2m=Ienv+πρenv+2+Ienv-πρenv-2,    (2)
-aanmle2m=Inuc+πρnuc+2+Inuc-πρnuc-2,    (3)
ipic=(Nenv+penv++Nenvpenv             +Nnucpnuc)c,    (4)
2=Nenv+ρenv+penv++Nenv-ρenv-penv-.    (5)

The fact that the muon has same spin as the electron, despite possessing a smaller Compton wavelength and the same number of triolets according to our chemical theory [Avner, Boillot, Richard, submitted], suggests that the angular momentum of envelope triolets could be a constant ρenv+penv+ ≃ ρenvpenvLtrlt,env, yielding from (5):

=2NenvLtrlt,env.    (6)

Ltrlt,env is possibly determined by the triangular substructure of envelope triolets made of three strongly interacting sparks, and could be at the basis of Planck's constant. Further constraints are also deduced (see Values of Observables) from Equations (1–5):

nenv=Nenv--Nenv+,    (7)
nenv=(Nenv-ηenv--Nenv+ηenv+),    (8)
aanmlnnuc=Nnuc+(ηnuc--ηnuc+),    (9)
Tenv+Tnucmc2=11benv(Nenv+ηenv++Nenv-ηenv-)                            +Nnucbnucηnuc,    (10)
benv=2Nenv,    (11)

where benv and bnuc are dimensionless numbers. Assuming that ηenv+ ≃ ηenv ≡ ηenv ≃ 1, we deduce (in Values of Observables) from Equations (10–11) that the kinetic energies of the envelope and nucleus are approximately equal Tenv+Tnuc ≃ +mc2/2, leading to relation:

bnucηnuc=2Nnuc.    (12)

Furthermore, the electromagnetic force acting on a nucleus triolet due to the envelope charge and current and the electromagnetic force exerted on an envelope triolet due to the net nucleus magnetic moment were derived but found to be negligible when compared to intra-component interactions. This suggests that each component is only loosely bound to the other, almost constituting an independent system, and thus verifies the Virial theorem independently (see Values of Observables), yielding for potential energies UenvUnuc ≃ –mc2 and total energies EenvEnuc ≃ –mc2/2. The system of Equations (9–11) further allows to determine ηnuc+ (Values of Observables) for each value of Nnuc+:

ηnuc+=Nnuc+bnuc[2-aanmlbnucnnuc2Nnuc+2+4+(aanmlbnucnnuc2Nnuc+2)2],    (13)

while ηnuc is then given by Equation (9).

System cohesion and stability can be formulated by ensuring triolets are at radial equilibrium. As triolets are electrically charged and travel at light velocity, we use Liénard-Wichert potentials from relativistic electrodynamics [41] to express the radial components of electric field Eij and magnetic field Bij emitted by triolet Tj of charge qj at retarded time t', radius ρj and retarded angle θ'j, and sensed at distance Rij—electromagnetic fields traveling at light velocity in vacuum—by triolet Ti arriving at the vertical (angle 0), radius ρi, at time t (Figure 3A). From known electrodynamical expressions [41] for these fields, using cylindrical unit vectors and coordinates, and Figure 3B, we deduce (Forces and Potentials):

Eij=qjsin γj4πε0Rijρi(1+sin γj)2ρ^,    (14)
Bij=-qj4πε0cRijρj(1+sin γj)2z^,    (15)

where Rij and γj are defined by:

Rij2=ρi2+ρj22ρiρjcos θj,    (16)
sin γj=ρiRijsin θj.    (17)

Note that these fields depend on position coordinates only, not velocities, thereby justifying the use of the Virial theorem. We then derive expressions (Forces and Potentials) for the net radial Lorentz force Fij due to triolet Tj exerted on triolet Ti belonging to the same component, and for the centrifugal force Fctfg,i experienced by triolet Ti:

Fij=qiqj4πε0Rij(1+sin γj)2[sin γjρi+1ρj]ρ^,    (18)
Fctfg,i=hcbiρi2ρ^,    (19)

where bi stands for benv (respectively, bnuc) when Ti belongs to the envelope (resp. the nucleus). In the electron at rest, assuming triolets remain at radial equilibrium, the net radial component of the Lorentz force exerted by other triolets should compensate the centrifugal force. Neglecting the small contribution of the envelope onto the nucleus and vice-versa, and expressing equilibrium for triolet Ti along the radial direction and rearranging (Triolets at Radial Equilibrium), we obtain for the envelope and nucleus:

1α-benvnenv2jenvNenv-1ρi2sgn(i.j)Rij(1+sin γj)2(sin γjρi+1ρj)   Gienv(ηi),    (20)
1α-bnucnnuc2jnucNnuc-1ρi2sgn(i.j)Rij(1+sin γj)2(sin γjρi+1ρj)   Ginuc(ηi),    (21)

where sgn(i·j) is the sign of the product of the charges of triolets Ti and Tj, and α is the fine-structure constant, which is found to be related to the ratio between the centrifugal force and the net radial electromagnetic force experienced by any single triolet inside the electron. We assume positive and negative triolets are intertwined and uniformly distributed along the orbits except—as negative triolets are more numerous at the envelope—consecutive negative envelope triolets, which presumably repel to produce stretches of alternatively charged triolets separated by empty space (Figure 2). Let denv designate the distance (using the number of missing triolets as units) between the stretches. The expressions under the sums in Equations (20–21) can be calculated by first considering the non-retarded angular positions θj of triolets distributed along the circular orbit, then by determining the corresponding retarded angles θ'j, as illustrated in Retarted Angles, using Newton's recursion method for instance onto transcendental equation:

(θj-θj)2=1-2ρiρjcos θj+(ρiρj)2,    (22)

and then deriving γj from Equation (17). Equations (20–21) will help us derive adequate values for Nenv, Nnuc, nenv, nnuc, benv, bnuc.

FIGURE 3
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Figure 3. Geometric diagrams. (A) The influence of electromagnetic fields due to triolet Tj onto Ti: let triolets Ti and Tj belong to the same component (envelope or nucleus). Triolet Tj rotates at light velocity along circular orbit of radius ρj and arrives at angle θj at time t, but was at position Tj' at retarded angle θj' and time t' when it emitted electromagnetic fields that reached triolet Ti revolving along coplanar circular orbit of radius ρi and arriving at angle 0 (vertical y axis) at time t. The retarded electromagnetic fields can be expressed using Liénard-Wichert potentials. This figure applies to all envelope and nucleus triolets. (B) Diagram showing vectors and angles involved in the demonstration of the expressions of electromagnetic fields and potentials. (C) Diagram depicting the case ρj= ρi.

The potential energy due to interactions between the nucleus and envelope being negligible, the total potential energy of our system is approximately UtotUenv + Unuc, where Uenv, and Unuc are, respectively, the envelope and nucleus potential energies, which are evaluated in Potential Energy:

Uenv2αmc2nenv2ienvNenvjiNenv-1sgn(i.j)Hij(1+sin γj),    (23)
Unuc2αmc2nnuc2inucNnucjiNnuc-1sgn(i.j)Hij(1+sin γj),    (24)

where Hij = Rijc. Assuming ηenv+ ≃ ηenv ≃ 1, we demonstrate (Potential Energy) from benv=2Nenv (11) and Equation (20), which expresses the radial stability of every envelope triolet, that Equation (23) yields Uenv≃ –mc2. Likewise, assuming ηnuc+ ≃ ηnuc, we demonstrate (Potential Energy) from bnucηnuc = 2Nnuc (12) and Equation (21), which expresses the radial stability of every nucleus triolet, that Equation (24) yields Unuc ≃ –mc2. Hence, we find: UtotUenv + Unuc ≃ −2mc2, as expected from the Virial theorem, and recover electron mass. As substructure stability implies radial equilibrium for all envelope and nucleus triolets (20–21), it allows predicting electron mass. We find it remarkable that the same number of triolets allows to recover both substructure stability and electron mass.

Determination of Suitable Configurations

The problem then reduces to determining triolet configurations, i.e., sets of values for {nenv, nnuc, Nenv+, Nenv, Nnuc, benv, bnuc, ηenv+, ηenv, ηnuc+, ηnuc, denv}, that verify radial equilibrium for every triolet and correctly predict the total energy. We shall estimate the stability and total energy in three different models of the envelope successively, each lying at a different level of approximation. The three models are: the one-orbit model, where all envelope triolets rotate on the same orbit ηenv+≃ηenv≡ηenv; the two-orbits model, where positively-charged envelope triolets revolve on orbit of radius ηenv+ and negative triolets at radius ηenv; the n-orbits model where every envelope triolet i rotates on a circular orbit of specific but fixed radius ηi.

We shall first estimate the number of triolets Nenv present in the envelope by considering the one-orbit model. Assuming ηenv+ ≃ ηenv and ηnuc+ ≃ ηnuc, we have Rij ≃ 2ρicosγj (Figure 3C) both at the envelope and nucleus, and Equations (20–21) can be approximated to:

1α-benv2nenv2jenvNenv-1sgn(i.j)cos γj(1+sin γj)Gienv(ηi),    (25)
1α-bnuc2nnuc2jnucNnuc-1sgn(i.j)cos γj(1+sin γj)Ginuc(ηi).    (26)

Recalling that benv is related to Nenv via benv = 2Nenv (11), and setting values for input parameters {nenv, denv}, the iteration over Nenv values in Equation (25) enabled us to determine values for benv and Nenv approximately verifying Equations (25) and (11) simultaneously. Due to the asymmetry in the arrangement of envelope triolets, we found these Equations were satisfied for different values of Nenv depending on the triolet Ti under consideration. In the case nenv = 6, denv = 0 for instance, we found positive triolets approximatively satisfied these conditions for Nenv ≃ 108, while negative triolets did so for Nenv ≃ 144, thus justifying the necessity of considering two distinct orbits in the envelope. Although these figures should be regarded as merely indicative, cases denv = 1 and denv = 2 also pointed at average value Nenv = 126, corresponding to Nenv+ = 60 and Nenv= 66, and we shall be considering only this case in the remainder of our analysis. For the nucleus, in the absence of a constraint like Equation (11), values for bnuc and ηnuc satisfying Equations (26) and (12) simultaneously were determined for every iterated value of Nnuc. However, when accounting for the correction due to envelope current (first two terms, Triolets at Radial Equilibrium):

Genv>inucbnucsgn(i)nnuc[ηnuc32+3ηnuc48],    (27)

the best estimate appeared to be Nnuc = 18 (Table 1). Note that input values other than nenv = 6, nnuc = 2 did not yield any possible solutions.

TABLE 1
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Table 1. Stability and energy of various nucleus configurations.

Now, considering nenv = 6, nnuc = 2, denv = 2, ηenv+ ≃ ηenv ≃ 1 (one-orbit model), and putting in the value obtained above for Nenv, we evaluated potential energies Uenv, Unuc using Equations (13, 23–24) and found Uenv = −0.997·mc2 (Table 2), Unuc ≃ −1.000·mc2 (Table 1). The total potential energy therefore amounts to Utot ≃ −1.997·mc2, close to our expected result. Hence, recalling that kinetic energies satisfy TenvTnuc ≃ +mc2/2, then TtotTenv + Tnuc ≃ +mc2, EtotTtot+ Utot ≃ –mc2, and the mass of the electron is deduced directly from our model substructure. Likewise, since the muon is seen as an excited state of the electron [6] according to our chemical theory [Avner, Boillot, Richard, submitted], presumably displaying a similar arrangement of triolets albeit on a smaller scale, muon mass can also be successfully calculated by replacing m by muon mass mμ in expressions (23–24), or equivalently ƛc by the reduced muonic Compton Wavelength ƛmuon.

TABLE 2
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Table 2. Stability and energy of various envelope models.

We next evaluated the cohesion and stability of individual triolets. For the symmetric nucleus, we computed the right-hand side Gnuc of Equation (21) for every triolet; for nnuc = 2, Nnuc = 18 for instance, accounting for the correction due to the envelope current, we obtained Gnucnuc+) ≃ 136.83, Gnucnuc) ≃ 137.24 (Table 1). For the asymmetric envelope, which can be divided into six identical stretches of 21 triolets in the case Nenv = 126, we computed the right-hand sides Genv of Equation (20) for every triolet belonging to the first stretch and compared the results with the left-hand side 1/α ≃ 137.036, which they should yield if triolets were truly at radial equilibrium. For the one-orbit model, setting nenv = 6, values of Genv disagreed with the expected value for all values of denv (the case denv = 2 is given in Table 3). Clearly, in these conditions at least, the centrifugal and net electromagnetic forces fail to compensate and to ensure radial equilibrium, one dominating over the other, and triolets would be moving radially as well as azimuthally. Therefore, we considered the two-orbits model with ηenv+≃ 0.977, ηenv ≃ 1.023, for which we obtained an acceptable energy value (Table 2). Once again, we found that radial equilibrium was not verified for many envelope triolets, especially for consecutive negative triolets or those adjacent to them (Table 3). Hence, we decided to complicate our model again and considered envelope triolets orbiting at various but fixed radii ρi (n-orbits model) instead of the probably too general ρenv+ and ρenv. We heuristically determined fixed radii exhibiting reasonable stability for all envelope triolets, then used an optimization algorithm, described in Optimization Algorithm, to make every triolet tend toward radial equilibrium, minimizing criterion K, the average absolute deviation from 1/α per triolet:

K=1NenviNenv|Gienv(ηi)-1α|,    (28)

which effectively constitutes a measure of global stability of envelope triolets. Our algorithm converged toward a solution yielding acceptable energy and global stability (Table 2). The stability values Genvi) of individual envelope triolets belonging to the first stretch in the n-orbits model are shown in Table 3: most values appeared to be close to 1/α. We found that our optimization algorithm nicely converged toward stable solutions. However, the latter were highly dependent on initial conditions, and a thorough optimization study is needed to ensure local minima are avoided.

TABLE 3
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Table 3. Stability of individual envelope triolets.

Discussion

In this study, we presented a relativistic electrodynamical model of the electron based on natural interpretations of its associated observables. Our electron model is composed of triolets that revolve along coplanar circular orbits constituting an envelope and nucleus, which could be responsible for its wavelike and corpuscular behaviors, respectively. These two components would thus constitute a natural solution to wave-corpuscle duality. Capturing the values of charge, spin, magnetic moments, Compton wavelength and kinetic energy, we created a triolet-based configuration that verified cohesion and stability without invoking Poincaré stresses, and predicted electron and muon mass, defined as electromagnetic cohesion energy, directly from substructure stability. Importantly, our model accounts for kinetic energy and presents a negative cohesion potential energy, in agreement with the Virial theorem. In our model, the numbers of triolets in the envelope and nucleus are the adjusting parameters, and the same numbers are found to account both for substructure stability and electron mass. Notably, electron mass can be derived directly from an expression of substructure stability. Our study therefore implements Lorentz' hypothesis, which advocates the electromagnetic origin of mass, from an objective criterion, even if satisfaction of the criterion itself relies on two parameters, i.e., the numbers of triolets in the envelope and nucleus. Noteworthy, these parameters are not arbitrary, but instead are strongly constrained by several relations (11, 12, 20, 21, 27) that fix their values in our model. Altogether, we believe our study establishes that deterministic electrodynamical models of subatomic particles can be constructed beneath the Compton scale, in agreement with an objectively realist conception of physics.

Envelope triolets could also fluctuate radially or otherwise in time, possibly constituting a periodic wave that revolves at light velocity. This system has not been investigated here, but is of interest because this periodic wave could correspond to the wave associated to the electron, first imagined by de Broglie and later represented by wavefunction |ψ> in Schrödinger's wave mechanics or Dirac's quantum mechanics. It is conceivable that a wave made of envelope triolets, if it exists, attracts and drives the nucleus in the manner of the de Broglie-Bohm guiding wave [23, 24], sensing the electromagnetic fields generated by the envelopes belonging to other particles. Hence, envelope triolets could undulate and incarnate wavefunction |ψ>, whose concrete existence has recently been reconsidered [42]. Note further that nucleus triolets could also form a wave, reminiscent of the second wave described in de Broglie's double solution theory [23]. Specifically, triolets could propagate in a highly dynamical manner and experience irregular fluctuations, as in the hydrodynamical model of Bohm and Vigier [25]. Importantly, it has been suggested that solutions of this type could account both for quantum phenomena [26] and for quantum principles [37]. Bell also wrote that such solutions were compatible with the predictions of quantum mechanics [43]. Further, it is conceivable that such a complex envelope can exhibit several stable states, much like modes for a vibrating rope. These could correspond to the eigenstates of quantum mechanics. In the general case, the envelope would be in an unstable state, but could converge toward one of its eigenstates upon measurement, which could be conceived as the sum of interactions between system subparticles and apparatus subparticles. Such propositions constitute an interpretation of von Neumann's reduction of the wave packet [44], and would provide a possible solution to the measurement problem of quantum mechanics [45].

These considerations suggest that quantum theories, which encompass all subatomic phenomena and whose standard interpretation states that everything is intrinsically probabilistic, could eventually emerge [46, 47] from a relativistic electrodynamical description in agreement with the deterministic paradigm, which supports the causality principle, objective reality, and governs macroscopic physics. In this perspective, Schrödinger and Dirac Equations would constitute high-level descriptions of the dynamics of envelope triolets. Our study therefore provides new insight regarding the unification of the two apparently irreconcilable paradigms in physics: the deterministic and quantum paradigms.

Now, how exactly does the electron appear to be point-like in corpuscular interactions? How does our model relate to the observation that the electron seems spherical [48], or that its spin, charge and orbital components seem to be separable [4951]? How would the moving electron, which exhibits a wave satisfying de Broglie relation p=h/λ, be described? Could our description be regarded as an attempt to create a corpuscular counterpart to wave mechanics? Could analogous electrodynamical models be similarly constructed for other subatomic particles [52]? Could our extended model of the electron bring insight to the nature of molecular bonding, or to the arrangement of electrons inside atoms? And finally, what would be the implications for the interpretation of quantum mechanics [45]? How would quantum properties, such as the existence of eigenstates, the measurement problem or entanglement, and quantum phenomena, such as the two-slits experiment or the one-dimensional potential well, be understood in the light of our model? We believe the aforementioned questions should stimulate discussion and foster novel investigations.

Methods

Values of Observables

Charge

The charge of the electron is given by the Nenv+ triolets of charge (+e/nenv), the Nenv triolets of charge (–e/nenv), the Nnuc+ triolets of charge (+e/nnuc) and the Nnuc triolets of charge (–e/nnuc):

-e=e[(-Nenv-nenv)+(Nenv+nenv)+(-Nnuc-nnuc)+(Nnuc+nnuc)].

Assuming the nucleus is neutrally charged (hypothesis B), implying Nnuc+= Nnuc, we deduce:

nenv=Nenv--Nenv+.    (A1)

Nucleus and Envelope Orbits

Let us suppose triolets of charges (+e/nenv), (–e/nenv), (+e/nnuc), (–e/nnuc) revolve along four coplanar circular orbits of radii:

{ρenv+=ηenv+ƛCρenv=ηenvƛCρnuc+=ηnuc+ƛCρnuc=ηnucƛC    (A2)

where ƛc = ℏ/mc is the reduced Compton wavelength, and η's are dimensionless real numbers.

Classical and Anomalous Magnetic Moments

Let us express the classical magnetic moment μB= –e/2m = ∑iIiAi= ∑iQiAi/ti, where Ii is the current generated by triolet Ti, Qi its charge, ti = c/2πρi the time taken to go through a full orbit at light velocity c, and Ai the area formed by this orbit. The magnetic moment is due to a net charge (–e) made of Nenv = Nenv++ Nenv triolets revolving in the same direction along envelope orbits of radii ρenv+ and ρenv:

μB=-e2m=Qenv+Aenv+tenv++Qenv-Aenv-tenv-,-e2m=Nenv+enenvcπ ρenv+22πρenv++Nenv-(-e)nenvcπρenv-22πρenv-,-e2m=ec2nenv(Nenv+ηenv+-Nenv-ηenv-)mc,(Nenv-ηenv--Nenv+ηenv+)=nenv.    (A3)

As the anomalous magnetic moment μnuc = –aanml(e/2m), with aanml ≃ 0.001159, is relatively small, let us assume it is produced by an equal number Nnuc+ = Nnuc of positive and negative triolets of charge (±e/nnuc) revolving in the same direction as envelope triolets along nucleus orbits of slightly different radii due to the net envelope charge:

μnuc=-aanmle2m=Qnuc+Anuc+tnuc++Qnuc-Anuc-tnuc-, Nnuc+(ηnuc--ηnuc+)=aanmlnnuc.    (A4)

Virial Theorem

The virial theorem states that if a system remains bound, and if its inner potentials do not depend on velocities but only on positions, then the kinetic and potential energies take on definite shares in the total energy, depending on the degree n of the forces that apply. As the electron is a bound system, and as in our system the magnetic force will be found to depend on position coordinates ρ and γ only, the theorem applies and, for electromagnetic interactions in r−2, it stipulates that:

{T=mc2U=2mc2E=T+U=mc2    (A5)

where T, U, and E, respectively, designate the internal kinetic energy, internal potential energy, and total internal energy of the system. Note that the potential and total energies are negative, as they should be for a bound system.

Kinetic Energy

The kinetic energy is given by:

T=mc2=ipic=Nenv+penv+c+Nenv-penv-c                 +Nnuc+pnuc+c+Nnuc-pnuc-c,    (A6)

suggesting:

{penv+=mc/Kenv+penv=mc/Kenvpnuc+=mc/Knuc+pnuc=mc/Knuc,    (A7)

where the K's remain to be determined, thus yielding from Equation (A6):

1=Nenv+Kenv++Nenv-Kenv-+Nnuc+Knuc-+Nnuc+Knuc-.    (A8)

Note that we may assume that nucleus triolets possess comparable momentum pnuc+pnuc = pnuc, and that their orbit radius is approximately ρnuc+ ≃ ρnuc = ρnuc, since (ρnuc+– ρnuc) is very small according to Equation (A4).

Spin

Since particles as different as quarks and leptons (which possess different numbers of sparks according to our chemical model [Avner, Boillot, Richard, submitted]) share same spin, the latter can be interpreted as being the total angular momentum the particle conveys to the objects it encounters, i.e., the sum of the angular momenta of its envelope triolets. For the electron, assuming all triolets revolve in the same positive direction, it is written using Equations (A2, A6):

S=+2=iρipi=Nenv+ρenv+penv++Nenv-ρenv-penv-,    (A9)
12=Nenv+ηenv+Kenv++Nenv-ηenv-Kenv-,    (A10)

Further, as the muon is composed of the same number of triolets as the electron according to our chemical model and exhibits a Compton length much smaller than that of the electron [Avner, Boillot, Richard, submitted], spin ℏ/2 is thus independent of the radii of triolets' orbits. A necessary and sufficient condition is then that variables K's be proportional to η's:

{Kenv+=benv+ηenv+Kenv=benvηenvKnuc+=bnucηnuc+Knuc=bnucηnuc    (A11)

where benv+, benv, bnuc+, bnuc are values independent of radii, in order that the η's cancel out in Equation (A10), yielding:

12=Nenv+benv++Nenv-benv-.    (A12)

The angular momentum of triolet i is given by:

Li=piρi=mcbiηi.ηimc=bi,    (A13)

implying for spin and kinetic energy:

benv2=Nenv++Nenv-,    (A14)
1=Nenv+benvηenv++Nenv-benvηenv-+Nnuc+bnucηnuc++Nnuc-bnucηnuc-.    (A15)

Definition of Planck's Constant

Supposing angular momentum Ltrlt,env is a constant common to every envelope triolet, the expression for the spin, from Equation (A9), due to the envelope is:

2=Nenv+Ltrlt,env+Nenv-Ltrlt,env=NenvLtrlt,env,

and thus:

=2NenvLtrlt,env,    (A16)

meaning that the constant angular momentum Ltrl,env common to every envelope triolet could be at the basis of Planck's constant.

Kinetic Energy of the Nucleus and Envelope

From Equations (A6, A7, A11), the kinetic energy of the nucleus is given by:

Tnuc=Nnuc+pnuc+c+Nnuc-pnuc-c            =mc2Nnuc+bnuc(1ηnuc++1ηnuc-).    (A17)

Likewise, the kinetic energy of the envelope is:

Tenv=Nenv+penv+c+Nenv-penv-c,Tenv=mc2benv(Nenv+ηenv++Nenv-ηenv-).    (A18)

Now, assuming ηenv+≃ ηenv ≃ 1 according to Schrödinger's Zitterbewegung, Tenv becomes, using Equation (A14):

Tenvmc2benv(Nenv-+Nenv+)12mc2,    (A19)

and thus:

Tnuc=T-Tenv12mc2.    (A20)

The forthcoming study of the interactions between the nucleus and envelope will show that they are negligible compared to intra-component forces (nucleus onto itself, envelope onto itself). The two components therefore almost behave as two bound independent systems, and thus presumably obey the Virial theorem separately. Hence, since we have TnucTenvmc2/2, we should also obtain UnucUenv≃ –mc2 so that the total energies amount to: EnucEenv ≃ –mc2/2 and Etot≃ –mc2.

Determination of ηnuc+ and ηnuc-

In order to determine ηnuc+ and ηnuc-, considering Equations (A17) and (A20), we have:

(1ηnuc++1ηnuc-)bnuc2Nnuc+.    (A21)

The latter expression, together with Equation (A4), can allow us to determine ηnuc+ and ηnuc in terms of Nnuc+, aanml, nnuc, and bnuc:

bnuc2Nnuc+1ηnuc++1(ηnuc++aanmlnnucNnuc+),1ηnuc+(bnucηnuc+2Nnuc+-1)=1ηnuc+(11+aanmlnnucNnuc+ηnuc+),1=(1+aanmlnnucNnuc+ηnuc+)(bnucηnuc+2Nnuc+-1),ηnuc+2(bnuc2Nnuc+)+ηnuc+(aanmlbnucnnuc2Nnuc+2-2)                    -(aanmlnnucNnuc+)=0,Δ=4+(aanmlbnucnnuc2Nnuc+2)2,

and taking the positive solution, we find:

ηnuc+=Nnuc+bnuc(2-aanmlbnucnnuc2Nnuc+2+4+(aanmlbnucnnuc2Nnuc+2)2),    (A22)

and ηnuc can then be derived from Equation (A4).

Forces and Potentials

Centrifugal Force of a Triolet

Assuming triolets travel at light velocity, the centrifugal force [16] of triolet Ti, revolving along orbit of radius ρi = ηiƛc, is in cylindrical coordinates:

Fctf,i=piviρi=mc.cbiηiρi=cbiρi2,    (B1)

where bi stands for benv (respectively, bnuc) when Ti belongs to the envelope (resp. the nucleus). This expression applies both to nucleus and envelope triolets.

Electromagnetic Force Exerted on Nucleus Triolet i Due to Current at Envelope

The electromagnetic force exerted onto nucleus triolet i is given by the Lorentz force written using scalar potential V and vector potential A:

Fenv±>i=sgn(i).ennuc[Venv±>itAenv±>i+cθ^                      ×(×Aenv±>i)],    (B2)

if all triolets revolve in the same positive direction. The expressions for the scalar and vector potentials and their derivatives must be determined.

As a net charge (–e) circulates around the envelope, the scalar potential and vector potential, for rnuc < renv and cos θ = 0 (since the orbit is in the plane z = 0), are given [41] by:

Venv±>inuc=Qenv±4πε0renv±l=0,2,4[Pl(0)]2(ρiρenv±)l,    (B3)
Aenv±>inuc=μ0Ienv±2l=1,3,5[Pl1(0)]2l(l+1)(ρiρenv±)l,    (B4)

where the Pl(x) and Pl1(x), respectively, designate the Legendre polynomials and associated Legendre polynomials, yielding:

Venv±>inucQenv±4πε0[1ρenv±+14ρi2ρenv±3+964ρi4ρenv±5],    (B5)
Venv±>inucρnuc±Qenv±4πε0[12ρiρenv±3+916ρi3ρenv±5].    (B6)

Recalling μ0= 1/(ε0c2), v = c and Qenv± = ±Nenv±e/nenv:

μ0Ienv±2=μ0Qenv±2tenv±12ε0c2(±Nenv±enenv)(c2πρenv±),    (B7)
Aenv±>inuc±Nenv±e4πε0cnenv[12ρiρenv±2+316ρi3ρenv±4],    (B8)
(Aenv±>inucθ^)tnuc=-Aenv±>inuccρnucρ^                                              -(±Nenv±e)4πε0nenv[12ρenv±2+316ρi2ρenv±4]ρ^,    (B9)
×A=|ρ^θ^k^ρρθz0Aenv±>inuc0|=(Aenv±>inuc)ρnuck^     (±Nenv±e)4πε0cnenv[12ρenv±2+916ρi2ρenv±4]k^.    (B10)

The electromagnetic force (B2) exerted on a nucleus triolet Ti by the envelope is then given by:

Fenv±>inucsgn(i)nnuc(±Nenv±e2)4πε0nenvρenv±2[12ρiρenv±+38ρi2ρenv±2                                   +916ρi3ρenv±3]ρ^.    (B11)

Electromagnetic Force Exerted on Envelope Triolet i Due to Current Flowing at Nucleus

According to Equation (A3), the magnetic moment due to the nucleus is:

μnuc=-aanmle2m=Nnuc+ec2nnuc(ρnuc+-ρnuc-).    (B12)

The vector potential and its derivatives are given [41] by:

Anuc>ienvμ04πμnucρi2θ^,    (B13)
Anuc>ienv18πε0cNnuc+ennuc(ρnuc+-ρnuc-)ρi2θ^,    (B14)
(Anuc>ienvθ^)ti=-Anuc>ienvcρiρ^                       =-18πε0Nnuc+ennuc(ρnuc+-ρnuc-)ρi3ρ^    (B15)
×A=|ρ^θ^k^ρρθz0Anuc>ienv0|=(Anuc>ienv)ρik^            =(-1)4πε0cNnuc+ennuc(ρnuc+-ρnuc-)ρi3k^.    (B16)

As the net nucleus charge is zero, and using Equation (A4), the force is defined by:

Fnuc>ienv=sgn(i).enenv[-tAnuc>i+cθ^j(×Anuc>i)],Fnuc>ienv=38πε0sgn(i)e2Nnuc+nnucnenv(ηnuc+-ηnuc-)ƛCρi3ρ^,Fnuc>ienv=-38πε0sgn(i)e2nenvaanmlƛCρi3ρ^.    (B17)

Electromagnetic Force Exerted on Triolet i at Radius ρi Due to Triolet j at Radius ρj

Every triolet experiences the fields emitted by all other triolets belonging to the same or adjacent orbit in the same component. Here we estimate the electromagnetic field and force exerted by a single triolet revolving on the same or adjacent orbit.

Let triolet Tj' (ρjsin θ'j, ρjcos θ'j) of charge qj, revolving at light velocity on circular orbit of radius ρj, be positioned at angle θ'j at retarded time t', and emitting an electromagnetic field received at time t by triolet Ti(0, ρi) of charge qi revolving at light velocity on circular orbit of radius ρi, and arriving at angle θi = 0 on vertical axis y (Figure 3A). We have:

                   TjTi(-ρj sin θjρi-ρj cos θj),TjTi2=ρi2+ρj2-2ρiρjcos θjRij2,    (B18)
n^ji=TjTiTjTi(-ρjRijsin θjρi-ρj cos θjRij),    (B19)
Rij=ρi2+ρj2-2ρiρjcos θj.    (B20)

The trajectory, velocity and acceleration of triolet Tj are, respectively, given by:

wj(t)=ρj(sin ωtx^+cos ωty^),    (B21)
vj(t)=ρjω(cos ωtx^-sin ωty^),    (B22)
aj(t)=-ρjω2(sin ωtx^+cos ωty^),    (B23)

with ω being the angular velocity, satisfying relations c = ρω and θ' = ωt'. Since v = c, β = v/c = 1, we also have:

ρ^j(sin θjcos θj0),        βj(cos θj-sin θj0),  β.j(-c sin θj/ρj-c cos θj/ρj0)                     =-cρj ρ^j    (B24)
g=1-βj.n^ji=1-cos (π2+γj)=1+sin γj.    (B25)

The electric and magnetic fields emitted by Tj and received by Ti are given [41] by:

Ej=qj4πε0[(n^ji-βj)(1-β2)g3Rij2+n^ji×[(n^ji-βj)×β.j]cg3Rij],    (B26)
Bj=μ0qj4π[(vj×n^ji)(1-β2)g3Rij2+(βj×n^ji)(β.j.n^ji)+gβ.j×n^jig3Rij].    (B27)

From Figure 3B, it can be seen that:

β.j.n^ji=cρjcos γj,    (B28)
cosγ¯j2|n^ji-βj|,    (B29)
cosγ¯j2=12(1+cos γ¯j)=12(1+sin γj).    (B30)

And thus:

n^ji.(n^ji-βj)=|n^ji-βj|.cosγ¯j2=2 cos2γ¯j2=1+sin γj.    (B31)

From Equations (B25, B28) and identity: a×(b×c) = (a·c)b–(a·b)c, we deduce:

n^ji×[(n^ji-βj)×β.j]=(β.j.n^ji)(n^ji-βj)-[n^ji.(n^ji-βj)]β.j,n^ji×[(n^ji-βj)×β.j]=cρjcos γj(n^ji-βj)-(1+sin γj)β.j,    (B32)

implying, since 1–β2 = 0 and using Equations (B24, B25, B32):

Ej=qj4πε0[n^ji×[(n^ji-βj)×β.j]cg3Rij],Ej=qj4πε0Rijρj[(n^ji-βj)cos γj(1+sin γj)3+ρ^j1(1+sin γj)2].    (B33)

From Figure 3B, we also have:

βj×n^ji=-sin (π2+γj)z^=-cos γjz^,    (B34)
β.j×n^ji=-cρ^jρj×n^ji=-cρjsin (π-γj)z^=-cρjsin γjz^,    (B35)

yielding, using Equations (B27, B28) and μ0 = 1/(ε0c2):

Bj=μ0qj4π[-cρjcos γjcos γj-cρj(1+sin γj)sin γj(1+sin γj)3Rij]z^,Bj=-qj4πε0cRijρj(1+sin γj)2z^.    (B36)

The magnetic force is directed along ρi since Bj is along z. But to express the equilibrium we need to find the component of Ej along ρi, and thus we need:

n^ji.ρ^i=1Rij(ρi-ρj cos θj),    (B37)
ρ^i.(-βj)=cos (θj-π2)=sin θj,    (B38)
ρ^i.ρ^j= cos θj,    (B39)

yielding from Equation (B33):

Eji=qj4πε0Rijρj[1Rij(ρiρjcos θj)cos γj(1+sin γj)3           +sin θjcos γj(1+sin γj)3+ cos θj(1+sin γj)2]ρ^.    (B40)

This can be rearranged by expressing θ'j as a function of γj and vice versa. From Equations (B19, B24):

cos γj=-ρ^j.n^ji=-sin θj(-ρjRijsin θj)            -cos θj(ρi-ρj cos θjRij),cos γj=1Rij(ρj-ρi cos θj).    (B41)

Similarly, from Equation (B28):

sin γj z^=ρ^j×n^ji=|x^y^z^sin θjcos θj0ρjRijsin θjρiρjcos θjRij0|,    sin γj=ρiRijsin θj.    (B42)

Relations (B41) and (B42) may be reversed:

sin θj=Rijρisin γj,    (B43)
cos θj=1ρi(ρj- Rijcos γj).    (B44)

Then, using these to rearrange Equation (B40) and developing:

1Rij2(ρi-ρj cos θj)(ρj-ρi cos θj)             =1Rij2[ρiρjsin2θj-Rij2cos θj],sin θjRij(ρj-ρi cos θj)+cos θj(1+ρiRijsin θj)               =cos θj+ρjRijsin θj,

we obtain using Equation (B42):

Eji=qjsin θj4πε0Rij2(1+sin γj)3[ρiRijsin θj+1]ρ^,Eji=qjsin γj4πε0Rijρi(1+sin γj)2ρ^.    (B45)

The Lorentz force is then:

 Fij=qi(Eij+cθ^i×Bij), Fij=qiqj4πε0Rijρi[sin γj(1+sin γj)2]ρ^           +qiqj4πε0Rijρj[1(1+sin γj)2]ρ^,  Fij=qiqj4πε0Rij(1+sin γj)2[sin γjρi+1ρj]ρ^.    (B46)

The scalar and vector Liénard-Wichert retarded electromagnetic potentials [41] are:

Vij=qj4πε0(Rij-βj.Rij)rtrd=qj4πε0Rij(1+sin γj),    (B47)
Aij=μ04π(qjvjθ^jRij-βj.Rij)rtrd=qjθ^j4πε0cRij(1+sin γj).    (B48)

Approximation ρij. When making this approximation (one-orbit model), from Figure 3C, Rij becomes:

Rij=2ρicos γj.    (B49)

Note that if ρi= ρj, Equation (B46) then becomes:

Fij=qiqj8πε0ρi2cos γj(1+sin γj)ρ^.    (B50)

Triolets at Radial Equilibrium

Equilibrium of Envelope Triolets

Envelope triolets are submitted to the centrifugal force (B1), the magnetic force due to the net nucleus magnetic moment (B17), and the net electromagnetic force due to the other envelope triolets (B46). Equilibrium for env– triolets can be written:

0=cbenvρenv-2+(-e)nenvjNenv-1enenv14πε0sgn(j)Rij(1+sin γj)2                    ×[sin γρenv-+1ρj]+38πε0e2nenvaanmlƛCρenv3. 

And rearranging to isolate the fine-structure constant:

4πε0ce2=1α=benvρenv2nenv[jNenv11nenvsgn(j)Rij(1+sin γj)2                 ×(sin γρenv+1ρj)3aanmlƛC2ρenv3].    (C1)

Likewise, equilibrium for env+ triolets can be written:

1α=benvρenv+2nenv[jNenv11nenvsgn(j)Rij(1+sin γj)2(sin γρenv++1ρj)   +3aanmlƛC2ρenv+3].    (C2)

Neglecting the term due to the nucleus magnetic moment, the equations become:

1α=-benvnenv2[jNenv-1ρi2sgn(i.j)Rij(1+sin γj)2(sin γρi+1ρj)]Genv.    (C3)

The fine structure constant therefore appears to be naturally related to the ratio between the centrifugal force and the net electromagnetic force experienced by a single triolet. Making the ρij approximation (B49), we obtain:

1α=-benv2nenv2[jNenv-1sgn(i.j)cos γj(1+sin γj)].    (C4)

Equilibrium of Nucleus Triolets

Nucleus triolets are submitted to the centrifugal force (B1), the electromagnetic force due to the envelope (B11), and the net electromagnetic force due to the other nucleus triolets (B46). Equilibrium for nuc– triolets is thus written:

1αbρnuc2nnuc[ρnuc2nenv(Nenv+ρenv+3Nenvρenv3)   +jNnuc11nnucsgn(j)Rij(1+sin γj)2(sin γρnuc+1ρj)].    (C5)

Similarly we have for the nuc+ triolets:

1αbρnuc+2nnuc[ρnuc+2nenv(Nenv+ρenv+3Nenvρenv3)    jNnuc11nnucsgn(j)Rij(1+sin γj)2(sin γρnuc++1ρj)].    (C6)

Neglecting the term due to the envelope current, the equations become:

1α=-bnucnnuc2[jNnuc-1ρi2sgn(i.j)Rij(1+sin γj)2(sin γρi+1ρj)]Gnuc.    (C7)

Making the ρij approximation (B49), we obtain:

1α=-bnuc2nnuc2[jNnuc-1sgn(i.j)cos γj(1+sin γj)].    (C8)

Also, the correction due to envelope current (first two terms) is:

Genv>inuc-bnucsgn(i)(-nenv)nnucnenv[ρnuc32ρenv3+3ρnuc48ρenv4]                             bnucsgn(i)nnuc[ηnuc32+3ηnuc48].    (C9)

Retarted Angles

Evaluating the Values of Retarded Angle θj' From Non-retarded Angle θj

If we suppose triolets are uniformly distributed along the circular orbits (this is certainly true of the nucleus since we have Nnuc+ = Nnuc, but is an approximation in the case of the envelope, as there are more negative than positive triolets), then angle θj (expressed in radians) determining the position of the jth triolet (starting at 1) at non-retarded time t on the orbit is defined by:

θjnuc=2πjNnuc.    (D1)

Note that, for the envelope, we also need to account for the empty space of length denv (using the number of missing triolets as units) separating the nenv stretches of triolets, yielding for triolets Tj belonging to the first stretch:

θj1stStretch=2πj(Nenv+nenvdenv).    (D2)

To evaluate θj' determining the angular position Tj' at retarded time t' when the electromagnetic field was emitted toward triolet Ti, which arrives at angle 0 (vertical y axis) at time t to receive the field, we use the following relation, derived from Figure 3A:

Rij=ρjδθj=ρj(θj-θj).    (D3)

Then squaring Equations (B20) and (D3) and equating, we obtain:

(θjθj)2=12(ρiρj)cos θj+(ρiρj)2.    (D4)

Given ρi, ρj, and θj, the retarded angles θ'j may be numerically determined by recurrence, using a computer program that implements Newton method for instance, to resolve transcendental Equation (D4) for all triolets of angular position θj expressed in radians. The corresponding values of γj are then estimated using Equation (B42).

Potential Energy

Electric Potential Energy

By definition, the electric potential energies at the envelope and nucleus are defined by:

Uelec,env=iNenvjiNenv-1qiVij=ienvjiqiqj4πε0Rij(1+sin γj),    (E1)
Uelec,env=αmc2nenv2ienvjisgn(i.j)Hij(1+sin γj),    (E2)

where Hij = Rijc. Likewise, we have:

Uelec,nuc=αmc2nnuc2inucjisgn(i.j)Hij(1+sin γj).    (E3)

Making the ρi = ρj approximation (B49), we obtain:

Uelec,env=αmc22nenv2ienvjisgn(i.j)ηjcos γj(1+sin γj),    (E4)
Uelec,nuc=αmc22nnuc2inucjisgn(i.j)ηjcos γj(1+sin γj).    (E5)

Magnetic Potential Energy

The magnetic potential energy Umag and electric potential energy Uelec are, respectively, the opposite of the magnetic work and electric work [41] given by:

Wmag=12μ0all spaceB2dτ=-Umag,    (E6)
Welec=ε02all spaceE2dτ=-Uelec.    (E7)

Now, the vector expression relating the magnetic field to the electric field:

B=1cn^×E    (E8)

holds in relativistic electrodynamics with particles going at light velocity, yielding:

Wmag=12μ0c2all spaceE2dτ,    (E9)

and since we know that c2 = 1/ε0μ0, we have:

Wmag=ε02all spaceE2dτ=Welec.    (E10)

Therefore:

Umag=Uelec.    (E11)

Total Potential Energy

Neglecting the potential energy of the envelope acting on the nucleus Uenv>nuc, and the potential energy of the nucleus acting on the envelope Unuc>env, the electron potential energy is approximately:

UtotUenv+Unuc,    (E12)

where Uenv is the envelope potential energy and Unuc the nucleus potential energy. Using Equations (E2, E11), we obtain:

Uenv=Uenv,mag+Uenv,elec=2Uenv,elec,    (E13)
Uenv=2αmc2nenv2ienvjisgn(i.j)Hij(1+sin γj),    (E14)

where Hij= Rijc. Likewise, using Equation (E.3) we have:

Unuc=2αmc2nnuc2inucjisgn(i.j)Hij(1+sin γj).    (E15)

Compatibility Between Potential Energies and Radial Equilibrium Equations

It can be shown that Equations (E14, E15) are compatible with Equations (C4, C8) if we assume ηnuc+≃ ηnuc and ηenv+ ≃ ηenv ≃ 1. Indeed, Equation (C4) becomes:

[jNenv-1sgn(i.j)2 cos γj(1+sin γj)]-nenv2αbenv.

Then, by replacing the relation above into Equation (98), since ηenv+≃ηenv≃1, we obtain:

Uenv=2αmc2nenv2ienvjisgn(i.j)2ηi cos γj(1+sin γj)           -2αmc2nenv2Nenvnenv2αbenv,Uenv-2Nenvmc2benv.

Since benv = 2Nenv, this yields: Uenv ≃ –mc2 as expected. Likewise, Equation (C8) becomes:

[jNnuc-1sgn(i.j)2 cos γj(1+sin γj)]-nnuc2αbnuc.

Then, by replacing the relation above into Equation (E15), we obtain:

Unuc=2αmc2nnuc2inucjisgn(i.j)2ηi cos γj(1+sin γj)-2αmc2nnuc2Nnucηnucnnuc2αbnuc,Unuc-2Nnucmc2bnucηnuc.

Since 2Nnuc = bnuc η nuc (A21), we obtain: Unuc ≃ –mc2 as expected.

Optimization Algorithm

An optimization algorithm has been devised and implemented to determine a set of optimum orbital radii for envelope triolets by minimizing average absolute deviation K, in the n-orbits model where each triolet possesses a specific radii ηi at the envelope. An approximate solution is determined heuristically before applying this algorithm. The algorithm next considers in turn every envelope triolet belonging to the first stretch, tries five different radii surrounding the current radius, and computes for each the stability of all envelope triolets. The radius yielding best overall stability is then attributed to the corresponding triolets in all six stretches. Once the procedure has been applied to all triolets of the first stretch, it is run again, considering five closer radii this time (thus slowly reducing the noise), until convergence toward an optimum solution is reached. The corresponding pseudocode is shown below. The algorithm was applied with the following values: delta = 0.00201, step = 0.00005, nenv= 6, Nenv+ = 60, Nenv = 66, denv = 2.

Data Availability Statement

All datasets generated for this study are included in the article/supplementary material.

Author Contributions

SA conceived the study, formed the hypotheses, constructed the model, wrote down and solved the equations, implemented the computations, and wrote the manuscript. FB helped solve the equations, devised the optimization algorithm, independently implemented the computations, and reviewed the manuscript. All authors approved the submitted version.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The authors wish to thank Patrick Richard (IFSTTAR, University Gustave Eiffel) for helpful advice and Gilles Salbert (IGDR, University of Rennes 1) for support and reading the manuscript.

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Keywords: electron substructure, fundamental constants, electromagnetic mass, wave-corpuscle duality, objective reality

Citation: Avner S and Boillot F (2020) Electron Mass Predicted From Substructure Stability in Electrodynamical Model. Front. Phys. 8:213. doi: 10.3389/fphy.2020.00213

Received: 24 March 2020; Accepted: 19 May 2020;
Published: 17 July 2020.

Edited by:

Ana Maria Cetto, National Autonomous University of Mexico, Mexico

Reviewed by:

Xing Lu, Beijing Jiaotong University, China
Kazuharu Bamba, Fukushima University, Japan

Copyright © 2020 Avner and Boillot. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Stéphane Avner, stephane.avner@univ-rennes1.fr